Inequalities for differential forms

Inequalities for Differential Forms Inequalities for Differential Forms Ravi P. Agarwal Florida Institute of Technol...

Author: Ravi P. Agarwal | Shusen Ding | Craig Nolder

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Inequalities for Differential Forms

Inequalities for Differential Forms

Ravi P. Agarwal Florida Institute of Technology Melbourne, FL, USA

Shusen Ding Seattle University Seattle, WA, USA and

Craig Nolder Florida State University Tallahassee, FL, USA

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Ravi P. Agarwal Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901 [email protected]

Shusen Ding Department of Mathematics Seattle University Seattle, WA 98122 [email protected]

Craig Nolder Department of Mathematics Florida State University Tallahassee, FL 32306 [email protected]

ISBN 978-0-387-36034-8 e-ISBN 978-0-387-68417-8 DOI 10.1007/978-0-387-68417-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009931765 Mathematics Subject Classification (2000): Primary: 26D10, 58A10, 35J60, Secondary: 26D15, 26D20, 30C65, 31B05, 46E35, 53A45 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated to Sadhna Agarwal, Yuhao Ding, Raymond W. Nolder, Laura Yang

Preface

Diﬀerential forms have been widely studied and used in many ﬁelds, such as physics, general relativity, theory of elasticity, quasiconformal analysis, electromagnetism, and diﬀerential geometry. They can be used to describe various systems of partial diﬀerential equations and to express diﬀerent geometrical structures on manifolds. Hence, diﬀerential forms have become invaluable tools for many ﬁelds. One of the purposes of this monograph is to present a series of estimates and inequalities for diﬀerential forms, particularly, for the forms satisfying the homogeneous A-harmonic equations, or the nonhomogeneous A-harmonic equations, or the conjugate A-harmonic equations in Rn , n ≥ 2. These estimates and inequalities are critical tools to investigate the properties of solutions to the nonlinear diﬀerential equations and to control oscillatory behavior in domains or on manifolds. These results can be further used to explore the global integrability of diﬀerential forms and to estimate the integrals of diﬀerential forms. Throughout this monograph we always keep in our mind that diﬀerential forms are the extensions of functions (functions are 0-forms). Hence, all results about diﬀerential forms presented in this monograph remain valid for functions deﬁned in Rn . In Chapter 1, we study various versions of the Hardy–Littlewood inequalities for diﬀerential forms satisfying the conjugate A-harmonic equation. We ﬁrst introduce some deﬁnitions and notation related to diﬀerential forms, which will be used in this monograph. Then, we discuss diﬀerent versions of the A-harmonic equations and weight classes. From Sections 1.5, 1.6, and 1.7, we present the local and global Hardy–Littlewood inequalities with diﬀerent weights in John domains and Ls (μ)-averaging domains, respectively. We also give the best integrable exponents in Section 1.8. Finally, we investigate the Hardy–Littlewood inequalities with Orlicz norms. In Chapter 2, we concentrate on the Lp -estimates for solutions of the nonhomogeneous A-harmonic equation. We also extend these estimates to the

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Ar (Ω)-weighted cases. We conclude Chapter 2 with the global norm comparison inequalities and some applications to the compositions of operators. Chapters 3 and 4 treat the Poincar´e inequalities and the Caccioppoli inequalities, respectively. Speciﬁcally, we present the Poincar´e inequalities with Lp -norms and Orlicz norms for diﬀerential forms. We provide some estimates for Green’s operator and the projection operator. As applications of the Poincar´e inequalities, we also obtain some estimates for Jacobians of the Sobolev mappings. We develop both local and global Caccioppoli-type estimates with diﬀerent weights in a domain or on a manifold in Chapter 4. Roughly speaking, these estimates provide upper bounds for the Ls -norm of ∇u (if u is a function) or du (if u is a form) in terms of the Ls -norm of diﬀerential form u. We also discuss Caccioppoli-type estimates with Orlicz norms. Chapters 5 and 6 are concerned with the imbedding inequalities and the reverse H¨older inequalities, respectively. The imbedding inequalities for functions can be found in almost every book on partial diﬀerential equations; see Sections 7.7 and 7.8 in [63], for example. Hence, we only study the imbedding inequalities for diﬀerential forms in Chapter 5. We also explore the imbedding inequalities for some operators applied to diﬀerential forms and discuss various weighted cases. In Chapter 6, various versions of the reverse H¨ older inequalities are established. Chapter 7 is devoted to the integral estimates for some related operators, such as the homotopy operator, Laplace–Beltrami operator, and the gradient operator. We also develop some estimates for the compositions of operators, including the Hardy–Littlewood maximal operator and the sharp maximal operator. We know that the Jacobian of a quasiconformal mapping satisﬁes a stronger estimate, the reverse H¨older inequality. Then, what kind of estimates can we expect for the Jacobian of a mapping in a Sobolev class? We discuss the integrability of Jacobians in Chapter 8. Finally, in Chapter 9, we develop norm comparison theorems related to BM O-norms and Lipschitz norms. We also prove that the integrability exponents described in the Lipschitz norm comparison theorem are the best possible. This monograph presents an up-to-date account of the advances made in the study of inequalities for diﬀerential forms and will hopefully stimulate further research in this area. We would like to express our deep gratitude to our colleagues and friends who gave us various help and support during the preparation of this monograph. In particular, we are grateful for the valuable discussions with Professor Janet Mills, Professor Wynne Guy, and Professor John Carter. During the preparation of this monograph, Professor Yuming Xing, Professor Bing Liu, and Professor Peilin Shi generously devoted considerable time and eﬀort in

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reading various versions of the manuscript and giving us many precious and thoughtful suggestions which led to substantial improvements in the text. We also want to thank Ms. Damle Vaishali at Springer Verlag, New York, for her support and cooperation.

Melbourne, Florida Seattle, Washington Tallahasse, Florida August, 2008

Ravi P. Agarwal Shusen Ding Craig A. Nolder

Contents

1

Hardy–Littlewood inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Diﬀerential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Deﬁnitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Poincar´e lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A-harmonic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 A-harmonic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 p-Harmonic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Two equivalent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Three-dimensional cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 The equivalent system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Some weight classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Ar (Ω)-weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Ar (λ, E)-weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Aλr (E)-weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Some classes of two-weights . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Inequalities in John domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Local inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Global inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Inequalities in averaging domains . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Averaging domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Ls (μ)-averaging domains . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Other weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Two-weight cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Local inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Global inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The best integrable condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 5 6 8 9 10 14 14 15 17 19 21 21 23 25 27 29 29 32 34 37 37 38 41 43 43 45 45 45 47 xi

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1.9 Inequalities with Orlicz norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Norm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Lp (log L)α -norm inequality . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Ar (Ω)-weighted case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Global Ls (log L)α -norm inequality . . . . . . . . . . . . . . . . . .

47 48 49 52 55

2

Norm comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The local unweighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Basic Lp -inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The local weighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Ls -estimates for d v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Ls -estimates for du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The norm comparison between d and d . . . . . . . . . . . . . 2.4 The global estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Global estimates for d v . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Global estimates for du . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Global Lp -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Global Ls -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Imbedding theorems for diﬀerential forms . . . . . . . . . . .

57 57 58 58 60 61 61 64 65 67 67 68 69 70 72 72

3

Poincar´ e-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Inequalities for diﬀerential forms . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Inequalities for harmonic forms . . . . . . . . . . . . . . . . . . . . . 3.2.4 Global inequalities in averaging domains . . . . . . . . . . . . 3.2.5 Aλr -weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Inequalities for Green’s operator . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Basic estimates for operators . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Weighted inequality for Green’s operator . . . . . . . . . . . . 3.3.3 Global inequality for Green’s operator . . . . . . . . . . . . . . 3.4 Inequalities with Orlicz norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Local inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The proof of the global inequality . . . . . . . . . . . . . . . . . . 3.5 Two-weight inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Statements of two-weight inequalities . . . . . . . . . . . . . . . 3.5.2 Proofs of the main theorems . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Aλr (Ω)-weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . 3.6 Inequalities for Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Some notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Two-weight estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 75 76 78 83 84 86 88 90 92 92 93 96 98 100 100 101 104 107 108 109

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3.7 Inequalities for the projection operator . . . . . . . . . . . . . . . . . . . . 3.7.1 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . 3.7.2 Inequality for Δ and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Other Poincar´e-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 116 116

4

Caccioppoli inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local and global weighted cases . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Ar (Ω)-weighted inequality . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Ar (λ, Ω)-weighted inequality . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Aλr (Ω)-weighted inequality . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Parametric version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Inequalities with two parameters . . . . . . . . . . . . . . . . . . . 4.3 Local and global two-weight cases . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 An unweighted inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Two-weight inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Inequalities with Orlicz norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Basic · Lp (log L)α (E) estimates . . . . . . . . . . . . . . . . . . . . 4.4.2 Weak reverse H¨older inequalities . . . . . . . . . . . . . . . . . . . 4.4.3 Ar (M )-weighted cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Inequalities with the codiﬀerential operator . . . . . . . . . . . . . . . . 4.5.1 Lq -estimate for d v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Two-weight estimate for d v . . . . . . . . . . . . . . . . . . . . . .

119 119 120 120 123 125 126 127 127 127 129 133 133 136 137 140 140 141

5

Imbedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Solutions to the nonhomogeneous equation . . . . . . . . . . . . . . . . 5.4 Imbedding inequalities for operators . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The gradient and homotopy operators . . . . . . . . . . . . . . . 5.4.2 Some special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Global imbedding theorems . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Other weighted cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Ls -estimates for T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Ls -estimates for ∇ ◦ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Ar (λ, Ω)-weighted imbedding theorems . . . . . . . . . . . . . . 5.5.4 Some corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Global imbedding theorems . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Aλr (Ω)-weighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Compositions of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Ar (Ω)-weighted estimates for T ◦ d ◦ G . . . . . . . . . . . . . . 5.6.2 Ar (Ω)-weighted estimates for ∇ ◦ T ◦ G . . . . . . . . . . . . . 5.6.3 Imbedding for T ◦ d ◦ G . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 145 146 147 148 152 153 154 154 156 158 159 162 164 165 165 169 171

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5.7 Two-weight cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Two-weight imbedding for the operator T . . . . . . . . . . . 5.7.2 Ar,λ (E)-weighted imbedding . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Ar (λ, E)-weighted imbedding . . . . . . . . . . . . . . . . . . . . . . 5.7.4 Global imbedding theorem . . . . . . . . . . . . . . . . . . . . . . . . .

172 173 178 181 184

6

Reverse H¨ older inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Gehring’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Inequalities for supersolutions . . . . . . . . . . . . . . . . . . . . . . 6.2 The ﬁrst weighted case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Ar (Ω)-weighted inequalities . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Inequalities in Ls (μ)-averaging domains . . . . . . . . . . . . . 6.2.3 Inequalities in John domains . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Parametric inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Estimates for du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The second weighted case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Inequalities with Aα r (Ω)-weights . . . . . . . . . . . . . . . . . . . . 6.3.2 Inequalities with two parameters . . . . . . . . . . . . . . . . . . . 6.3.3 Aλr (Ω)-weighted inequalities for du . . . . . . . . . . . . . . . . . 6.4 The third weighted case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Local inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Global inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Analogies for du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Two-weight inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Ar,λ (Ω)-weighted cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Ar (λ, Ω)-weighted cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Two-weight inequalities for du . . . . . . . . . . . . . . . . . . . . . 6.6 Inequalities with Orlicz norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Elementary inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Lp (log L)α -norm inequalities . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 188 189 190 193 195 196 200 201 201 205 209 209 210 212 213 214 214 216 218 219 219 220

7

Inequalities for operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Some basic estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Estimates related to Green’s operator . . . . . . . . . . . . . . . 7.2.2 Estimates for ∇ ◦ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Estimates for d ◦ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Compositions of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Estimates for ∇ ◦ T ◦ G and T ◦ Δ ◦ G . . . . . . . . . . . . . . 7.3.2 Global estimates on manifolds . . . . . . . . . . . . . . . . . . . . . 7.3.3 Ls -estimates for T ◦ G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Local imbedding theorems for T ◦ G . . . . . . . . . . . . . . . . 7.3.5 Global imbedding theorems for T ◦ G . . . . . . . . . . . . . . . 7.3.6 Some special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.7 Ls -estimates for Δ ◦ G ◦ d . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 226 229 234 236 236 242 244 248 250 251 252

Contents

xv

7.4 Poincar´e-type inequalities for operators . . . . . . . . . . . . . . . . . . . . 7.4.1 Poincar´e-type inequalities for T ◦ G . . . . . . . . . . . . . . . . . 7.4.2 Poincar´e-type inequalities for G ◦ T . . . . . . . . . . . . . . . . . 7.5 The homotopy operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Basic estimates for T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Ar (Ω)-weighted estimates for T . . . . . . . . . . . . . . . . . . . . 7.5.3 Poincar´e-type imbedding for T . . . . . . . . . . . . . . . . . . . . . 7.5.4 Two-weight Poincar´e-type imbedding for T . . . . . . . . . . 7.6 Homotopy and projection operators . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Basic estimates for T ◦ H . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Ar (Ω)-weighted inequalities for T ◦ H . . . . . . . . . . . . . . . 7.6.3 Other single weighted cases . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Inequalities with two-weights in Ar (λ, Ω) . . . . . . . . . . . . 7.6.5 Inequalities with two-weights in Aλr (Ω) . . . . . . . . . . . . . . 7.6.6 Basic estimates for H ◦ T . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.7 Weighted inequalities for H ◦ T . . . . . . . . . . . . . . . . . . . . 7.6.8 Two-weight inequalities for H ◦ T . . . . . . . . . . . . . . . . . . 7.6.9 Some global inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Compositions of three operators . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Basic estimates for T ◦ d ◦ H . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Ar (Ω)-weighted inequalities for T ◦ d ◦ H . . . . . . . . . . . . 7.7.3 Cases of other weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.4 Cases of two-weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Estimates for T ◦ H ◦ d . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.6 Estimates for H ◦ T ◦ d . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 The maximal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Global Ls -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 The norm comparison theorem . . . . . . . . . . . . . . . . . . . . 7.8.3 The fractional maximal operator . . . . . . . . . . . . . . . . . . . 7.9 Singular integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

256 256 275 281 281 283 284 285 288 288 290 291 292 293 295 297 299 302 304 304 306 307 308 311 312 313 313 315 316 318

8

Estimates for Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Global integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Lp (log L)α (Ω)-integrability . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 The norm comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323 323 324 325 327 329 331 333

9

Lipschitz and BM O norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 BMO spaces and Lipschitz classes . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Some recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Sharpness of integrability exponents . . . . . . . . . . . . . . . .

339 339 340 340 342

xvi

Contents

9.3 Global integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Estimates for du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Estimates for du+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Lipschitz and BM O norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Estimates for Lipschitz norms . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Lipschitz norms of T ◦ G . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Lipschitz norms of G ◦ T . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Lipschitz norms of T ◦ H and H ◦ T . . . . . . . . . . . . . . . . 9.4.5 Estimates for BM O norms . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Weighted norm inequalities . . . . . . . . . . . . . . . . . . . . . . . . 9.4.7 Estimates in averaging domains . . . . . . . . . . . . . . . . . . . . 9.4.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

344 344 345 346 346 350 353 354 355 357 360 366

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Chapter 1

Hardy–Littlewood inequalities

In this ﬁrst chapter, we discuss various versions of the Hardy–Littlewood inequality for diﬀerential forms, including the local cases, the global cases, one weight cases, and two-weight cases. We know that diﬀerential forms are generalizations of the functions, which have been widely used in many ﬁelds, including potential theory, partial diﬀerential equations, quasiconformal mappings, nonlinear analysis, electromagnetism, and control theory; see [1–19], for example. During recent years new interest has developed in the study of the Lp theory of diﬀerential forms on manifolds [20, 21]. For p = 2, the Lp theory has been well studied. However, in the case of p = 2, the Lp theory is yet to be fully developed. The development of the Lp theory of diﬀerential forms makes it possible to transport all notations of diﬀerential calculus in Rn to the ﬁeld of diﬀerential forms. The outline of this chapter is ﬁrst to provide background materials, such as the deﬁnitions of diﬀerential forms and A-harmonic equations, some classes of weight functions and domains, and then, introduce diﬀerent versions of Hardy–Littlewood inequalities on various domains with some speciﬁc weights or norms.

1.1 Diﬀerential forms We ﬁrst introduce some preliminaries about diﬀerential forms. But, we do not try to include all basic properties and results related to diﬀerential forms in this section. We only brieﬂy review some basic deﬁnitions and terminology related to diﬀerential forms which are needed in later chapters.

1.1.1 Basic elements Throughout this monograph Ω is used to denote an open subset of Rn , n ≥ 2, and R = R1 . Balls are denoted by B and σB is the ball with the same center R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 1,

1

2

1 Hardy–Littlewood inequalities

as B with diam(σB) = σdiam(B). We do not distinguish the balls from cubes. The n-dimensional Lebesgue measure of a set E ⊆Rn is denoted by |E|. We call w a weight if w ∈ L1loc (Rn ) and w > 0 a.e. For 0 < p < ∞, we denote the weighted Lp norm of a measurable function f over E by

1/p |f (x)| wdx

||f ||p,E,w =

p

E

if the above integral exists. Let e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . ., en = (0, 0, . . . , 1) be the standard unit basis of Rn . For l = 0, 1, . . . , n, the linear space of l-vectors, spanned by the exterior products eI = ei1 ∧ ei2 ∧ · · · ∧ eil , corresponding to all ordered l-tuples I = (i1 , i2 , . . . , il ), 1 ≤ i1 < i2 < · · · < il ≤ n, is denoted by ∧l = ∧l (Rn ). The Grassman algebra ∧ = ∧(Rn ) = ⊕nl=0 ∧l (Rn ) is a graded algebra with respect to the exterior products. For α = and β = β I eI ∈ ∧, the inner product in ∧ is given by < α, β >= αI β I

αI eI ∈ ∧

with summation over all l-tuples I = (i1 , i2 , . . . , il ) and all integers l = 0, 1, . . . , n. A diﬀerential l-form ω on Ω is a Schwartz distribution on Ω with values in ∧l (Rn ). We use D (Ω, ∧l ) to denote the space of all diﬀerential l-forms and Lp (Ω, ∧l ) to denote the l-forms ω(x) = ωI (x)dxI = ωi1 i2 ···il (x)dxi1 ∧ dxi2 ∧ · · · ∧ dxil I

with ωI ∈ Lp (Ω, R) for all ordered l-tuples I. Thus Lp (Ω, ∧l ) is a Banach space with norm |ω(x)| dx

||ω||p,Ω =

1/p p

Ω

=

( Ω

1/p |ωI (x)| )

2 p/2

dx

.

I

It is easy to see that {dxi1 ∧ dxi2 ∧ · · · ∧ dxil , 1 ≤ i1 < i2 < · · · < il ≤ n} is a basis of the space ∧l , and hence dim(∧l ) = dim(∧l (Rn )) =

dim(∧) =

n l=0

dim(∧l (Rn )) =

n n

l=0

l

= 2n .

n

and l

1.1 Diﬀerential forms

3

We should also notice that dxi ∧ dxj = −dxj ∧ dxi , i = j and dxi ∧ dxi = 0. We denote the exterior derivative by d : D (Ω, ∧l ) → D (Ω, ∧l+1 ) for l = 0, 1, . . . , n − 1. The exterior diﬀerential can be calculated as follows: dω(x) =

n

k=1 1≤i1 <···
∂ωi1 i2 ···il (x) dxk ∧ dxi1 ∧ dxi2 ∧ · · · ∧ dxil . ∂xk

Its formal adjoint operator d which is called the Hodge codiﬀerential is deﬁned by d = (−1)nl+1 d : D (Ω, ∧l+1 ) → D (Ω, ∧l ), where l = 0, 1, . . . , n − 1 and is the Hodge star operator that will be deﬁned in Deﬁnition 1.1.1. For ω ∈ D (Ω, ∧l ) the vector-valued diﬀerential form ∂ω ∂ω ,..., ∇ω = ∂x1 ∂xn consists of diﬀerential forms ∂ω ∈ D (Ω, ∧l ), ∂xi where the partial diﬀerentiation is applied to the coeﬃcients of ω. Similarly, W 1,p (Ω, ∧l ) is used to denote the Sobolev space of l-forms which equals Lp (Ω, ∧l ) ∩ Lp1 (Ω, ∧l ) with norm ωW 1,p (Ω,∧l ) = diam(Ω)−1 ωp,Ω + ∇ωp,Ω . 1,p 1,p The notations Wloc (Ω, R) and Wloc (Ω, ∧l ) are self-explanatory. For 0 < p < ∞ and a weight w(x), the weighted norm of ω ∈ W 1,p (Ω, ∧l ) over Ω is denoted by

ωW 1,p (Ω,∧l ),w = diam(Ω)−1 ωp,Ω,w + ∇ωp,Ω,w .

(1.1.1)

We deﬁne the Hodge star operator : ∧ → ∧ as follows. Deﬁnition 1.1.1. If ω = αi1 i2 ···ik (x1 , x2 , . . . , xn )dxi1 ∧dxi2 ∧· · ·∧dxik , i1 < i2 < · · · < ik , is a diﬀerential k-form, then

ω = sign(π)αi1 i2 ···ik (x1 , x2 , . . . , xn )dxj1 ∧ · · · ∧ dxjn−k , where π = (i1 , . . . , ik , j1 , . . . , jn−k ) is a permutation of (1, . . . , n) and sign(π) is the signature of the permutation.

4

1 Hardy–Littlewood inequalities

It should be noticed that Hodge star operator can be deﬁned equivalently as follows. Deﬁnition 1.1.2. If ω = αi1 i2 ···ik (x1 , x2 , . . . , xn )dxi1 ∧ dxi2 ∧ · · · ∧ dxik = αI dxI , i1 < i2 < · · · < ik , is a diﬀerential k-form, then

ω = αi1 i2 ···ik dxi1 ∧ dxi2 ∧ · · · ∧ dxik = (−1) (I) αI dxJ , where I = (i1 , i2 , . . . , ik ), J = {1, 2, . . . , n} − I, and

k(k + 1) + ij . 2 j=1 k

(I) =

For example,

dx1 = (−1)2 dx2 ∧ dx3 = dx2 ∧ dx3 in ∧1 (R3 ). Applying to a 2-form ω = a12 dx1 ∧ dx2 + a13 dx1 ∧ dx3 + a23 dx2 ∧ dx3 in ∧2 (R3 ), we have

ω = (a12 dx1 ∧ dx2 + a13 dx1 ∧ dx3 + a23 dx2 ∧ dx3 ) = (−1)

2(2+1) +1+2 2

a12 dx3 +(−1)

2(2+1) +1+3 2

a13 dx2 +(−1)

2(2+1) +2+3 2

a23 dx1

= a12 dx3 − a13 dx2 + a23 dx1 . Let u = u1 dx1 + u2 dx2 + · · · + un dxn be a 1-form in ∧1 (Rn ). Then, the (n − 1)-form u can be found as

u = u1 dx2 ∧ dx3 ∧ · · · ∧ dxn − · · · + (−1)n+1 un dx1 ∧ dx2 ∧ · · · ∧ dxn−1 . Thus, we have d( u) = u1x1 dx1 ∧ dx2 ∧ · · · ∧ dxn + · · · + unxn dx1 ∧ dx2 ∧ · · · ∧ dxn = (u1x1 + u2x2 + · · · + unxn )dx1 ∧ dx2 ∧ · · · ∧ dxn . Therefore, we ﬁnd that d u = −( d )u = −(u1x1 + u2x2 + · · · + unxn ) = −div u. The Hodge star operator has the following properties: (1) maps k-forms to (n − k)-forms for 0 ≤ k ≤ n. (2) If e1 , e2 , . . . , en is the standard unit basis of Rn and α, β ∈ ∧, then

1 = e1 ∧ e2 ∧ · · · ∧ en

and

α ∧ β = β ∧ α =< α, β > ( 1).

1.1 Diﬀerential forms

5

Hence the norm of α ∈ ∧ is given by the formula |α|2 = < α, α > = (α ∧ α) ∈ ∧0 = R. The Hodge star is an isometric isomorphism on ∧ with (from ∧l to ∧n−l ) and (−1)l(n−l) (from ∧l to ∧l ).

1.1.2 Deﬁnitions and notations Let us recall some basic deﬁnitions, terms, and results that will be used frequently throughout this monograph. Let X and Y be the vector spaces of dimensions n and m, respectively. We denote the space of all linear transformations from X to Y by L(X, Y ). We write X = L(X, R) and Y = L(Y, R). For any T ∈ L(X, Y ), the dual transformation T ∈ L(Y , X ) is deﬁned by the formula T (g) = g ◦ T for g ∈ Y = L(Y, R). We say T (g) is the pullback of g, which is induced via the mapping T : X → Y . The concept of the pullback extends naturally to l-covectors, T : ∧l (Y ) → ∧l (X), l = 1, 2, . . . , n. Speciﬁcally, for any ω ∈ ∧l (Y ), we deﬁne T ω ∈ ∧l (X) by the rule (T ω)(v1 , v2 , . . . , vl ) = ω(T v1 , T v2 , . . . , T vl ), where v1 , v2 , . . . , vl ∈ X. The operation of pulling back has many nice properties. Here we list some of them: T (ω ∧ ϕ) = (T ω) ∧ (T ϕ), ω, ϕ ∈ ∧l (Y ), (T ◦ S) (ω) = S (T ω), (T −1 ) = (T )−1 if T is a linear isomorphism, (kT ) = k l T on l-covectors, k ∈ R. Next, we assume that Ω ⊂ Rn and Ω ⊂ Rm . Let f : Ω → Ω be a mapping and its diﬀerential Df (x) : Rn → Rm be deﬁned for almost all x ∈ Ω. Then, the associate pullback [Df (x)] : ∧l (Rm ) → ∧l (Rn ) is well deﬁned pointwise almost everywhere. Let ω be a diﬀerential form in Ω , ω(y) =

I

ω I (y)dyI =

ωi1 i2 ···il (y)dyi1 ∧ dyi2 ∧ · · · ∧ dyil ,

6

1 Hardy–Littlewood inequalities

where ω I are functions deﬁned at each point y ∈ Ω . Then, the pullback of ω via f , denoted by f (ω), is a diﬀerential form in Ω deﬁned pointwise almost everywhere by f (ω)(x) = [Df (x)] ω(y), where y = f (x) and ω(y) is the l-covector in ∧l (Rm ). We should notice that f ◦ d = d ◦ f which is a very useful property. See [22, 23] for more properties of the pullback of diﬀerential forms. Let V be a vector space over R. An algebra with unit 1 generated by the elements of V over R satisfying the relation u ∧ v = −v ∧ u for arbitrary u, v ∈ V is denoted by ∧(V ) and called an exterior algebra of V or a Grassmann algebra. Here ∧ stands for the product of this algebra. Let {e1 , e2 , . . . , en } be the basis for V and {e1 , e2 , . . . , en } be the basis for the dual space V . Then, the elements 1, e1 , . . . , en generate the algebra ∧(V ) and the covectors 1, e1 , . . . , en , e1 ∧ e2 , . . . , en−1 ∧ en , . . . , e1 ∧ e2 ∧ · · · ∧ en act as the induced basis of ∧(V ). A convenient notation in exterior algebra has many applications in the study of diﬀerential forms and determinants. For example, for covectors v j = v1j e1 + · · · + vnj en ∈ V , j = 1, 2, . . . , l, we have 1 vi1 vi12 · · · vi1l v2 v2 · · · v2 i1 i2 il . v1 ∧ v2 ∧ · · · ∧ vl = .. .. ei1 ∧ · · · ∧ eil . .. .. . . . 1≤i1 <···
2

l

1.1.3 Poincar´ e lemma Let ω = ω(x)dx1 ∧ dx2 ∧ · · · ∧ dxn be an n-form. After choosing an orientation of Rn , we can deﬁne the integral by ω= ω(x)dx Ω

Ω

1.1 Diﬀerential forms

7

if the coeﬃcient function ω is integrable. Let f : Ω → Ω be an orientationpreserving diﬀeomorphism and f : L1 (Ω , ∧n ) → L1 (Ω, ∧n ) be the pullback induced by f . Then, the relation f (ω) = ω Ω

Ω

plays an important role in the theory of integration on manifolds. In fact, the operation f is equivalent to “substitution of variables.” The following Poincar´e lemma is ubiquitous in the study of diﬀerential forms. Poincar´ e Lemma. Let M be a contractible diﬀerentiable manifold, and u be a diﬀerentiable k-form in M with du = 0. Then, u is exact, i.e., there exists a (k − 1)-form v in M such that dv = u. It has been proved that diﬀerential forms have the following Hodge decomposition properties. Theorem 1.1.3. If ω ∈ Lp (Rn , ∧k ), 1 < p < ∞, then there is a (k −1)-form α and a (k + 1)-form β such that ω = dα + d β and dα + d β ∈ Lp (Rn , ∧k ). Moreover, the forms dα and d β are unique up to a constant form and α ∈ Kerd ∩ Lp1 (Rn , ∧k−1 ), β ∈ Kerd ∩ Lp1 (Rn , ∧k+1 ). Also, we have the uniform estimate αLp1 (Rn ) + βLp1 (Rn ) ≤ Cp (k, n)ωp for some constant Cp (k, n) independent of ω. Here d is the Hodge codiﬀerential operator (the formal adjoint of d), k = 1, . . . , n − 1. See [22, 24] for the proof of the Poincar´e lemma. For more operational properties and results about diﬀerential forms, see [22–26]. We conclude this section with the following generalized H¨ older’s inequality which will be frequently used throughout this monograph. Theorem 1.1.4. Let 0 < α < ∞, 0 < β < ∞, and s−1 = α−1 + β −1 . If f and g are measurable functions on Rn , then f g s,E ≤ f α,E · g β,E for any E ⊂ Rn .

8

1 Hardy–Littlewood inequalities

1.2 A-harmonic equations The developments of the theory about the A-harmonic equation are closely related to the theory of quasiconformal and quasiregular mappings. A series of results about the solutions to diﬀerent versions of the A-harmonic equation and their applications in ﬁelds such as quasiconformal mappings and the theory of elasticity have been found recently; see [1, 2, 27–52]. In fact, many properties of solutions to the A-harmonic equation are generalizations of quasiconformal and quasiregular mappings in Rn . Quasiconformal and quasiregular mappings in Rn are natural extensions of conformal and analytic functions of one complex variable, respectively. In the two-dimensional case these mappings were introduced by H. Gr¨ otzsch [53] in 1928 and the higher dimensional case was ﬁrst studied by M.A. Lavrent ev [54] in 1938. Farreaching results were obtained also by O. Teichm¨ uller [55] and L.V. Ahlfors [56]. The systematic study of quasiconformal mappings in Rn was furnished by F.W. Gehring [5] and J. V¨ ais¨al¨ a [57] in 1961, and the study of quasiregular mappings by Yu. G. Reshetnyak in 1966 [58]. In a highly signiﬁcant series of papers published during the period 1966–1969, Reshetnyak proved the fundamental properties of quasiregular mappings by exploiting tools from diﬀerential geometry, non-linear partial diﬀerential equation theory, such as the p-harmonic equation div(∇u|∇u|p−2 ) = 0

(1.2.1)

for functions in Rn , and the theory of Sobolev spaces. The p-harmonic equation (1.2.1) is a special case of the A-harmonic equation div A(x, ∇u) = 0

(1.2.2)

for functions in Rn , where A: Rn ×Rn → Rn is a mapping which satisﬁes certain structural assumptions. See [59, Chapter 3] for detailed results about A-harmonic equation of functions in Rn . The A-harmonic equation has also been studied extensively in [60–86, 51]. One of the main purposes of this monograph is to summarize the recent developments of A-harmonic equations for diﬀerential forms. During the period 1969–1972, O. Martio, S. Rickman, and J. V¨ ais¨ al¨ a [87– 90] gave a second approach to the theory of quasiregular mappings which was based on the fact that a nonconstant quasiregular mapping is discrete and open. On the other hand, their approach made use of tools from the theory of quasiconformal mappings, such as curve families and moduli of curve families. The extremal length and modulus of a curve family were introduced by L. V. Ahlfors and A. Beurling in their paper [91] on conformal invariants in 1950. A third approach was developed by B. Bojarski and T. Iwaniec in 1983. Their methods are real analytic in nature and largely independent of Reshetnyak s work. A fourth approach was suggested by M. Vuorinen [92],

1.2 A-harmonic equations

9

which is a ramiﬁcation of the curve family method (see [87–89]), in which conformal invariants play a central role. Each of the above four approaches yields a theory covering the whole spectrum of results of the theory of quasiregular mappings. Particularly, the fourth approach introduced by M. Vuorinen in [93–95] has been applied mainly to distortion theory. This work has been continued in [43, 96–98], where some qualitative distortion theorems were discovered. These papers also include results which are sharp as the maximal dilation K approaches 1.

1.2.1 Quasiconformal mappings 1,p Let f : Ω →Rn , f = (f 1 , . . . , f n ) be a mapping of Sobolev class Wloc (Ω,Rn ), i 1 ≤ p < ∞, whose distributional diﬀerential Df = [∂f /∂xj ] : Ω → GL(n) is locally integrable function on Ω with values in the space GL(n) of all 1,p (Ω,Rn ) n × n-matrices. A homeomorphism f : Ω →Rn of Sobolev class Wloc is said to be K-quasiconformal, 1 ≤ K < ∞, if its diﬀerential matrix Df (x) and the Jacobian determinant J = J(x, f ) = detDf (x) satisfy

|Df (x)|n ≤ KJ(x, f ),

(1.2.3)

where |Df (x)| = max{|Df (x)h| : |h| = 1} denotes the norm of the Jacobi matrix Df (x) : Rn → Rn , as a linear transformation and the constant K ≥ 1 is independent of x ∈ Ω. The operator norm |Df (x)| stands for the magnitude of inﬁnitesimal deformation of one-dimensional objects, the cofactors D f (x) ∈Rn×n of the diﬀerential matrix characterize inﬁnitesimal deformations of (n−1)-dimensional objects, and the n-form J(x, f )dx = df 1 ∧· · ·∧df n represents an inﬁnitesimal change of volume at the point x ∈ Ω. Hadamard’s inequality gives the bounds |J(x, f )| ≤ |D f (x)|n/(n−1) ≤ |Df (x)|n . The mapping f is said to be K-quasiregular if the injectivity in the above deﬁnition is dropped. We say that f is orientation preserving (reversing) if its Jacobian determinant J(x, f ) is nonnegative (nonpositive) almost everywhere. A Sobolev mapping f : Ω →Rn is said to have ﬁnite distortion if (i) its Jacobian determinant is locally integrable and does not change sign and (ii) there exists a measurable function K = K(x) ≥ 1, ﬁnite almost everywhere, such that (1.2.3) |Df (x)|n = K(x)|J(x, f )|. Equation (1.2.3) is called the distortion equation and K(x) is called the outer distortion function of f . It is well known that every mapping of ﬁnite distortion solves a nonlinear system of ﬁrst-order PDEs, the so-called Beltrami system

10

1 Hardy–Littlewood inequalities 2

D f (x)Df (x) = |J(x, f )| n G, where D f (x) stands for the transpose of the diﬀerential matrix and = G(x), called the conformal distortion tensor of f , is acquired from the Cauchy–Green tensor by proper re-scaling, such that detG(x) ≡ 1 [99]. This in turn gives rise to a degenerate elliptic equation of the second order, an analog of the familiar A-harmonic equation. The matrix function G = G(x) = [Gij ] ∈Rn×n is often called the conformal structure or the Riemannian metric in Ω. This may be understood as saying that f is conformal with respect to this metric. G

One diﬀerence between conformal and quasiconformal mappings is that the later need not be diﬀerentiable in the usual sense. However, by a theorem due to A. Mori, F.W. Gehring, and J. V¨ ais¨al¨ a, every quasiconformal mapping f : Ω → Rn is diﬀerentiable almost everywhere and its Jacobian determinant J(x, f ) is locally integrable. Moreover, quasiconformality of f can be expressed by the diﬀerential inequality (1.2.3). Hence quasiconformal mappings can be treated by the methods of measure and integration. Lp -integrability for quasiconformal and quasiregular mappings in certain domains in Rn is an interesting and active branch of quasiconformal and quasiregular mappings. As early as in the 1930s, G.H. Hardy and J.E. Littlewood in [100, 101] studied the Lp -integrability of the real part and the imaginary part of a holomorphic function which is deﬁned in the unit disk in R2 . They also proved that the Lp -norms of the real part and the imaginary part of such a holomorphic function are comparable, see Section 1.5. This is a well-known result and is called Hardy–Littlewood theorem for conjugate harmonic functions in the unit disk. This work opened a new direction of research which we will see later in this chapter.

1.2.2 A-harmonic equations Historically, the origin of quasiconformal mappings is connected with the developments of the methods of complex functions. Since the power of this concept was ﬁrst realized, quasiconformal mappings have engaged the attention of many prominent mathematicians and the theory has been greatly expanded to higher dimensions. In higher dimensions one might consider the coordinate function of a quasiconformal mapping. These are rather special solutions of an A-harmonic equation [they are coupled by a system of the ﬁrst-order partial diﬀerential equations (the Beltrami system in Rn ) like conjugate harmonic functions are coupled by the Cauchy–Riemann system]: d A(x, dω) = 0

(1.2.4)

1.2 A-harmonic equations

11

for diﬀerential forms, where A : Ω × ∧l (Rn ) → ∧l (Rn ) satisﬁes the following conditions: |A(x, ξ)| ≤ a|ξ|p−1

and

< A(x, ξ), ξ > ≥ |ξ|p

(1.2.5)

for almost every x ∈ Ω and all ξ ∈ ∧l (Rn ). Here a > 0 is a constant and 1 < p < ∞ is a ﬁxed exponent associated with (1.2.4). Equation (1.2.4) is often called a homogeneous A-harmonic equation. A solution ω to (1.2.4) is 1,p (Ω, ∧l−1 ) such that an element of the Sobolev space Wloc < A(x, dω), dϕ >= 0 Ω

for all ϕ ∈ W 1,p (Ω, ∧l−1 ) with compact support. Deﬁnition 1.2.1. We call u an A-harmonic tensor in Ω if u satisﬁes the A-harmonic equation (1.2.4) in Ω. A diﬀerential l-form u ∈ D (Ω, ∧l ) is called a closed form if du = 0 in Ω. Similarly, a diﬀerential (l + 1)-form v ∈ D (Ω, ∧l+1 ) is called a coclosed form if d v = 0. The equation (1.2.6) A(x, du) = d v is called the conjugate A-harmonic equation. For example, du = d v is an analog of a Cauchy–Riemann system in Rn . Clearly, the A-harmonic equation is not aﬀected by adding a closed form to u and coclosed form to v. Therefore, any type of estimates between u and v must be modulo such forms. If u is a solution to (1.2.4) in Ω, then at least locally in a ball B, there exists a form v ∈ W 1,q (B, ∧l+1 ), p1 + 1q = 1, such that (1.2.6) holds. Throughout this chapter, we shall assume that p1 + 1q = 1. Deﬁnition 1.2.2. When u and v satisfy (1.2.6) in Ω, and A−1 exists in Ω, we call u and v conjugate A-harmonic tensors in Ω. Equation (1.2.2) is the homogeneous form of the nonhomogeneous A-harmonic equation divA(x, ∇u) = B(x, ∇u) (1.2.7) for functions in Rn , where A: Rn ×Rn → Rn and B: Rn ×Rn → R are measurable and satisfy |A(x, ξ)| ≤ a|ξ|p−1 , < A(x, ξ), ξ > ≥ |ξ|p , and |B(x, ξ)| ≤ b|ξ|p−1 (1.2.8) for almost every x ∈ Ω and all ξ ∈ Rn . Here 1 + < B(x, ∇u), ϕ >= 0 (1.2.9) Ω

12

1 Hardy–Littlewood inequalities

for all ϕ ∈ C0∞ (Ω). Similarly, the nonhomogeneous A-harmonic equation for diﬀerential forms is written as d A(x, dω) = B(x, dω)

(1.2.10)

1,p (M, ∧l−1 ) and a solution to (1.2.10) is an element of the Sobolev space Wloc such that < A(x, dω), dϕ > + < B(x, dω), ϕ >= 0 M 1,p Wloc (M, ∧l−1 )

for all ϕ ∈ with compact support. The study of the nonhomogeneous A-harmonic equation (1.2.10) has just begun; see [72, 102, 103]. The global integrability of the solutions to (1.2.10) was proved recently by C. Nolder in [72]. Choosing A to be special operators, we obtain important examples of A-harmonic equations. For example, let A(x, ξ) = ξ|ξ|p−2 . Then, A satisﬁes condition (1.2.8), and we ﬁnd that (1.2.4) and (1.2.6) reduce to the following p-harmonic equation (1.2.11) d (du|du|p−2 ) = 0 and the conjugate p-harmonic equation du|du|p−2 = d v,

(1.2.12)

respectively, where v also satisﬁes the q-harmonic equation d(d v|d v|q−2 ) = 0 with

1 p

+

1 q

(1.2.13)

= 1.

If we choose p = 2 in (1.2.12), then (1.2.12) reduces to the following system: du = d v, which can be considered as an extension of the Cauchy–Riemann system in the plane. In addition to above harmonic equations for functions and diﬀerential forms, another type of the conjugate harmonic equation A(x, g + du) = h + d v

(1.2.14)

has also received much attention in recent years; see [104, 105], where u, v, g, and h are diﬀerential forms. Deﬁnition 1.2.3. If a pair of (l − 1)-form u and (l + 1)-form v satisfy (1.2.12), then u and v are called conjugate p-harmonic diﬀerential forms (or conjugate p-harmonic tensors).

1.2 A-harmonic equations

13

Example 1.2.4. Let f (x) = (f 1 , f 2 , . . . , f n ) be K-quasiregular in Rn , then u = f l df 1 ∧ df 2 ∧ · · · ∧ df l−1 and v = f l+1 df l+2 ∧ · · · ∧ df n , l = 1, 2, . . . , n − 1, are conjugate A-harmonic tensors. Let f (x) = (f 1 , f 2 , . . . , f n ) : Ω → Rn be a K-quasiregular mapping, K ≥ 1. Then, each of the functions u = f i (x) (i = 1, 2, . . . , n) or u = log |f (x)|, is a generalized solution of the quasilinear elliptic equation divA(x, ∇u) = 0, A = (A1 , A2 , . . . , An ), where

⎛ Ai (x, ξ) =

n

(1.2.15)

⎞n/2

∂ ⎝ θi,j (x)ξi ξj ⎠ ∂ξi i,j=1

(1.2.16)

and θi,j are some functions, which can be expressed in terms of the diﬀerential matrix Df (x) and satisfy C1 (K)|ξ|2 ≤

n

θi,j ξi ξj ≤ C2 (K)|ξ|2

(1.2.17)

i,j

for some constants C1 (K), C2 (K) > 0. This is a good example which shows the natural connection between two large sections of analysis: quasiregular mappings theory and the theory of partial diﬀerential equations. A signiﬁcant progress has been made in the study of equation (1.2.15) with condition 1,n (Ω) is a solution (1.2.17). From [106], we know that the function u ∈ Wloc of some equation of the form (1.2.15) with condition (1.2.17) if and only if there exists a diﬀerential (n − 1)-form θ(x) =

n

ˆ i ∧ · · · ∧ dxn ∈ Ln/(n−1) (Ω) θi (x)dx1 ∧ · · · ∧ dx loc

i=1

with properties (the sign ˆ means that the expression under ˆ is omitted): a) For every function g ∈ W 1,n (Ω) with compact support dg ∧ θ = 0. Ω

14

1 Hardy–Littlewood inequalities

b) Almost everywhere on Ω the following inequalities are true: ν1 |du(x)|n ≤ (du(x) ∧ θ(x)), where denotes the orthogonal complement of a form, and |θ(x)| ≤ ν2 |du(x)|n−1 with constants ν1 , ν2 > 0.

1.3 p-Harmonic equations In the last section, we have introduced several kinds of harmonic equations, including the p-harmonic equation for diﬀerential forms and functions, and the conjugate p-harmonic equations. Using the Hodge star operator, here we will develop a method to ﬁnd conjugate harmonic tensors in the three-dimensional case.

1.3.1 Two equivalent forms We recall the p-harmonic equation d (du|du|p−2 ) = 0

(1.3.1)

for u and the q-harmonic equation for v (the conjugate of u) d(d v|d v|q−2 ) = 0

(1.3.2)

that were introduced in the last section. A solution of the p-harmonic equation is called the p-harmonic tensor and a solution of the q-harmonic equation is called the q-harmonic tensor. Also, note that if u is a function, equation (1.2.11) becomes the usual p-harmonic equation div(∇u|∇u|p−2 ) = 0.

(1.3.3)

For the results related to the p-harmonic equation (1.3.3), see [59]. Let ∇u = (ux1 , ux2 , . . . , uxn ), w = |∇u|p−2 = (u2x1 + u2x2 + · · · + u2xn )(p−2)/2 , then div(∇u|∇u|p−2 ) = div(∇uw) = (wux1 )x1 + (wux2 )x2 + · · · + (wuxn )xn .

(1.3.4)

1.3 p-Harmonic equations

15

By a simple calculation, we obtain (wuxk )xk

= (p − 2)|∇u|

p−4

n

uxk uxi uxk xi + |∇u|p−2 uxk xk

(1.3.5)

i=1

for any k with 1 ≤ k ≤ n. Thus, equation (1.3.3) is equivalent to (p − 2)

n n

uxk uxi uxk xi + |∇u|2 Δu = 0.

(1.3.6)

k=1 i=1

Therefore, we have obtained two equivalent forms, (1.3.3) and (1.3.6), either of them is called the p-harmonic equation for functions. The solutions of (1.3.3) or (1.3.6) are called p-harmonic functions. The work related to the p-harmonic equation can be found in [107–118].

1.3.2 Three-dimensional cases We ﬁrst develop a method to ﬁnd conjugate harmonic tensors in R3 for p = 2, and then consider the general case. If p = 2, then equation (1.2.12) reduces to the following simple form: du = d v.

(1.3.7)

Here u = u(x1 , x2 , x3 ) is any 0-form (function) and v is a 2-form deﬁned by v = v1 dx1 ∧ dx2 + v2 dx1 ∧ dx3 + v3 dx2 ∧ dx3 ,

(1.3.8)

where v1 = v1 (x1 , x2 , x3 ), v2 = v2 (x1 , x2 , x3 ) and v3 = v3 (x1 , x2 , x3 ) are diﬀerentiable functions in R3 . Since n = 3 and l = 1, d = d . Thus, we have d v = d v = d( v) = d(v1 dx3 − v2 dx2 + v3 dx1 ) ∂v1 ∂v1 ∂v1 = ∂x dx1 + ∂x dx2 + ∂x dx3 ∧ dx3 1 2 3 ∂v2 ∂v2 ∂v2 ∧ dx2 − ∂x dx + dx + dx 1 2 3 ∂x2 ∂x3 1 ∂v3 ∂v3 ∂v3 ∧ dx . + ∂x dx + dx + dx 1 2 3 1 ∂x2 ∂x3 1

(1.3.9)

We know that dxi ∧ dxj = −dxj ∧ dxi for i = j and dxi ∧ dxi = 0. Hence, we obtain ∂v1 ∂v2 ∂v3 ∂v1 ∂v2 ∂v3 dx1 + dx2 − dx3 . + − + d v = ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2

16

1 Hardy–Littlewood inequalities

Thus, the equation du = d v is equivalent to the system ∂u = ∂x1

∂v1 ∂v2 + ∂x2 ∂x3

∂u = ∂x2

∂v3 ∂v1 − ∂x3 ∂x1

(1.3.10)

∂u ∂v2 ∂v3 =− − . ∂x3 ∂x1 ∂x2 When q = 2, the conjugate q-harmonic equation (1.2.13) reduces to d(d v) = 0.

(1.3.11)

From (1.3.9), we obtain 2 ∂ v1 ∂ 2 v2 ∂ 2 v1 ∂ 2 v3 dx1 ∧ dx2 − + d(d v) = − 2 − ∂x2 ∂x2 ∂x3 ∂x21 ∂x1 ∂x3 2 ∂ v2 ∂ 2 v3 ∂ 2 v1 ∂ 2 v2 dx1 ∧ dx3 − + + + ∂x21 ∂x1 ∂x2 ∂x2 ∂x3 ∂x23 ∂ 2 v2 ∂ 2 v3 ∂ 2 v1 ∂ 2 v3 dx2 ∧ dx3 . + − − + − ∂x1 ∂x2 ∂x22 ∂x1 ∂x3 ∂x23 So the equation d(d v) = 0 is equivalent to the following system: −

∂ 2 v1 ∂ 2 v2 ∂ 2 v1 ∂ 2 v3 − − + =0 ∂x22 ∂x2 ∂x3 ∂x21 ∂x1 ∂x3

∂ 2 v2 ∂ 2 v3 ∂ 2 v1 ∂ 2 v2 + + + =0 2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x23 −

(1.3.12)

∂ 2 v3 ∂ 2 v1 ∂ 2 v3 ∂ 2 v2 − + − = 0. 2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x3 ∂x23

Thus, in order to ﬁnd harmonic tensors, we only need to solve system (1.3.10) or (1.3.12). This gives us a method to ﬁnd conjugate harmonic tensors for the case p = q = 2. Similar to the case of p = 2, the equation du|du|p−2 = d v is equivalent to (we use the notation ∇u = du)

1.3 p-Harmonic equations

17

∂u |∇u|p−2 = ∂x1

∂v1 ∂v2 + ∂x2 ∂x3

∂u |∇u|p−2 = ∂x2

∂v3 ∂v1 − ∂x3 ∂x1

(1.3.13)

∂u ∂v2 ∂v3 |∇u|p−2 = − − ∂x3 ∂x1 ∂x2 with p = 2. Note that ∇u = du =

∂u ∂u ∂u dx1 + dx2 + dx3 . ∂x1 ∂x2 ∂x3

Thus, we only need to ﬁnd a 2-form v deﬁned by (1.3.8) and a p-harmonic function u satisfying (1.3.13). Such pairs of u and v are conjugate p-harmonic tensors. Clearly, system (1.3.13) has inﬁnitely many solutions.

1.3.3 The equivalent system Now we will develop a method to ﬁnd p-harmonic tensors in Rn . Similar to the case of n = 3, we try to get the equivalent system for the equation du|du|p−2 = d v in Rn . Let v=

vij dxi ∧ dxj

(1.3.14)

i<j

be a 2-form in Rn , where vij = vji , 1 ≤ i, j ≤ n, then

v =

(−1)

2(2+1) +i+j 2

vij dxI−{i,j} ,

i<j

where I = {1, 2, . . . , n}, that is i+j+1 (−1) vij dxI−{i,j} .

v = i<j

Thus, d( v) =

i+j+1

(−1)

i<j

i−1 ∂vij j−2 ∂vij (−1) dxI−{j} + (−1) dxI−{i} . ∂xi ∂xj

Therefore, we have d v = d( v)

18

1 Hardy–Littlewood inequalities

=

i+j+1

i<j

(−1)

(−1)

n(n−1) n(n+1) + 2 2

∂v ∂v × (−1)−j+i−1 ∂xiji dxj + (−1)−i+j ∂xijj dxi =

n ∂vij ∂xi dxj

(−1)

i<j

= (−1)

n

= (−1)

n

= (−1)

n

+ (−1)

∂vij ∂xi dxj

−

∂vij i<j ∂xi dxj

−

i<j

n j=2

n+1 ∂vij ∂xj dxi

∂vij ∂xj dxi

∂vij i<j ∂xj dxi

j−1

∂vij i=1 ∂xi dxj

n

−

j−1

∂vij i=1 ∂xj dxi

j=2

.

Since vij = vji , the second sum can be written as −

j−1

n

∂vij i=1 ∂xj dxi

j=2

=− =−

n−1 n n−1 n

∂vji i=j+1 ∂xi dxj

j=1

n

=

∂vij j=i+1 ∂xj dxi

i=1

∂vji n−1 n 1i − ∂v + − dxj . dx 1 j=2 i=j+1 ∂xi ∂xi

i=2

Similarly, we have j−1 n ∂vij j=2

i=1

∂xi

dxj =

n−1 i=1

j−1 n−1 ∂vin ∂vij dxj . dxn + ∂xi ∂xi j=2 i=1

Hence, we obtain d v n

= (−1)

n−1 ∂vin i=1

n

−

= (−1)

∂xi

dxn +

j=2 i=1

n ∂v1i i=2

j−1 n−1 ∂vij

∂xi

dx1 +

∂xi

dxj −

i=2

j−1 n−1 ∂vij j=2

i=1

n ∂v1i

∂xi

−

∂xi

dx1 −

n ∂vji i=j+1

n−1 n ∂vji j=2 i=j+1

∂xi

dxj +

∂xi

n−1 ∂vin i=1

∂xi

dxj

dxn .

By (1.2.12), we have the following equation: du|du|p−2 n

= (−1)

−

n ∂v1i i=2

∂xi

dx1 +

j−1 n−1 ∂vij j=2

i=1

∂xi

−

n ∂vji i=j+1

∂xi

(1.3.15)

dxj +

n−1 ∂vin i=1

∂xi

dxn .

1.3 p-Harmonic equations

19

If u = u(x1 , x2 , . . . , xn ) is a function, then (1.3.15) can be written as the following system: n ∂u p−2 1i , = (−1)n − i=2 ∂v ∂x1 |∇u| ∂xi ∂u p−2 ∂xj |∇u|

= (−1)n

∂u p−2 ∂xn |∇u|

= (−1)n

j−1 ∂vij i=1 ∂xi

n−1 i=1

−

n

∂vji i=j+1 ∂xi

, j = 2, 3, . . . , n − 1,

∂vin ∂xi .

(1.3.16) Thus, we see that (1.2.12) is equivalent to system (1.3.16). If we choose p = 2 and n = 2 in the above system, then (1.3.16) reduces to the wellknown Cauchy–Riemann equations. Thus, system (1.3.16) can be considered as a natural extension of the Cauchy–Riemann equations. In order to ﬁnd p-harmonic tensors, we only need to ﬁnd u and v satisfying (1.3.16). We conclude this section with the following example of a pair of the conjugate p-harmonic tensors, which have nice symmetric properties.

1.3.4 An example In this section, we introduce an example of a pair of the conjugate harmonic tensors in R3 . Example 1.3.1. Let u=

3 3 = 2 ρ x1 + x22 + x23

be a harmonic function in R3 and v be a 2-form in R3 deﬁned by v = v3 dx1 ∧ dx2 + v2 dx1 ∧ dx3 + v1 dx2 ∧ dx3 . Note that we have switched v1 with v3 in (1.3.8). Then, (1.3.10) becomes ∂u = ∂x1

∂v3 ∂v2 + ∂x2 ∂x3

∂u = ∂x2

∂v1 ∂v3 − ∂x3 ∂x1

∂u ∂v2 ∂v1 = − − ∂x3 ∂x1 ∂x2 and (1.3.13) reduces to ∂u |∇u|p−2 = ∂x1

∂v3 ∂v2 + ∂x2 ∂x3

(1.3.17)

20

1 Hardy–Littlewood inequalities

∂u |∇u|p−2 = ∂x2

∂v1 ∂v3 − ∂x3 ∂x1

(1.3.18)

∂u ∂v2 ∂v1 |∇u|p−2 = − − . ∂x3 ∂x1 ∂x2 Here v1 , v2 , and v3 are deﬁned as follows: x2 x3 x2 x3 x4 − x4 x22 − x23 , v 1 = 2 2 2 3 2 = 2 xi i<j (xi + xj ) x1 + x22 + x23 (x21 + x22 )(x21 + x23 ) x1 x3 x1 x3 x4 − x4 x21 − x23 v 2 = 2 1 2 3 2 = 2 , 2 2 2 xi i<j (xi + xj ) x1 + x2 + x3 (x1 + x22 )(x22 + x23 ) x1 x2 x1 x2 x4 − x4 x21 − x22 . v 3 = 2 1 2 2 2 = 2 2 xi i<j (xi + xj ) x1 + x22 + x23 (x1 + x23 )(x22 + x23 ) Then, u and v form a pair of conjugate harmonic tensors. Now, we check that u, v1 , v2 , and v3 deﬁned above satisfy (1.3.17). For u = 3/ρ, system (1.3.17) reduces to ∂v3 ∂v2 −3x1 = + , ∂x2 ∂x3 (x21 + x22 + x23 )3/2 (x21

−3x2 = + x22 + x23 )3/2

∂v1 ∂v3 − , ∂x3 ∂x1

(1.3.19)

∂v2 ∂v1 −3x3 = − − . ∂x1 ∂x2 (x21 + x22 + x23 )3/2 By a long computation and simpliﬁcation, we have ∂v1 x3 (x22 − x23 ) 2x3 x22 = 2 + 2 , 2 2 2 2 2 3/2 2 ∂x2 (x1 + x2 )(x1 + x2 + x3 ) (x1 + x2 ) (x21 + x22 + x23 )1/2 x2 (x22 − x23 ) −2x2 x23 ∂v1 = 2 + , 2 2 2 2 2 2 ∂x3 (x1 + x3 )(x1 + x2 + x3 )3/2 (x1 + x3 )2 (x21 + x22 + x23 )1/2 x3 (x21 − x23 ) 2x3 x21 ∂v2 = 2 + , ∂x1 (x1 + x22 )(x21 + x22 + x23 )3/2 (x21 + x22 )2 (x21 + x22 + x23 )1/2 x1 (x21 − x23 ) −2x1 x23 ∂v2 = 2 + , ∂x3 (x2 + x23 )(x21 + x22 + x23 )3/2 (x22 + x23 )2 (x21 + x22 + x23 )1/2 x2 (x21 − x22 ) 2x2 x21 ∂v3 = 2 + , ∂x1 (x1 + x23 )(x21 + x22 + x23 )3/2 (x21 + x23 )2 (x21 + x22 + x23 )1/2 x1 (x21 − x22 ) −2x1 x22 ∂v3 = 2 + 2 . 2 2 2 2 2 3/2 2 ∂x2 (x2 + x3 )(x1 + x2 + x3 ) (x2 + x3 ) (x21 + x22 + x23 )1/2

1.4 Some weight classes

21

Thus, it follows that ∂v3 ∂x2

+

∂v2 ∂x3

=

x1 (2x21 −(x22 +x23 )) (x22 +x23 )(x21 +x22 +x23 )3/2

=

−3x1 (x22 +x23 ) (x22 +x23 )(x21 +x22 +x23 )3/2

=

−3x1 . (x21 +x22 +x23 )3/2

−

2x1 (x22 +x23 )(x21 +x22 +x23 )1/2

Hence, the ﬁrst equation of system (1.3.19) holds. By the same method, we can check that second and third equations of (1.3.19) hold too. Again by a long computation and simpliﬁcation, it follows that ∂v2 ∂v3 ∂v1 + + = 0, ∂x1 ∂x2 ∂x3 so that v is a closed 2-form.

1.4 Some weight classes In this section, we present various classes of the weight functions. These weight functions will be used in later chapters to establish weighted norm inequalities. We also discuss some two-weight classes.

1.4.1 Ar (Ω)-weights We study some basic elements of Ar (Ω)-weights which were introduced by Muckenhoupt (1972) who used this weight class to explore the properties of the Hardy–Littlewood maximal operator. We also discuss some other weight classes. Recall that a function w(x) is called a weight if w > 0 a.e. and w ∈ L1loc (Rn ). Deﬁnition 1.4.1. w is called a doubling weight and write w ∈ D(Ω) if there exists a constant C such that μ(2B) ≤ Cμ(B) for all balls B with 2B ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx. If this condition holds only for all balls B with 4B ⊂ Ω, then w is weak doubling and we write w ∈ W D(Ω). Deﬁnition 1.4.2. Let σ > 1. It is said that w satisﬁes a weak reverse H¨older inequality and write w ∈ W RHβ (Ω) when there exist constants C, β, γ with 0 < γ < β such that

22

1 Hardy–Littlewood inequalities

1 |B|

1/β

≤C

wβ dx B

1 |B|

1/γ

wγ dx

(1.4.1)

σB

for all balls B with σB ⊂ Ω. We say that w satisﬁes a reverse H¨older inequality when (1.4.1) holds with σ = 1, and write w ∈ RHβ (Ω). Remark. From [119], we know that if (1.4.1) holds for some γ with 0 < γ < β and any ball B with σB ⊂ Ω, then it also holds for all γ with 0 < γ < ∞, that is, if 1/β 1/γ0 1 1 wβ dx ≤C wγ0 dx |B| B |B| σB holds for some γ0 with 0 < γ0 < β, then

1 |B|

1/β

≤C

β

w dx B

1 |B|

1/γ

(1.4.1)

γ

w dx σB

holds for all γ with 0 < γ < ∞. We should also notice that the space W RHβ (Ω) is independent of σ > 1, see [120]. We next deﬁne Muckenhoupt weights. Deﬁnition 1.4.3. A weight w satisﬁes the Ar (Ω)-condition in a subset Ω ⊂ Rn , where r > 1, and write w ∈ Ar (Ω) when sup B

1 |B|

wdx B

1 |B|

r−1

< ∞,

w1/(1−r) dx B

where the supremum is over all balls B ⊂ Ω. Notice it follows from H¨ older’s inequality that if w ∈ Ar (Ω) and 0 < α ≤ 1, then wα ∈ Ar (Ω). Indeed sup B

1 |B|

≤ sup B

1 |B|

α

B

1 |B|

wdx B

r−1

wα dx

wα/(1−r) dx B

1 |B|

α(r−1)

w

1/(1−r)

dx

< ∞.

(1.4.2)

B

If w satisﬁes the Ar -condition for all balls B with 2B ⊂ E, we write w ∈ loc loc Aloc r (E). Also we write A∞ (E) = ∪r>1 Ar (E) and A∞ (E) = ∪r>1 Ar (E). It is well known that w ∈ A∞ (Ω) if and only if w ∈ RHβ (Ω) for some β > 1. This is also true for Aloc ∞ (Ω) and W RHβ (Ω) for some β > 1. Moreover (Ω) ⊂ W D(Ω). Aloc ∞

1.4 Some weight classes

23

1.4.2 Ar (λ, E)-weights We have discussed the Ar (E)-weights and their properties in section 1.4.1. Now, we introduce the Ar (λ, E)-weights. These weights have properties similar to those of Ar (E). The following Ar (λ, E)-weight class was introduced in [121]. Deﬁnition 1.4.4. Let w be a locally integrable nonnegative function in E ⊂ Rn and assume that 0 < w < ∞ almost everywhere. We say that w belongs to the Ar (λ, E) class, 1 < r < ∞ and 0 < λ < ∞, or that w is an Ar (λ, E)-weight, write w ∈ Ar (λ, E) or w ∈ Ar (λ) when it will not cause any confusion, if sup B

1 |B|

λ

w dx B

1 |B|

1/(r−1) r−1 1 dx <∞ w B

for all balls B ⊂ E ⊂ Rn . It is clear that Ar (1) is the usual Ar -class, see [122, 123, 59] for more properties of Ar -weights. Similar to proofs in [123], we prove some properties of the Ar (λ)-weights. The following theorem says that Ar (λ) is an increasing class with respect to r. Theorem 1.4.5. If 1 < r < s < ∞, then Ar (λ) ⊂ As (λ). Proof. Let w ∈ Ar (λ). Since 1 < r < s < ∞, by H¨older’s inequality

1 1/(s−1) B

w

s−1 dx

≤

1 1/(r−1) B

w

= |B|s−r = so that

1 |B|

|B|s−1 |B|r−1

dx

r−1

1 1/(r−1)

B

11/(s−r) dx

s−r

r−1 dx B w

r−1 1 1/(r−1) dx , B w

1/(s−1) s−1 1/(r−1) r−1 1 1 1 dx ≤ dx . w |B| w B B

Thus, we ﬁnd sup B

1 |B|

wλ dx B

1 |B|

1/(s−1) (s−1) 1 dx w B

24

1 Hardy–Littlewood inequalities

≤ sup B

1 |B|

λ

w dx B

1/(r−1) (r−1) 1 dx <∞ w B

1 |B|

for all balls B ⊂ Rn since w ∈ Ar (λ). Therefore, w ∈ As (λ), and hence Ar (λ) ⊂ As (λ). The following result shows that Ar (λ)-weights have the property similar to the strong doubling property of Ar -weights. Theorem 1.4.6. If w ∈ Ar (λ), λ ≥ 1, and the measure μ is deﬁned by dμ = w(x)dx, then |E|r μ(E) ≤ Cr,λ,w , (1.4.3) |B|λ+r−1 μ(B)λ where B is a ball in Rn and E is a measurable subset of B. Proof. By H¨ older’s inequality, we have |E| = E dx = E w1/r w−1/r dx

1/r

(r−1)/r w1/(1−r) dx ≤ E wdx E

(r−1)/r 1/r = (μ(E)) w1/(1−r) dx . E This implies

r−1

|E| ≤ μ(E) r

w

1/(1−r)

dx

.

(1.4.4)

E

Note that λ ≥ 1, by H¨older’s inequality again, we have 1 |B| so that 1=

1 μ(B)

wdx ≤ B

wdx ≤ B

1 |B|

|B| μ(B)

Hence, we obtain

1/λ

wλ dx

,

B

1 |B|

1/λ

wλ dx

.

B

μ(B)λ ≤ |B|λ−1

wλ dx.

(1.4.5)

B

Since w ∈ Ar (λ), there exists a constant Cr,λ,w such that

1 |B|

wλ dx B

1 |B|

1/(r−1) r−1 1 dx ≤ Cr,λ,w . w B

Combining (1.4.4), (1.4.5), and (1.4.6), we deduce that

(1.4.6)

1.4 Some weight classes

25

r−1 |E|r μ(B)λ ≤ μ(E)|B|λ−1 B wλ dx E w1/(1−r) dx 1 1/(r−1) r−1 1 1 λ ≤ μ(E)|B|λ+r−1 |B| w dx dx |B| B B w ≤ Cr,λ,w μ(E)|B|λ+r−1 . Hence

μ(E) |E|r ≤ Cr,λ,w . |B|λ+r−1 μ(B)λ

If we put λ = 1 in Theorem 1.4.6, then (1.4.3) becomes |E|r μ(E) , ≤ Cr,w |B|r μ(B) which is called the strong doubling property of Ar -weights; see [59]. It is well known that an Ar -weight w satisﬁes the following reverse H¨older inequality. Lemma 1.4.7. If w ∈ Ar , r > 1, then there exist constants β > 1 and C, independent of w, such that w β,Q ≤ C|Q|(1−β)/β w 1,Q for all cubes Q ⊂ Rn .

1.4.3 Aλ r (E)-weights The following class of Aλr -weights was introduced in [124] and several new weighted integral inequalities for diﬀerential forms were proved. Deﬁnition 1.4.8. We say that the weight w(x) > 0 satisﬁes the Aλr (E)condition, r > 1 and λ > 0, and write w ∈ Aλr (E), if sup B

1 |B|

wdx B

1 |B|

λ(r−1)

w1/(1−r) dx

<∞

B

for any ball B ⊂ E. Here E is a subset of Rn . The Aλr (E)-weights have similar properties as we have discussed for Ar (λ, E)-weights. We leave it to readers to establish the properties of the Aλr (E)-weights.

26

1 Hardy–Littlewood inequalities

Example 1.4.9. Let Ω ⊂ R2 be a bounded domain. For any x ∈ Ω, we deﬁne ⎧ 1 ⎨ |x|α , x = 0, w(x) = ⎩ 1, x = 0, where α is a positive constant with α < 2. Then, w(x) is an Aλr (Ω)-weight for any constant λ > 1. To show this let B ⊂ Ω be a disc with center x0 and radius r0 . Then, r0 ≤ diam(Ω) < ∞ since Ω is a bounded domain. We may assume that x0 = 0. Using the polar coordinate substitution, we have 2π r w(x)dx = 0 dθ 0 0 ρ−α ρdρ B r = 2π 0 0 ρ1−α dρ = C1 r02−α . Thus, 1 |B|

wdx = B

C2 . r0α

(1.4.7)

Using the polar coordinate substitution again, it follows that B

1 w(x)

1/(r−1) 2+α/(r−1)

dx = C3 r0

.

Hence, we ﬁnd 1 1 1/(r−1) λ(r−1) dx ≤ C4 r0αλ . |B| B w(x)

(1.4.8)

Combining (1.4.7) and (1.4.8), we obtain sup B

1 |B|

wdx B

1 |B|

λ(r−1)

w

1/(1−r)

dx

α(λ−1) < ∞. 1. Similarly, we have the following example of the Aλr (Ω)-weights with λ > 0. Example 1.4.10. Let Ω ⊂ R2 be a bounded domain. For any x ∈ Ω, we deﬁne w(x) = |x|β , where β is a constant with 0 < β < 2(r − 1) and r is the constant in the deﬁnition of the Aλr (Ω)-weights. Then, w(x) is an Aλr (Ω)weight for any constant λ with λ > 0.

1.4 Some weight classes

27

1.4.4 Some classes of two-weights The following class of Ar,λ (E)-weights (or the two-weight) appeared in [125]. It is easy to see that the class of Ar,λ (E)-weights is an extension of the usual class of Ar -weights [59], and also class of Ar (λ)-weights [121]. See [126–133] for applications of the two-weight. Deﬁnition 1.4.11. A pair of weights (w1 , w2 ) satisfy the Ar,λ (E)-condition in a set E ⊂ Rn , write (w1 , w2 ) ∈ Ar,λ (E) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1, if sup B⊂E

1 |B|

1/λr

w1λ dx B

1 |B|

B

1 w2

1/λr

λr /r

< ∞.

dx

Deﬁnition 1.4.12. A pair of weights (w1 , w2 ) satisfy the Aλr (E)-condition in a set E ⊂ Rn , and write (w1 , w2 ) ∈ Aλr (E) for some r > 1 and λ > 0, if sup B

1 |B|

w1 dx B

1 |B|

B

1 w2

λ(r−1)

1/(r−1)

<∞

dx

for any ball B ⊂ E. Deﬁnition 1.4.13. A pair of weights (w1 , w2 ) satisfy the Ar (λ, E)-condition in a set E ⊂ Rn , and write (w1 , w2 ) ∈ Ar (λ, E) for some r > 1 and λ > 0, if sup B

1 |B|

w1λ dx B

1 |B|

B

1 w2

r−1

1/(r−1) dx

<∞

for any ball B ⊂ E. Example 1.4.14. Let Ω ⊂ R2 be a bounded domain. For any x ∈ Ω, we assume that d(x, ∂Ω) is the distance from x to Ω and deﬁne w1 (x) = and w2 (x) =

1 dα (x, ∂Ω)

⎧ 1 , ⎨ |x|α/λ d(x,∂Ω)

x = 0,

⎩

x = 0,

1,

where α is a constant with 0 < α < 1 and λ is the parameter appearing in the deﬁnition of Aλr -weights. Then, (w1 , w2 ) is a pair of the Aλr (Ω)weights.

28

1 Hardy–Littlewood inequalities

To show it, let B ⊂ Ω be a ball with center x0 and radius r0 . We may assume that x0 = 0. Otherwise, we can move the center to the origin by a simple transformation. Then, for any x ∈ B, we have d(x, ∂Ω) ≥ r0 − |x|. Hence, 1 1 ≤ w1 (x) = α d (x, ∂Ω) (r0 − |x|)α and

1 = |x|α/λ d(x, ∂Ω) ≤ C1 |x|α/λ , x = 0 w2 (x)

since Ω is bounded. Using the polar coordinate substitution, we have 2π r w (x)dx ≤ 0 dθ 0 0 (r0 − ρ)−α ρdρ B 1 (1.4.9) r = 2π 0 0 (r0 − ρ)−α ρdρ. In (1.4.9) let r0 − ρ = t, to ﬁnd 1 |B| B w1 (x)dx =

1 πr02

B

r0

w1 (x)dx t−α (r0 − t)dt

≤

2 r02

=

2 1 (1−α)(2−α) r0α

=

C2 r0α .

0

(1.4.10)

By the same method, we obtain 1 |B|

B

1 w2

1/(r−1) dx

C1 |x|α/λ

1/(r−1)

≤

1 |B|

≤

C3 λ(r−1) α/(λ(r−1)) α+2λ(r−1) r0

B

α/(λ(r−1))

= C4 r0

dx (1.4.11)

.

Thus, we have 1 1 1/(r−1) λ(r−1) dx ≤ C5 r0α . |B| B w2

(1.4.12)

Combining (1.4.10) and (1.4.12), we ﬁnd that (w1 , w2 ) satisfy the conditions of Aλr (Ω)-weights, and hence (w1 , w2 ) is a pair of the Aλr (Ω)-weights. The following result which is proved in [119] can be used to extend the local results to the global cases.

1.5 Inequalities in John domains

29

Theorem 1.4.15. If V is a collection of cubes in Rn and CQ are nonnegative numbers associated with the cubes Q ∈ V and w ∈ Ar , dμ(x) = w(x)dx, then for 1 ≤ p < ∞ and N ≥ 1, it follows that ⎛ ⎝ Rn

(

⎞1/p

⎛

≤ Bp ⎝

CQ · χN Q )p dμ(x)⎠

( Rn

Q∈V

⎞1/p CQ · χQ )p dμ(x)⎠

,

Q∈V

where Bp is independent of the collection V and the numbers CQ .

1.5 Inequalities in John domains

In 1927, M. Riesz proved the following result in [134] for conjugate harmonic functions. Let f (z) = u(z) + iv(z), v(0) = 0, be analytic in the unit disk D. Then, for every p > 1, 2π 2π |u(z)|p dϕ ≤ C |v(z)|p dϕ (z = reiϕ , r < 1), 0

0

where C = C(p) depends on p only. In 1932, Hardy and Littlewood (see [100]) proved the following inequality for conjugate harmonic functions in the unit disk D. Theorem 1.5.1. For each p > 0, there is a constant C such that |u − u(0)|p dxdy ≤ C |v − v(0)|p dxdy D

D

for all analytic functions f = u + iv in the unit disk D.

1.5.1 Local inequalities This opened a new research direction in analysis. During the last 80 years, their work has attracted several mathematicians and various versions of the Hardy–Littlewood inequalities or theorems have been established. Considering the length of this monograph, we cannot include all of them here; however, we encourage the reader to see [149, 155–158, 177, 103, 178] for these inequalities and their applications. In this section, we ﬁrst introduce the Hardy–Littlewood inequalities locally, then consider the weighted case in Section 1.5.2. Finally, we extend the result to the global case in δ-John domains in Section 1.5.3.

30

1 Hardy–Littlewood inequalities

A proper subdomain Ω ⊂ Rn is called a δ-John domain, δ > 0, if there exists a point x0 ∈ Ω which can be joined with any other point x ∈ Ω by a continuous curve γ ⊂ Ω so that d(ξ, ∂Ω) ≥ δ|x − ξ| for each ξ ∈ γ. Here d(ξ, ∂Ω) is the Euclidean distance between ξ and ∂Ω. In [119] Theorem 1.5.1 is generalized to the following global Hardy– Littlewood inequality for K-quasiregular mappings. Let f = (f 1 , f 2 , . . . , f n ) be a K-quasiregular mapping in a John domain Ω. If w satisﬁes the Ar condition and 0 < p < ∞, then j # f i # p,Ω,w ≤ Cp f p,Ω,w

for i, j = 1, 2, . . . , n, where Cp is a constant, independent of f . Here f # p,E,w = inf f − ap,E,w . a∈R

In fact, the above inequality was proved in a class of domains, which is more general than the class of John domains. The following result was obtained in [99]. Let D ⊂ Rn be a bounded, convex domain. To each y ∈ D there corresponds a linear operator Ky : C ∞ (D, ∧l ) → C ∞ (D, ∧l−1 ) deﬁned by

1

tl−1 ω(tx + y − ty; x − y, ξ1 , . . . , ξl−1 )dt

(Ky ω)(x; ξ1 , . . . , ξl−1 ) = 0

and a decomposition ω = d(Ky ω) + Ky (dω). A homotopy operator T : C ∞ (D, ∧l ) → C ∞ (D, ∧l−1 ) is deﬁned by averaging Ky over all points y in D, i.e., Tω = ϕ(y)Ky ωdy , (1.5.1) D

where ϕ ∈ C0∞ (D) is normalized by D ϕ(y)dy = 1. Then, there is also a decomposition ω = d(T ω) + T (dω). (1.5.2) The l-form ωD ∈ D (D, ∧l ) is deﬁned by ωD = |D|−1 ω(y)dy, l = 0, and ωD = d(T ω), l = 1, 2, . . . , n D

1.5 Inequalities in John domains

31

for all ω ∈ Lp (D, ∧l ), 1 ≤ p < ∞. By substituting z = tx + y − ty and t = v/(1 + v), we have ω(z, ζ(z, x − z), ξ)dz, (1.5.3) T ω(x, ξ) = D

where the vector function ζ : D × Rn →Rn is given by ∞ ζ(z, h) = h v l−1 (1 + v)n−l ϕ(z − vh)dv. 0

The following imbedding inequality appears in [99]: For each s > 1, the integral in (1.5.3) deﬁnes a bounded operator T : Ls (D, ∧l ) → W 1,s (D, ∧l−1 ), l = 1, 2, . . . , n with the norm estimated by T uW 1,s (D) ≤ C|D|us,D .

(1.5.4)

The following Ls -norm estimates for diﬀerential forms were also proved in [99]: (1) Let u ∈ Lsloc (D, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form in a bounded, convex domain D ⊂ Rn . Then, ∇(T u)s,D ≤ C|D|us,D ,

(1.5.5)

T us,D ≤ C|D|diam(D)us,D .

(1.5.6)

(2) Let u ∈ D (D, ∧l ) be such that du ∈ Ls (D, ∧l+1 ). Then u − uD is in W 1,s (D, ∧l ) and (1.5.7) u − uD W 1,s (D) ≤ C|D|dus,D . Here B is any ball or cube and C is a constant independent of u. In [71] appears the following local Hardy–Littlewood inequality for solutions to the conjugate A-harmonic equation. Theorem 1.5.2. Let u and v be conjugate A-harmonic tensors in Ω ⊂ Rn , σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u and v, such that q/p ||u − uQ ||s,Q ≤ C|Q|β ||v − c1 ||t,σQ and

||v − vQ ||t,Q ≤ C|Q|−βp/q ||u − c2 ||s,σQ p/q

32

1 Hardy–Littlewood inequalities

1,q for all cubes Q with σQ ⊂ Ω. Here c1 is any form in Wloc (Ω, ∧) with d c1 = 1,p 0, c2 is any form in Wloc (Ω, ∧) with dc2 = 0, and β = 1/s + 1/n − (1/t + 1/n)q/p.

Theorem 1.5.3 (Covering lemma). Each Ω has a modiﬁed Whitney cover of cubes V = {Qi } such that ∪i Qi = Ω, √ Q∈V χ 5 Q ≤ N χΩ 4

for all x ∈Rn and some N > 1, and if Qi ∩ Qj = ∅, then there exists a cube R (this cube need not be a member of V) in Qi ∩ Qj such that Qi ∪ Qj ⊂ N R. Moreover, if Ω is δ-John, then there is a distinguished cube Q0 ∈ V which can be connected with every cube Q ∈ V by a chain of cubes Q0 , Q1 , . . . , Qk = Q from V and such that Q ⊂ ρQi , i = 0, 1, 2, . . . , k, for some ρ = ρ(n, δ). In [71] the covering lemma is used to combine the local results into the following global Hardy–Littlewood inequality in John domains.

Theorem 1.5.4. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be conjugate A-harmonic tensors and 0 < s, t < ∞. If Ω is a δ-John domain, q ≤ p, v − c ∈ Lt (Ω, ∧2 ) and s = Φ(t) =

npt , nq + t(q − p)

0 < t < ∞,

(1.5.8)

then u ∈ Ls (Ω, ∧0 ) and moreover, there exists a constant C, independent of u and v, such that q/p

u − uQ0 s,Ω ≤ C v − c t,Ω . 1,q (Ω, ∧) with d c = 0 and Q0 is the cube appearing Here c is any form in Wloc in Theorem 1.5.3.

The covering lemma has been further used in [179–182] [71, 74, 196, 197] to extend the local results into global results.

1.5.2 Weighted inequalities In [179], Theorems 1.5.2 and 1.5.4 have been generalized to the following weighted versions (Theorems 1.5.5 and 1.5.6). Theorem 1.5.5. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Ar (Ω). Let s = Φ(t) be deﬁned by (1.5.8). Then, there exists a constant C, independent of u and v, such that

1.5 Inequalities in John domains

33

1/s |u − uQ |s wdx

q/pt

≤C

Q

|v − c|t wpt/qs dx σQ

for all cubes Q with σQ ⊂ Ω ⊂ Rn 1,q (Ω, ∧) with d c = 0. Wloc

and σ > 1. Here c is any form in

Proof. By Lemma 1.4.7, there exist constants α > 1 and C1 , independent of w, such that (1.5.9) w α,σQ ≤ C1 |Q|(1−α)/α w 1,σQ . Since 1/αs + (α − 1)/αs = 1/s, by the H¨ older inequality, we ﬁnd that 1/s Q

u − uQ s,Q,w ≤ w α,

· u − uQ αs/(α−1),

Q.

(1.5.10)

By Theorem 1.5.2, there is a constant C2 , independent of u and v, such that for any t > 0, we have u − uQ αs/(α−1),

Q

q/p σQ,

≤ C2 |Q|β · v − c t ,

(1.5.11)

where β = (α − 1)/αs + 1/n − (1/t + 1/n)q/p. Combining (1.5.10) and (1.5.11), we obtain

1/s Q

u − uQ s,Q,w ≤ C2 |Q|β · w α,

q/p σQ .

· v − c t ,

(1.5.12)

Now, choose t = t/k, where k is to be determined later, and note that |v − c| = w−p/qs |v − c|wp/qs . By the H¨older inequality, we get v−c

t ,

σQ

≤

1/t ||(1/w)pt/qs ||1/(k−1), σQ

·

1/t |v − c| w t

pt/qs

dx

.

σQ

(1.5.13) From (1.5.9), (1.5.12), and (1.5.13), we ﬁnd that

1/s

q/pt

u − uQ s,Q,w ≤ C3 |Q|β +(1−α)/αs w 1,σQ ||( w1 )pt/qs ||1/(k−1), q/pt × σQ |v − c|t wpt/qs dx .

σQ

(1.5.14) We choose k = 1 + pt(r − 1)/qs, so that (k − 1)qs/pt = r − 1, and since w ∈ Ar , it follows that 1/s

q/pt

w 1,σQ ||( w1 )pt/qs ||1/(k−1),σQ 1 1/(r−1) r−1 1/s 1/s+(k−1)q/pt 1 1 = |σQ| dx |σQ| σQ wdx |σQ| σQ w 1/s+(k−1)q/pt

≤ C4 |Q|

. (1.5.15)

34

1 Hardy–Littlewood inequalities

By (1.5.14) and (1.5.15), we obtain u − uQ s,Q,w ≤ C5 |Q|

q/pt

γ

|v − c| w t

pt/qs

dx

,

(1.5.16)

σQ

where γ = β +(1−α)/αs+1/s+q(k −1)/pt = −(nq +t(q −p))/npt+1/s = 0 by (1.5.8). So (1.5.16) is the same as

q/pt

u − uQ s,Q,w ≤ C

|v − c| w t

pt/qs

dx

,

σQ

that is

1/s |u − uQ |s wdx

q/pt

≤C

Q

|v − c|t wpt/qs dx

.

σQ

1.5.3 Global inequalities We discuss the global case in John domains. Using the local result and the properties of John domains, we obtain the following global inequality.

Theorem 1.5.6. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be conjugate Aharmonic tensors. Let q ≤ p, v − c ∈ Lt (Ω, ∧2 ), w ∈ Ar (Ω), and s = Φ(t) be deﬁned in (1.5.8). Then, there exists a constant C, independent of u and v, such that 1/s q/pt |u − uQ0 |s wdx ≤C |v − c|t wpt/qs dx Ω

Ω

1,q for any δ-John domain Ω ⊂ Rn . Here c is any form in Wloc (Ω, ∧) with d c = 0 and Q0 ⊂ Ω is the cube appearing in Theorem 1.5.3.

Proof. Since w ∈ Ar , we can write dμ(x) = w(x)dx. From Theorem 1.5.5, we have qs/pt s t pt/qs |u − uQ | dμ(x) ≤ C |v − c| w dx . (1.5.17) Q

σQ

We use the notations and the covering V deﬁned in Theorem 1.5.3, and the properties of the measure dμ(x) = w(x)dx: if w ∈ Ar , then μ(N Q) ≤ M N nr μ(Q)

(1.5.18)

for each cube Q with N Q ⊂ Rn (see [123]) and max(μ(Qi ), μ(Qi+1 )) ≤ M N nr μ(Qi ∩ Qi+1 )

(1.5.19)

1.5 Inequalities in John domains

35

for the sequence of cubes Qi , Qi+1 , i = 0, 1, . . . , k − 1, described in Theorem 1.5.3. We use the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ) for all s > 0. In particular, we have |u − uQ0 |s wdx Ω = Ω |u − uQ0 |s dμ(x) ≤ 2s Q∈V Q |u − uQ |s dμ(x) + 2s Q∈V Q |uQ0 − uQ |s dμ(x). (1.5.20) The ﬁrst sum can be estimated by (1.5.17) and Theorem 1.5.3 as s Q∈V Q |u − uQ | dμ(x) qs/pt ≤ C1 Q∈V σQ |v − c|t wpt/qs dx ≤ C1 N

Ω

|v − c|t wpt/qs dx

qs/pt

(1.5.21)

.

Now we estimate the second sum in (1.5.20). Fix a cube Q ∈ V and let Q0 , Q1 , . . . , Qk = Q be the chain in Theorem 1.5.3. Then, |uQ0 − uQ | ≤

k−1

|uQi − uQi+1 |.

(1.5.22)

i=0

From (1.5.17) and (1.5.19), we get |uQi − uQi+1 |s = ≤

1 μ(Qi ∩Qi+1 )

Qi ∩Qi+1

M N nr max(μ(Qi ),μ(Qi+1 ))

≤ C2 ≤ C3

Qi ∩Qi+1

i+1

1 j=i μ(Qj )

Qj

i+1

1 j=i μ(Qj )

|uQi − uQi+1 |s dμ(x) |uQi − uQi+1 |s dμ(x)

|u − uQj |s dμ(x)

|v − c|t wpt/qs dx σQj

qs/pt .

Since Q ⊂ N Qj for j = i, i + 1, 0 ≤ i ≤ k − 1, we ﬁnd |uQi − uQi+1 | χQ (x) ≤ C3 s

i+1 χN Qj (x) j=i

μ(Qj )

qs/pt |v − c| w t

pt/qs

dx

.

σQj

Thus, by (1.5.22) we obtain (note |a + b|1/s ≤ 21/s (|a|1/s + |b|1/s )) |uQ0 − uQ |χQ (x) ≤ C4

R∈V

1 μ(R)

qs/pt 1/s

|v − c| w t

σR

pt/qs

dx

· χN R (x)

36

1 Hardy–Littlewood inequalities

for every x ∈ Rn . Hence, s Q∈V Q |uQ0 − uQ | dμ(x) s

qs/pt 1/s 1 t pt/qs dμ(x). ≤ C5 Rn R∈V μ(R) |v − c| w dx χ (x) N R σR (1.5.23) If 0 ≤ s ≤ 1, we use the inequality | tα |s ≤ |tα |s , (1.5.18), and Theorem 1.5.3 to get s Q∈V Q |uQ0 − uQ | dμ(x) ≤ C6 ≤ C7

R∈V

R∈V

μ(N R) μ(R)

σR

σR

|v − c|t wpt/qs dx

|v − c|t wpt/qs dx

qs pt

qs pt

.

p Note that qs/pt ≥ 1 and tα ≤ ( tα )p for p ≥ 1 and tα > 0. Thus, s Q∈V Q |uQ0 − uQ | dμ(x)

qs/pt ≤ C7 R∈V Ω |v − c|t wpt/qs χσR (x)dx

qs/pt ≤ C7 Ω |v − c|t wpt/qs R∈V χσR (x)dx (1.5.24)

qs/pt ≤ C7 Ω |v − c|t wpt/qs N χΩ (x)dx

qs/pt ≤ C8 Ω |v − c|t wpt/qs dx . Combination of (1.5.20), (1.5.21), and (1.5.24) proves the theorem for the case 0 < s ≤ 1. Now for the case 1 < s < ∞, by (1.5.23) and Theorem 1.4.16, we have s Q∈V Q |uQ0 − uQ | dμ(x) s

qs/pt 1/s 1 t pt/qs dμ(x). ≤ C9 Rn R∈V μ(R) |v − c| w dx χ (x) R σR Note that

R∈V

χR (x) ≤

R∈V

χσR (x) ≤ N χΩ (x)

N N and thus with the elementary inequality | i=1 ti |s ≤ N s−1 i=1 |ti |s and Theorem 1.5.3, we obtain s Q∈V Q |uQ0 − uQ | dμ(x)

qs/pt 1 t pt/qs |v − c| w dx χ (x) dμ(x) ≤ C10 Rn R R∈V μ(R) σR

1.6 Inequalities in averaging domains

37

|v − c|t wpt/qs dx

qs/pt ≤ C11 Ω |v − c|t wpt/qs dx . = C10

R∈V

qs/pt

σR

(1.5.25)

Finally, combination of (1.5.20), (1.5.21), and (1.5.25) proves the theorem for the case 1 < s < ∞.

1.6 Inequalities in averaging domains In the previous section, we have discussed weighted global Hardy–Littlewood inequalities in John domains. One may ask the question, whether there are more general kind of domains such that the Hardy–Littlewood inequality also holds in these domains? We will answer this question in this section.

1.6.1 Averaging domains In 1989, Susan G. Staples [183] introduced the following Ls -averaging domains. Deﬁnition 1.6.1. A proper subdomain Ω ⊂Rn is called an Ls -averaging domain, s ≥ 1, if there exists a constant C such that

1 |Ω|

1/s

|u − uΩ | dx

≤ C sup

s

B⊂Ω

Ω

1 |B|

1/s

|u − uB | dx s

B

for all u ∈ Lsloc (Ω). Here |Ω| is the n-dimensional Lebesgue measure of Ω. In [183] these domains are characterized in terms of the quasihyperbolic metric. Particularly, Staples proved that any John domain is Ls -averaging domain. Recently, S. Ding and C. Nolder [184] have extended the Ls -averaging domains to the following Ls (μ)-averaging domains. Deﬁnition 1.6.2. We call a proper subdomain Ω ⊂Rn an Ls (μ)-averaging domain, s ≥ 1, if there exists a constant C such that

1 μ(B0 )

|u − uB0 ,μ | dμ s

Ω

1/s

≤ C sup

4B⊂Ω

1 μ(B)

1/s

|u − uB,μ | dμ s

B

for some ball B0 ⊂ Ω and all u ∈ Lsloc (Ω; μ). Here the supremum is over all balls B with 4B ⊂ Ω. The factor 4 here is for convenience; in fact, these domains are independent of this expansion factor, see [183]. For further properties of Ls (μ)-averaging domains, see [185].

38

1 Hardy–Littlewood inequalities

The Ls (μ)-averaging domain can also be characterized by quasihyperbolic metric, which was introduced by F.W. Gehring in 1970s. Deﬁnition 1.6.3. The quasihyperbolic distance between x and y in Ω is given by 1 ds, k(x, y) = k(x, y; Ω) = inf γ d(z, ∂Ω) γ where γ is any rectiﬁable curve in Ω joining x to y. Here d(z, ∂Ω) is the Euclidean distance between z and ∂Ω. F.W. Gehring and B. Osgood [11] have proved that for any two points x and y in Ω there is a quasihyperbolic geodesic arc joining them. For a function u we denote the μ-average over B by 1 udμ. uB,μ = μ(B) B

1.6.2 Ls(μ)-averaging domains The results presented in this section show that diﬀerent versions of Hardy– Littlewood inequality hold both locally and globally in Ls (μ)-averaging domains. In [184], S. Ding and C. Nolder proved the following version of the local Hardy–Littlewood inequality. Theorem 1.6.4. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Aloc r (Ω). Let s = Φ(t) be deﬁned by (1.5.8). Then, there is a constant C, independent of u and v, such that 1/s 1/(pκ) 1 |u − uB |s w dx ≤ C|B|γ |v − c|qκ dx μ(B) B

σB

for all balls B with 2σB ⊂ Ω. Here γ = 1/(βs) + 1/n − (1/t + 1/n)q/p, σ and c are as in Theorem 1.5.5, κ = βts/(βqs − pt), where β is the exponent in the weak reverse H¨ older inequality for Ar -weights and βqs − pt > 0. Proof. By Theorem 1.5.5 and H¨ older’s inequality, we have μ(B)−1/s

1/s

|u − uB,μ |s w dx

B

−1/s

≤ C1 μ(B)

×|B|γ

1/βs β

w dx

σB

σB

|v − c|βqts/(βqs−pt) dx

(βqs−pt)/βpts .

1.6 Inequalities in averaging domains

39

Since w ∈ Aloc older inequality (1.4.1) and r (Ω), it satisﬁes the weak reverse H¨ the weak doubling condition, and so μ(B)−1/s

1/βs wβ

≤ C2 μ(B)−1/s |B|(1−β)/(βs) μ(σB)1/s

σB

≤ C3 |B|

1−β βs

μ(B)−1/s μ(B)1/s = C3 |B|

1−β βs

and from this Theorem 1.6.4 follows. In [184], S. Ding and C. Nolder also obtained the following result which plays an important role in the proof of showing that any John domain is an Ls (μ)-averaging domain. Theorem 1.6.5. Let w ∈ W D(Ω). Suppose that s and q are positive constants and that μ({x ∈ B : |u(x) − uB,μ | > t}) ≤ se−qt μ(B)

(1.6.1)

for each t > 0 and each ball B with 4B ⊂ Ω. Then, there exists a constant C = C(s, q, n) such that |uB(x),μ − uB(y),μ | ≤ C(k(x, y) + 1)

(1.6.2)

for all x and y in Ω. Here B(x) is the ball B(x, d(x, ∂Ω)/4). Proof. For each z ∈ Ω, notice that 4B(z) ⊂ Ω. Fix x and y ∈ Ω and choose a quasihyperbolic geodesic arc γ joining x and y in Ω. We deﬁne an ordered sequence of points {zj } on γ by induction as follows. First let z1 = x. Next suppose that z1 , . . . , zj have been deﬁned, and let βj = γ(zj , y) denote that part of γ from zj to y and γj the component of βj ∩ B(zj ) which contains zj . Deﬁne zj+1 as the other endpoint of γj . We simplify notation as follows: Bj = B(zj ), dj = d(zj , ∂Ω), rj = dj /4 = radius of Bj , and y = zm+1 . From the deﬁnition of {zj }, we have |zj+1 − zj | = rj , j = 1, . . . , m − 1, and |zm+1 − zm | ≤ rm .

(1.6.3)

For j = 1, . . . , m−1, pick zj ∈ ∂Ω so that d(zj , ∂Ω) = |zj −zj |. If z ∈ γj ⊂ Bj , then d(z, ∂Ω) ≤ |z − zj | ≤ |z − zj | + |zj − zj | ≤ 4dj /3. Hence,

γj

3 1 ds ≥ d(z, ∂Ω) 4dj

ds ≥ γj

Summing the above inequality over j gives

3|zj+1 − zj | 3 . = 4dj 16

40

1 Hardy–Littlewood inequalities

m−1 3(m − 1) 1 1 ≤ ds ≤ ds = k(x, y), 16 d(z, ∂Ω) d(z, ∂Ω) γj γ 1

(1.6.4)

where m < ∞. Consider now the relative size of neighboring balls. Fix j and choose z, z ∈ ∂Ω so that dj = |z − zj | and dj+1 = |z − zj+1 |. Then, dj ≤ dj+1 + rj and dj+1 ≤ dj + rj , with the ﬁrst inequality yielding rj+1 = dj+1 /4 ≥ (dj − rj )/4 ≥ 3rj /4, and the second leading to rj ≥ 3rj+1 /4 if rj+1 ≥ rj . Hence, 3rj /4 ≤ rj+1 ≤ 4rj /3.

(1.6.5)

μ(Bj ∩ Bj+1 ) > c1 (μ(Bj ) + μ(Bj+1 ))

(1.6.6)

Next we shall show that

for a constant c1 . Fix j and let v = (zj + zj+1 )/2, s = (max(rj , rj+1 ))/8, and B be the ball of radius s centered at v. In the case rj ≥ rj+1 we have s = rj /8 ≤ rj+1 /6; similarly, if rj+1 ≥ rj we have s ≤ rj /6. Thus, for z ∈ B we obtain 1 |z − zj | ≤ |z − v| + |v − zj | ≤ s + |zj − zj+1 | ≤ rj , 2 and a similar argument shows |z − zj+1 | ≤ rj+1 . Hence we conclude that B ⊂ Bj ∩ Bj+1 . Since μ is doubling, we have 1 Bj ≥ N1 (w)μ(Bj ) μ(Bj ∩ Bj+1 ) ≥ μ(B) ≥ μ 8

and μ(Bj ∩ Bj+1 ) ≥ μ(B) ≥ μ Let c1 =

1 2

1 Bj+1 8

≥ N2 (w)μ(Bj+1 ).

min{N1 (w), N2 (w), 4s}. Then, it follows that μ(Bj ∩ Bj+1 ) ≥ 12 (N1 (w)μ(Bj ) + N2 (w)μ(Bj+1 )) ≥ c1 (μ(Bj ) + μ(Bj+1 )).

For j = 1, 2, . . . , m + 1, let Ej = {x ∈ Bj : |u − uBj ,μ | > t}, where t = (log(2s/c1 ))/q. From (1.6.1), we obtain μ(Ej ) ≤ c1 μ(Bj )/2.

(1.6.7)

1.6 Inequalities in averaging domains

41

Combination of (1.6.6) and (1.6.7) yields μ((Bj ∩ Bj+1 )\(Ej ∪ Ej+1 )) > 0. Therefore, there exists x ∈ (Bj ∩ Bj+1 )\(Ej ∪ Ej+1 ), and hence |uBj ,μ − uBj+1 ,μ | ≤ |u − uBj ,μ | + |u − uBj+1 ,μ | ≤ 2t. Summing the above inequality and using (1.6.4), we conclude that m |uB(x),μ − uB(y),μ | ≤ 1 |uBj ,μ − uBj+1 ,μ | ≤ 2mt ≤ 32tk(x, y)/3 + 2t ≤

32t 3 (k(x, y)

=

32(log(2s/c1 )) (k(x, y) 3q

+ 1) + 1)

= C(k(x, y) + 1). Finally, we state the following result from [184] which extends Theorem 1.5.6 into the case of Ls (μ)-averaging domains.

Theorem 1.6.6. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be conjugate Aharmonic tensors in an Ls (μ)-averaging domain Ω ⊂Rn . If v−c ∈ Lt (Ω, ∧2 ), w ∈ Aloc r (Ω) and nqsβ , t = Ψ (s) = np + (p − q)sβ then there exists a constant C, independent of u and v, such that ⎛ ⎝ 1 μ(Ω)

⎞1/s |u − uB0 ,μ |s w dx⎠

⎛ ≤C⎝

Ω

⎞1/(pκ) |v − c|qκ dx⎠

,

Ω

where κ = βts/(βqs − pt) is again as above and d c = 0. Here B0 is some ﬁxed ball.

1.6.3 Other weighted inequalities Recently, G. Bao [186] proved the following local Ar (λ, Ω)-weighted Hardy– Littlewood integral inequality. Theorem 1.6.7. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Ar (λ, Ω) for some r > 1 and λ > 1. Let 0 < s, t < ∞. Then, there exists a constant C, independent of u and v, such that

42

1 Hardy–Littlewood inequalities

1/s |u − uB |s wλ/p dx

q/pt

≤ C|B|γ

|v − c|t wt/qs dx

B

σB

for all balls B with σB ⊂ Ω ⊂ Rn and σ > 1. Here c is any form in 1,q (Ω, ∧) with d c = 0 and γ = 1/s + 1/n − (1/t + 1/n)q/p. Wloc Note that the above inequality can be written in the following symmetric form:

1 |B|

1/qs

|u − uB |s wλ/p dx

≤ C|B|( nq − np ) 1

1

B

1 |B|

t

|v − c|t w qs dx

pt1 .

σB

In [180], parametric versions of the Ar (Ω)-weighted Hardy–Littlewood inequality were established. Corollary 1.6.8 follows from Theorem 1.5.5 using equation (1.4.2). Corollary 1.6.8. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Ar (Ω) for some r > 1. Let 0 < s, t < ∞. Then, there exists a constant C, independent of u and v, such that

1/s |u − uQ | w dx s

α

≤ C|Q|

q/pt |v − c| w

γ

Q

t

ptα/qs

dx

σQ

for all cubes Q with σQ ⊂ Ω ⊂ Rn , σ > 1, and 0 < α ≤ 1. Here c is any 1,q (Ω, ∧) with d c = 0 and γ = 1/s + 1/n − (1/t + 1/n)q/p. form in Wloc Note that α ∈ (0, 1] is arbitrary in Corollary 1.6.8. Hence, for particular values of α, we get diﬀerent versions of the Hardy–Littlewood inequality. For example, if we let α = qs, qs ≤ 1 in Theorem 1.6.8, we obtain the following symmetric inequality. Corollary 1.6.9. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Ar (Ω) for some r > 1. Let 0 < t < ∞ and qs ≤ 1. Then, there exists a constant C, independent of u and v, such that Q

1/qs |u − uQ |s wqs dx

≤ C|Q|γ

1/pt |v − c|t wpt dx

σQ

for all cubes Q with σQ ⊂ Ω ⊂ Rn and σ > 1. Here c is any form in 1,q (Ω, ∧) with d c = 0 and γ = (1/s + 1/n)/q − (1/t + 1/n)/p. Wloc Similarly, if we choose α = 1/pt, pt ≥ 1 in Theorem 1.6.8, we obtain the following symmetric inequality. Corollary 1.6.10. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂ Rn and w ∈ Ar (Ω) for some r > 1. Let 0 < s < ∞ and pt ≥ 1. Then, there exists a constant C, independent of u and v, such that

1.7 Two-weight cases

43

1/qs |u − uQ |s w1/pt dx

≤ C|Q|γ

1/pt |v − c|t w1/qs dx

Q

σQ

for all cubes Q with σQ ⊂ Ω ⊂ Rn and σ > 1. Here c is any form in 1,q (Ω, ∧) with d c = 0 and γ = (1/s + 1/n)/q − (1/t + 1/n)/p. Wloc

1.7 Two-weight cases In the last section, we have discussed diﬀerent versions of the Hardy– Littlewood inequality, including the Ar -weighted Hardy–Littlewood inequality. These inequalities may be extended into the two-weight case. In order to prove the two-weight inequality, one only needs to replace w(x) on the lefthand side by w1 (x) and on the right-hand side by w2 (x) in single weighted inequalities and slightly adjust the conditions on the weight if necessary. For example, the following is one version of two-weight Hardy–Littlewood inequality that was obtained recently.

1.7.1 Local inequalities Theorem 1.7.1. Let u and v be conjugate A-harmonic tensors in a domain Ω ⊂Rn and (w1 , w2 ) ∈ Ar,λ (Ω). Let s = Φ(t) be as in (1.5.8). Then, there exists a constant C, independent of u and v, such that

1/s |u − uB |s w1α dx

q/pt αpt/qs

≤C

|v − c|t w2

B

dx

(1.7.1)

σB

for all cubes B with σB ⊂ Ω ⊂ Rn and σ > 1. Here c is any form in 1,q (Ω, ∧) with d c = 0, 0 < α < t and α < λ, where λ ≥ 1. Wloc Proof. Let δ = sλ/(λ − α) so that δ > s, and

1/s |u − uB |s w1α dx 1/s α/s = B (|u − uB |w1 )s dx

B

≤

|u − uB |δ dx B

= u − uB δ,B

1/δ

αδ/(δ−s)

w B 1

αδ/(δ−s)

B

w1

(δ−s)/sδ dx

(1.7.2)

(δ−s)/sδ dx

.

By Theorem 1.5.2, for q ≤ p and 0 < δ, ξ < ∞, we have q/p

u − uB δ,B ≤ C1 |B|β v − c1 ξ,σB ,

(1.7.3)

44

1 Hardy–Littlewood inequalities

where β = 1/δ + 1/n − (1/ξ + 1/n)q/p and 0<ξ=

stλq . (sq − αp)t + λsq

Note that α < t leads to qs > pt so that qs > pα, and hence condition (1.5.8) gives ξ < t. Thus, by the H¨ older inequality q/p

v − c1 ξ,σB =

αp/sq

(|v − c1 |w2 σB

≤ C2

αp sq

σB

q/ξp

−αp/sq ξ

w2

) dx

(|v − c1 |w2 )t dx

1/t

( 1 )(αp/sq)(ξt/(t−ξ)) dx σB w2

(t−ξ) ξt

q/p . (1.7.4)

Substitution of (1.7.3) and (1.7.4) into (1.7.2) gives

1/s |u − uB |s w1α dx q/pt αp/sq t ≤ C3 |B|β σB (|v − c1 |w2 ) dx

B

×

( 1 )(αp/sq)(ξt/(t−ξ)) dx σB w2

(t−ξ)q/ξtp

αδ/(δ−s)

w B 1

(δ−s)/sδ dx

.

(1.7.5) From the choice of δ and ξ, and sq/αp > 1, it follows that r = sq/αp, and hence (t−ξ)q/ξtp (δ−s)/sδ αδ/(δ−s) 1 (αp/sq)(ξt/(t−ξ)) ( ) dx w dx 1 σB w2 B ≤ C4 |σB|β1 ≤ C5 |B|

β1

1 |σB|

1 |σB|

( 1 )αpλ/(sq−αp) dx σB w2

( 1 )λ/(r−1) dx σB w2

(sq−αp) λsp

(r−1)/rλ

1 |σB|

1 |σB|

wλ dx σB 1

wλ dx σB 1

α/sλ

1/λr q/p

≤ C6 |B|β1 , (1.7.6) where β1 = (1/ξ − 1/t)q/p + (1/s − 1/δ). Substituting (1.7.6) into (1.7.5) and using β + β1 = 1/δ + 1/n − (1/ξ + 1/n)q/p + (1/ξ − 1/t)q/p + (1/s − 1/δ) = 0 in view of condition (1.5.8), we ﬁnally obtain |u − uB |

s

B

1/s

w1α dx

≤ C7

q/pt |v −

σB

αpt/sq c1 | w2 dx t

.

1.8 The best integrable condition

45

1.7.2 Global inequalities Theorem 1.7.2. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be conjugate Aharmonic tensors. If Ω is a δ-John domain, q ≤ p, v − c ∈ Lt (Ω, ∧2 ), and s = Φ(t) as in (1.5.8), α > 0, and weight (w1 , w2 ) ∈ Ar,λ . Then, q/p

u − uQ0 s,Ω,w1α ≤ Cv − c

αpt/qs

t,Ω,w2

,

(1.7.7)

1,q where c is any form in Wloc (Ω, ∧) with d c = 0 and Q0 is a ﬁxed cube.

There are several other versions of the two-weight Hardy–Littlewood inequality, e.g., see [187, 188]. Some work related to the Hardy–Littlewood inequality can also be found in [189–193, 148, 67–69, 73, 74].

1.8 The best integrable condition In this section, we show that condition (1.5.8) in Theorem 1.5.4 is the best possible.

1.8.1 An example We illustrate the following example in the three-dimensional space. Example 1.8.1. Let f (x) = x|x|β = (x1 |x|β , x2 |x|β , x3 |x|β ) be a K-quasiregular mapping in R3 . Here β = −1 is a real number. From Example 1.2.4 with l = 1, we know that u = f 1 = x1 |x|β and v = f 2 df 3 = x2 |x|β d(x3 |x|β ) are conjugate A-harmonic tensors. It is easy to compute that v

=

βx1 x2 x3 |x|2β−2 dx2 ∧ dx3 − βx22 x3 |x|2β−2 dx1 ∧ dx3 +|x|2β−2 x2 (|x|2 + βx23 )dx1 ∧ dx2 ,

so that |u| = |x1 ||x|β

46

1 Hardy–Littlewood inequalities

and |v| = |x|2β−1 |x2 |(|x|2 + (β 2 + 2β)x23 )1/2 . Now applying the following spherical ⎧ ⎨ x1 = x = ⎩ 2 x3 =

coordinate transformation r cos θ sin ϕ r sin θ sin ϕ r cos ϕ,

where 0 < r ≤ 1, 0 ≤ ϕ ≤ π, 0 ≤ θ ≤ 2π, we obtain |u|s = | sin ϕ|s | cos θ|s r(β+1)s . Let Ω = B3 \{0}, where B3 denotes the unit ball in R3 , so that

1/s

|u|s dx Ω

us,Ω = =

limε→0

2π 0

| cos θ| dθ s

π 0

| sin ϕ| sin ϕdϕ s

1 ε

1/s r

(β+1)s+2

dr

1/s (β+1)s+3 = C(s, β) limε→0 (1 − ε ) , where C(s, β) is independent of u and v. Similarly, we have vt,Ω =

1/t |v|t dx Ω

1/t = B(t, β) limε→0 (1 − ε(2β+1)t+3 ) , where B(t, β) is also independent of u and v. 3 and vt,Ω < Obviously, one sees that us,Ω < ∞ if and only if s > − β+1 3 1 ∞ if and only if t > − 2β+1 for β > − 2 . Since n = 3, p = nl = 3 and n 3 3 q = n−l = 32 we ﬁnd that Φ(− 2β+1 ) = − β+1 in (1.5.8). Thus, condition (1.5.8) in Theorem 1.5.4 is the best possible. Also, from Theorem 3.1 in [71] and viewing the example in another way by comparing integrability exponents, we have

ε(β+1)s+3

Thus,

1/s

q/tp ∼ ε(2β+1)t+3 .

1.9 Inequalities with Orlicz norms

47

1 (2β + 1)t + 3 (β + 1)s + 3 = s 2 t and hence s = 3t/(3/2 − t/2) . On the other hand, substituting p = 3 and q = 3/2 into Φ(t) deﬁned in (1.5.8), we have s = Φ(t) =

npt = 3t/(3/2 − t/2) . nq + t(q − p)

This again implies that condition (1.5.8) in Theorem 1.5.4 is the best possible.

1.8.2 Remark Remark. (1) The above example shows that condition (1.5.8) is the best possible in R3 when p = 3 and q = 3/2. There exists an example in R2 which shows that this condition is also best possible for q > 2 and all p, see [71] for details. (2) The above example can be used to show the exactness of exponents in the following result about Lipschitz conditions of conjugate A-harmonic tensors. The result was proved in [71]. If 0 ≤ k, l ≤ 1 satisfy p(k − 1) = q(l − 1), then there exists a constant C such that uplocLipk ,Ω /C ≤ vqlocLipl ,Ω ≤ CuplocLipk ,Ω for all conjugate A-harmonic tensors u and v in Ω. The detailed discussion of this topic is in Chapter 9. (3) The above example can be used to show that the similar results in Ls (μ)averaging domains discussed in Section 1.6 are also sharp.

1.9 Inequalities with Orlicz norms In this section, we introduce the Orlicz norm and discuss various versions of the Hardy–Littlewood inequality with the Orlicz norm. A continuous and increasing function ϕ : [0, ∞] → [0, ∞] with ϕ(0) = 0 and ϕ(∞) = ∞ is called an Orlicz function. The Orlicz Lϕ (Ω) consists space of all measurable functions f on Ω such that Ω ϕ |fλ| dx < ∞ for some λ = λ(f ) > 0. Lϕ (Ω) is equipped with the nonlinear Luxemburg functional

48

1 Hardy–Littlewood inequalities

f ϕ = inf

λ>0:

ϕ Ω

|f | λ

dx ≤ 1 .

A convex Orlicz function ϕ is often called a Young function. If ϕ is a Young function, then · ϕ deﬁnes a norm in Lϕ (Ω), which is called the Luxemburg norm or Orlicz norm. For ϕ(t) = tp logα (e + t), 0 < p < ∞ and α ≥ 0 (note that ϕ is convex for 1 ≤ p < ∞ and any real α with α ≥ 1 − p), we have |f | dx ≤ k p . |f |p logα e + f Lp logα L = f Lp logα L(Ω) = inf k : k Ω Let 0 < p < ∞ and α ≥ 0 be real numbers and E be any subset of Rn . We deﬁne the functional of a measurable function f over E by |f | p1 dx , |f |p logα e + [f ]Lp (log L)α (E) = ||f ||p E where ||f ||p =

E

|f (x)|p dx

1/p

.

1.9.1 Norm comparison theorem It has been proved that the norm f Lp logα L is equivalent to the norm [f ]Lp (log L)α (Ω) if 1 ≤ p < ∞ and α ≥ 0. Following [194], we prove that the norm f Lp logα L is also equivalent to [f ]Lp (log L)α (Ω) for 0 < p < 1 and α ≥ 0. Theorem 1.9.1. For each f ∈ Lp (log L)α (Ω), 0 < p < ∞ and α ≥ 0, we have f p ≤ f Lp logα L(Ω) ≤ [f ]Lp (log L)α (Ω) ≤ Cf Lp logα L(Ω) , α 1/p α is a constant independent of f . where C = 2α/p 1 + ep Proof. Let K = f Lp logα L(Ω) . Then, by the deﬁnition of the Luxemburg norm, we have 1/p |f | dx |f |p logα e + . K= K Ω It is clear that K ≥ f p and

|f | log

K≤

p

Ω

α

|f | e+ f p

1/p dx

= [f ]Lp (log L)α (Ω) ,

1.9 Inequalities with Orlicz norms

49

that is f Lp logα L(Ω) ≤ [f ]Lp (log L)α (Ω) . On the other hand, using K ≥ f p and the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ), s ≥ 0, we obtain |f | α p e + |f | log

f p dx Ω = ≤

α p e+ |f | log Ω

|f | K

|f |p log e +

|f | K

Ω

≤ 2α

Ω

·

K

f p

dx

|f |p logα e +

= 2α K p + 2α f pp logα

+ log

|f | K

K

f p

dx + 2α

K

f p

α dx

Ω

|f |p logα

K

f p

dx

.

Now since the function h(t) = tp logα Kt , 0 < t ≤ K, has its maximum α α value ep K p at t = K/eα/p , it follows that f pp logα

K α α ≤ K p. f p ep

Combination of the last two inequalities gives α α |f | K p, dx ≤ 2α 1 + |f |p logα e + f ep p Ω which is equivalent to [f ]Lp (log L)α (Ω) ≤ Cf Lp logα L(Ω) , α 1/p α . where C = 2α/p 1 + ep

1.9.2 Lp(log L)α-norm inequality Here, we shall prove the following Hardy–Littlewood inequality with Lp (log L)α norm. Theorem 1.9.2. Let u and v be solutions to the conjugate A-harmonic equation (1.2.6) in Ω ⊂ Rn , σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u and v, such that

50

1 Hardy–Littlewood inequalities q/p

||u − uB ||Ls (log L)α (B) ≤ C|B|γ ||v − c||Lt (log L)β (σB)

(1.9.1)

for all balls or cubes B with σB ⊂ Ω and all α with 0 < α < s and β ≥ 0. 1,q (Ω, ∧) with d c = 0 and γ = 1/s + 1/n − (1/t + Here c is any form in Wloc 2 1/n)q/p − α/s . Proof. First, using the H¨ older inequality with 1/s = 1/(s2 /(s − α)) + 2 1/(s /α), we ﬁnd that

|u − uB |s logα e + B

=

B

≤

B

|u−uB | ||u−uB ||s

|u − uB | logα/s e +

|u − uB |

s2 (s−α)

dx

= u − uB s2 /(s−α),B

dx

|u−uB | ||u−uB ||s

(s−α) 2 s

B

B

1s

log

logs e +

s dx

s

e+

1s |u−uB | ||u−uB ||s

|u−uB | ||u−uB ||s

dx

α2

(1.9.2)

s

α/s2 dx

.

Choosing parameter m with 0 < m < t and using the Hardy–Littlewood inequality (Theorem 1.5.2), we have q/p

u − uB s2 /(s−α),B ≤ C1 |B|γ1 v − cm,σB ,

(1.9.3)

where γ1 = (s − α)/s2 + 1/n − (1/m + 1/n)q/p and c is any coclosed form. Applying the H¨ older inequality with 1/m = 1/t + (t − m)/mt, we obtain v − cm,σB β |v − c| log t e + = σB ≤

σB

|v − c|t logβ e +

×

σB

log

−βm/(t−m)

|v−c| ||v−c||t

|v−c| ||v−c||t

e+

log

−β t

e+

|v−c| ||v−c||t

m dx

m1

1/t dx

|v−c| ||v−c||t

(1.9.4)

(t−m)/mt dx

1/t (t−m)/mt |v−c| β t e + dx |v − c| log 1dx ||v−c||t σB σB 1t |v−c| β (t−m)/mt t e + ≤ C2 |B| |v − c| log . ||v−c||t dx σB ≤

Combination of (1.9.2), (1.9.3), and (1.9.4) yields 1s |u−uB | α s e + |u − u | log B ||u−uB ||s dx B q(t−m) β t e+ ≤ C3 |B|γ1 + mpt |v − c| log σB

|v−c| ||v−c||t

q/pt dx

1.9 Inequalities with Orlicz norms

×

B

logs e +

51 |u−uB | ||u−uB ||s

α/s2 dx

(1.9.5)

.

Now since x > log(e + x) if x ≥ e, we have |u−uB | s e + log ||u−uB ||s dx B |u−uB | = {B: |u−uB | <e} logs e + ||u−u dx B ||s ||u−uB ||s |u−uB | + {B: |u−uB | ≥e} logs e + ||u−u dx B ||s ||u−uB ||s s |u−uB | dx ≤ C4 + B ||u−u B ||s = C4 + ||u−u1 B ||s B |u − uB |s dx

(1.9.6)

s

= C5 . Substituting (1.9.6) into (1.9.5), we ﬁnd that

1s |u−uB | dx |u − uB |s logα e + ||u−u || B s q(t−m) ≤ C6 |B|γ1 + mpt |v − c|t logβ e + σB B

|v−c| ||v−c||t

q/pt dx

(1.9.7) .

A simple calculation gives γ1 +

1 1 1 1q α q(t − m) = + − + − . mpt s n n t p s2

(1.9.8)

Substituting (1.9.8) into (1.9.7), we have 1s |u−uB | α s e + |u − u | log B ||u−uB ||s dx B q/pt |v−c| dx ≤ C6 |B|γ σB |v − c|t logβ e + ||v−c|| , t

(1.9.9)

where γ = 1/s + 1/n − (1/t + 1/n)q/p − α/s2 . By the equivalence of the norm f Lp logα L and the functional [f ]Lp (log L)α (Ω) , we ﬁnd that (1.9.9) is equivalent to (1.9.1). Thus, from Theorem 1.9.2, we have the following estimate for the composition of the homotopy operator T deﬁned in (1.5.1) and the diﬀerential operator d: q/p

||T (du)||Ls (log L)α (B) ≤ C|B|γ ||v − c||Lt (log L)β (σB) , provided conditions of Theorem 1.9.2 are satisﬁed.

52

1 Hardy–Littlewood inequalities

1.9.3 Ar (Ω)-weighted case Parallel to Section 1.5.2, weighted inequalities, we prove the following Ar (Ω)weighted Hardy–Littlewood inequality with Ls (log L)α -norm. Theorem 1.9.3. In addition to the conditions of Theorem 1.9.2, assume that w(x) ∈ Ar (Ω) for some r > 1. Then, q/p

||u − uB ||Ls (log L)α (B,w) ≤ C|B|γ ||v − c||Lt (log L)β (σB,wpt/qs )

(1.9.10)

for any ball B, where γ=

1 1 1q α(λ − 1) 1 + − + − s n n t p λs2

and λ > 1 is the constant appearing in reverse H¨ older inequality for w. Proof. By the reverse H¨older inequality, there exist constants λ > 1 and C1 , independent of w, such that w λ,B ≤ C1 |B|(1−λ)/λ w 1,B .

(1.9.11)

For any constants ki > 0, i = 1, 2, 3, there are constants m > 0 and M > 0 such that x x x ≤ log e + ≤ M log e + m log e + k1 k2 k3 for all x > 0. Therefore, we obtain m B |u|t logα e + ≤

≤M

|u| k1

α t e+ |u| log B B

dx |u| k2

|u|t logα e +

1t

dx |u| k3

1t

dx

(1.9.12) 1t

.

By properly selecting constants ki , we will have diﬀerent inequalities that we need. Choose k = λs/(λ − 1). Applying the H¨ older inequality with 1/s = 1/k + (k − s)/ks and using (1.9.11) and (1.9.12), we ﬁnd that

|u − uB |s logα e + B

≤

|u−uB | ||u−uB ||s

αk/s k e+ |u − u | log B B

wdx

1s

|u−uB | ||u−uB ||s

dx

k1

wk/(k−s) dx B

(k−s)/ks

1.9 Inequalities with Orlicz norms

≤

53

|u − uB |k logαk/s e +

B

≤ C2 |B|(1−λ)/λs

|u−uB | ||u−uB ||s

dx

k1

αk/s k e+ |u − u | log B B

wλ dx

B

|u−uB | ||u−uB ||k

1/λs

dx

k1

1/s

w1,B .

(1.9.13) Next, choose m = qst/(qs + pt(r − 1)). Then, 0 < m < t. From Theorem 1.9.2, we have

|u − uB |k log B

≤ C3 |B|γ where

αk s

e+

|u−uB | ||u−uB ||k

|v − c|m logβ σB

dx

e+

k1

|v−c| ||v−c||m

q/pm dx

(1.9.14) ,

γ = 1/k + 1/n − (1/n + 1/m)q/p − α/ks

and the parameter β will be determined later. Using the H¨ older inequality again with 1/m = 1/t + (t − m)/mt and (1.9.12), we obtain σB

= ≤

|v − c|m logβ

|v − c|t logβ σB

/m

t/m

e+ e+

1 w

σB

|v − c|t logβ σB

1/m dx

|v−c| ||v−c||m |v−c| ||v−c||m

wp/qs w−p/qs

wpt/qs dx

m

1/m dx

1/t (1.9.15)

(t−m)/mt

t/m

1/(r−1) σB

dx

×

mpt/qs(t−m)

×

|v−c| ||v−c||m

e+

|v − c| logβ

σB

≤ C3

1 w

e+

|v−c| ||v−c||t

wpt/qs dx

1/t

p(r−1)/qs dx

.

Since w ∈ Ar , it follows that 1/s

w1,B

1/(r−1) 1 w

σB

≤ |σB|r/s

≤ C4 |B|r/s .

1 |σB|

(r−1)/s dx

wdx σB

1 |σB|

r−1 1/s

1/(r−1)

σB

1 w

dx

(1.9.16)

54

1 Hardy–Littlewood inequalities

Combining (1.9.13), (1.9.14), (1.9.15), and (1.9.16), to conclude that B

|u − uB |s logα e +

≤ C5 |B|γ

|v − c|t logβ σB

where γ = γ + Selecting β =

βm t

t/m

wdx

1s

e+

|v−c| ||v−c||m

wpt/qs dx

q/pt

(1.9.17) ,

1 1 1 1q α(λ − 1) 1−λ r + = + − + − . λs s s n n t p λs2

and using (1.9.12), we have

|v − c|t logβ σB

≤ C6

|u−uB | ||u−uB ||s

σB

t/m

e+

|v−c| ||v−c||m

|v − c|t logβ e +

wpt/qs dx

|v−c| ||v−c||t

q/pt

wpt/qs dx

q/pt

(1.9.18) .

Substitution of (1.9.18) into (1.9.17) yields

|u − uB |s logα e + B

≤ C7 |B|γ

|u−uB | ||u−uB ||s

wdx

β t e+ |v − c| log σB

1s

|v−c| ||v−c||t

wpt/qs dx

q/pt

(1.9.19) ,

which is the same as q/p

||u − uB ||Ls (log L)α (B,w) ≤ C|B|γ ||v − c||Lt (log L)β (σB,wpt/qs ) by the equivalence of the norm f Lp logα L and the functional [f ]Lp (log L)α (Ω) .

Theorem 1.9.4. In addition to the conditions of Theorem 1.9.2, assume that w ∈ Ar (Ω) for some r > 1. Then, q/p

||u − uB ||Ls (log L)α (B,wη ) ≤ C|B|γ ||v − c||Lt (log L)β (σB,wηpt/qs )

(1.9.20)

for any ball B, where 0 < η ≤ 1, γ = 1s + n1 − n1 + 1t pq − α(λ−1) λs2 , and λ > 1 is the constant appearing in reverse H¨ older inequality for w.

1.9 Inequalities with Orlicz norms

55

Proof. From [59], we know that if w ∈ Ar (E) and 0 < η ≤ 1, then wη ∈ Ar (E). Therefore, from Theorem 1.9.3, we obtain (1.9.20) immediately.

1.9.4 Global Ls(log L)α-norm inequality Finally, we prove the following global Hardy–Littlewood inequality with Ls (log L)α -norm in δ-John domains.

Theorem 1.9.5. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be solutions to the conjugate A-harmonic equation (1.2.6) in a δ-John domain Ω ⊂ Rn . Let q ≤ p, v − c ∈ Lt (log L)β (Ω, ∧2 ), and s > 0 and t > 0 satisfy 1/s + 1/n − (1/t + 1/n)q/p − α/s2 = 0. Then, there exists a constant C, independent of u and v, such that q/p

||u − uB0 ||Ls (log L)α (Ω) ≤ C||v − c||Lt (log L)β (Ω) ,

(1.9.21)

1,q where 0 < α < s and β > 0 are constants, c is any form in Wloc (Ω, ∧) with d c = 0 and B0 ⊂ Ω is a ﬁxed ball.

Proof. Applying the H¨ older inequality with 1/s = 1/(s2 /(s−α))+1/(s2 /α), we obtain ||u − uB0 ||Ls (log L)α (Ω) = ≤

α s e+ |u − u | log B 0 Ω

Ω

|u − uB0 |s

×

Ω

2

logs e +

/(s−α)

|u−uB0 | ||u−uB0 ||s,Ω

dx

1s

(s−α)/s2 dx

|u−uB0 | ||u−uB0 ||s,Ω

= u − uB0 s2 /(s−α),Ω

(1.9.22)

α/s

2

dx

s e+ log Ω

|u−uB0 | ||u−uB0 ||s,Ω

α/s2 dx

.

Since (s − α)/s2 + 1/n − (1/m + 1/n)q/p = 0, using Theorem 1.5.4, we have q/p

u − uB0 s2 /(s−α),Ω ≤ C1 v − ct,Ω .

(1.9.23)

Combination of (1.9.22) and (1.9.23) yields ||u − uB0 ||Ls (log L)α (Ω) q/p

≤ C1 v − ct,Ω ·

s e+ log Ω

|u−uB0 | ||u−uB0 ||s,Ω

α/s2 dx

.

(1.9.24)

56

1 Hardy–Littlewood inequalities

Similar to (1.9.6), we can prove that |u − uB0 | dx ≤ C2 . logs e + (1.9.25) ||u − uB0 ||s,Ω Ω |v−c| Substituting (1.9.25) into (1.9.24) and using logβ e + ||v−c|| > 1, we ﬁnd t,Ω that ||u − uB0 ||Ls (log L)α (Ω) q/p

≤ C3 v − ct,Ω ≤ C3

β t e+ |v − c| log Ω

|v−c| ||v−c||t,Ω

q/pt dx

q/p

≤ C3 ||v − c||Lt (log L)β (Ω) . Notes to Chapter 1. In this chapter, we have focused our attention on Hardy–Littlewood inequalities on diﬀerential forms satisfying the conjugate A-harmonic equation. However, many diﬀerent versions of the Hardy– Littlewood inequality have been obtained during the recent years. For example, C. Nolder proved Hardy–Littlewood inequalities for solutions of elliptic equations in divergence form in [155] and for quasiregular maps on Carnot groups in [153]. A Carnot group is a connected, simply connected nilpotent Lie group G of topological dimG = N ≥ 2 equipped with a graded Lie algebra G = V1 ⊕ V2 ⊕ · · · ⊕ Vr so that [V1 , Vi ] = Vi+1 for i = 1, 2, . . . , r − 1 and [V1 , Vr ] = 0. This deﬁnes an r-step Carnot group. B. Osikiewiczand and A. Tonge developed an interpolation approach to Hardy–Littlewood inequalities for forms of operators on sequence spaces [158]. See [195–200, 187–190, 18, 148, 67–74] for further results about the Hardy–Littlewood inequality.

Chapter 2

Norm comparison theorems

In the previous chapter, we have discussed various versions of the Hardy– Littlewood inequality for a pair of solutions u and v of the conjugate A-harmonic equation. The purpose of this chapter is to present some norm comparison inequalities for diﬀerential forms satisfying the conjugate A-harmonic equations, which have been recently established in [104]. Since the proofs display a general method to obtain Lp -estimates, we include most of them in this chapter. Also, we always assume that 1 < p < ∞ and p−1 + q −1 = 1 throughout this chapter.

2.1 Introduction In the ﬁrst chapter, we have introduced the following conjugate A-harmonic equation: (2.1.1) A(x, du) = d v. In this chapter, we study a more general type of the conjugate A-harmonic equation: the nonhomogeneous conjugate A-harmonic equation A(x, g + du) = h + d v

(2.1.2)

for diﬀerential forms, where g, h ∈ D (Ω, ∧l ) and A : Ω × ∧l (Rn ) → ∧l (Rn ) satisﬁes the following conditions: |A(x, ξ)| ≤ a|ξ|p−1

and

< A(x, ξ), ξ > ≥ |ξ|p

(2.1.3)

for almost every x ∈ Ω and all ξ ∈ ∧l (Rn ). Here a > 0 is a constant and 1 < p < ∞ is a ﬁxed exponent associated with (2.1.2). Some results related to equation (2.1.2) have been obtained in [105]. In this chapter, we ﬁrst present some local norm inequalities in Section 2.2. Then, we obtain some Ar (Ω)-weighted estimates and the global norm inequalities for solutions of R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 2,

57

58

2 Norm comparison theorems

the nonhomogeneous conjugate A-harmonic equation in Sections 2.3 and 2.4, respectively. Finally, as applications of the results discussed in Sections 2.2, 2.3, and 2.4, we prove the global Sobolev–Poincar´e-type imbedding inequality and derive some global Lp -estimates for the gradient operator ∇ and the homotopy operator T from the Banach space Ls (D, ∧l ) to the Sobolev space W 1,s (D, ∧l−1 ), l = 1, 2, . . . , n. Some of the results presented in this chapter have nice symmetric properties. These results enrich the Lp -theory of diﬀerential forms and can be used to estimate the integrals of diﬀerential forms and to study the integrability of diﬀerential forms, also to explore the properties of the operators ∇ and T .

2.2 The local unweighted estimates We develop some unweighted norm inequalities for solutions of conjugate A-harmonic equations (2.1.1) and (2.1.2). These inequalities have rich symmetric properties and play important role in this chapter.

2.2.1 Basic Lp-inequalities In this section, we present some local norm inequalities which will be extended into the weighted cases and the global cases in next two sections. We ﬁrst discuss the following norm comparison theorem which describes the relationship between the norm of du and the norm of d v. Theorem 2.2.1. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a domain Ω ⊂ Rn . If g ∈ Lp (B, ∧l ) and h ∈ Lq (B, ∧l ), then du ∈ Lp (B, ∧l ) if and only if d v ∈ Lq (B, ∧l ). Moreover, there exist constants C1 , C2 , independent of u and v, such that d vqq,B ≤ C1 (hqq,B + gpp,B + dupp,B ),

(2.2.1)

dupp,B ≤ C2 (hqq,B + gpp,B + d vqq,B )

(2.2.2)

for all balls B with B ⊂ Ω ⊂ Rn . Here p−1 + q −1 = 1. Proof. We only need to prove (2.2.1) and (2.2.2). From equation (2.1.2) and condition (2.1.3), we have |h + d v| = |A(x, g + du)| ≤ a|g + du|p−1 . Hence, we obtain

(2.2.3)

2.2 The local unweighted estimates

59

|d v| = |d v + h − h| ≤ |h| + |d v + h| ≤ |h| + a|g + du|p−1 .

(2.2.4)

Using the elementary inequality |

N

ti | ≤ N s

s−1

i=1

N

|ti |s ,

(2.2.5)

i=1

we ﬁnd that |d v|q ≤ (|h| + a|g + du|p−1 )q ≤ 2q−1 (|h|q + aq |g + du|(p−1)q ) = 2q−1 (|h|q + aq |g + du|p )

(2.2.6)

≤ 2q−1 (|h|q + 2p−1 aq (|g|p + |du|p )) ≤ C1 (|h|q + |g|p + |du|p ). Integrating the above inequality over B, we obtain d vqq,B ≤ C1 (hqq,B + gpp,B + dupp,B ). This completes the proof of inequality (2.2.1). From (2.1.3) and Schwartz inequality, we have |g + du|p ≤< A(x, g + du), g + du > =< h + d v, g + du > ≤ |h + d v| · |g + du|. Therefore, we obtain |g + du|p−1 ≤ |h + d v| ≤ |h| + |d v|.

(2.2.7)

Hence, |g + du|p = |g + du|q(p−1) ≤ (|h| + |d v|)q . Using the elementary inequality (2.2.5) again, we ﬁnd that |du|p = |du + g − g|p ≤ 2p−1 (|du + g|p + |g|p ) ≤ 2p−1 (|g|p + (|h| + |d v|)q ) ≤ 2p−1 (|g|p + 2q−1 (|h|q + |d v|q )) ≤ C2 (|g|p + |h|q + |d v|q ). Integrating inequality (2.2.8) over B, we obtain (2.2.2).

(2.2.8)

60

2 Norm comparison theorems

Note that (2.2.1) and (2.2.2) can be used to estimate the integrals of diﬀerential forms and to study the integrability of diﬀerential forms. From the proof of Theorem 2.2.1, the following corollary is immediate. Corollary 2.2.2. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a domain Ω ⊂ Rn . Then, h + d vqq,B ≤ C1 g + dupp,B , g + dupp,B ≤ C2 h + d vqq,B for all balls B with B ⊂ Ω ⊂ Rn .

2.2.2 Special cases Applying Theorem 2.2.1 with g = 0 and h = 0, we obtain the following corollary immediately. Corollary 2.2.3. Let u and v be a pair of solutions to the conjugate A-harmonic equation (2.1.1) in a domain Ω ⊂ Rn . Then, du ∈ Lp (B, ∧l ) if and only if d v ∈ Lq (B, ∧l ). Moreover, there exist constants C1 , C2 , independent of u and v, such that C1 dupp,B ≤ d vqq,B ≤ C2 dupp,B

(2.2.9)

for all balls B with B ⊂ Ω ⊂ Rn . Theorem 2.2.4. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Then, there exists a constant C, independent of u and v, such that d vqq,B ≤ Cdiam(B)−p u − cpp,σB

(2.2.10)

for all balls B with σB ⊂ Ω ⊂ Rn . Here c is any closed form and σ is a constant with σ > 1. Note that (2.2.10) can be written as d vq,B ≤ Cdiam(B)−p/q u − cp,σB . p/q

(2.2.11)

Proof. Since u and v are a pair of solutions to the conjugate A-harmonic equation A(x, du) = d v, it follows that u is a solution to the A-harmonic equation d A(x, du) = 0.

2.3 The local weighted estimates

61

Hence, the Caccioppoli inequality is applicable to u. Using Theorem 2.4.1 with α = 1 in [180] and Theorem 2.2.1 with g = 0 and h = 0, we obtain d vqq,B ≤ Cdupp,B ≤ Cdiam(B)−p u−cpp,σB . Theorem 2.2.5. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Then, there exists a constant C, independent of u and v, such that u − uB pp,B ≤ Cdiam(B)p d vqq,σB

(2.2.12)

for all balls B with σB ⊂ Ω ⊂ Rn , where σ is a constant with σ > 1. Proof. Similar to the case in Theorem 2.2.4, using Poincar´e inequality (Theorem 2.12 with w(x) = 1 in [201]) to u, and Theorem 2.2.1 with g = 0 and h = 0, we obtain u − uB pp,B ≤ Cdiam(B)p dupp,σB ≤ Cdiam(B)p d vqq,σB . Combining Theorems 2.2.4 and 2.2.5, we have the following norm comparison inequality which has nice symmetric properties. It can be used to estimate the norm of d v in terms of u or u − c. Theorem 2.2.6. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Then, there exist constants C1 , C2 , independent of u and v, such that C1 diam(B)−p u − uB pp,B ≤ d vqq,σ1 B ≤ C2 diam(B)−p u − cpp,σ2 B for all balls B with σ2 B ⊂ Ω ⊂ Rn . Here c is any closed form and p, q, σ1 , σ2 are constants with σ2 > σ1 > 1.

2.3 The local weighted estimates Here we generalize the inequalities established in the previous section to the Ar (Ω)-weighted versions. We begin with Ls -estimates for d v.

2.3.1 Ls-estimates for dv We ﬁrst discuss Ls -estimates for d v in terms of g, h and du that appear in equation (2.1.2). Theorem 2.3.1. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a domain Ω ⊂ Rn . Assume that

62

2 Norm comparison theorems

w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that d vs,B,wα ≤ C|B|αr/s (ht,B,wαt/s + |g|p/q t,B,wαt/s + |du|p/q t,B,wαt/s )

(2.3.1)

for all balls B with B ⊂ Ω ⊂ Rn . Here α is any positive constant with 1 > αr, s = (1 − α)q, and t = s/(1 − αr) = qs/(s − αq(r − 1)). Note that (2.3.1) can be written in the following symmetric form: |B|−1/s d vs,B,wα ≤ C|B|−1/t (ht,B,wαt/s + |g|p/q t,B,wαt/s + |du|p/q t,B,wαt/s ). (2.3.1) Proof. Since s = (1 − α)q < q, using the H¨ older inequality, we have d vs,B,wα = ≤

B

|d v|wα/s

|d v|q dx B

≤ d vq,B

s

1/s dx

1/q

B

B

wdx

α/s

wα/s

qs/(q−s)

(q−s)/qs dx

(2.3.2)

.

N N Applying the elementary inequality | i=1 ti |τ ≤ N τ −1 i=1 |ti |τ and Theorem 2.2.1, we obtain p/q p/q (2.3.3) d vq,B ≤ C1 hq,B + gp,B + dup,B . Since t = qs/(s − αq(r − 1)) > q, using the H¨ older inequality again, we ﬁnd that

q 1/q hq,B = B |h|wα/s w−α/s dx ≤

|h|t wαt/s dx B

= ht,B,wαt/s Now, choose

1/t 1 αqt/s(t−q) B

w

1 1/(r−1) B

w

(t−q)/qt dx

α(r−1)/s dx

.

αpt(r − 1) sp + αpt(r − 1) =p+ , s s so that k > p. Once again using the H¨ older inequality, we have k=

(2.3.4)

2.3 The local weighted estimates

gp,B =

B

≤

63

|g|p wαt/ks w−αt/ks dx

|g|k wαt/s dx B

≤ gk,B,wαt/s

1/p

1/k 1 αpt/s(k−p) B

w

1 1/(r−1) w

B

(k−p)/kp dx

(2.3.5)

(k−p)/kp dx

.

After a simple computation, we ﬁnd that α(r − 1) st αq(r − 1) k−p = · = , kp s ps + αpt(r − 1) ps and hence p/q gp,B

≤

p/q gk,B,wαt/s

α(r−1)/s 1/(r−1) 1 · dx . w B

(2.3.6)

Note that

p/q

gk,B,wαt/s =

|g|k wαt/s dx

B

|g|(sp+αpt(r−1))/s wαt/s dx

=

=

p/kq

B

|g|pt/q wαt/s dx B

ps/(pqs+αpqt(r−1)) (2.3.7)

1/t

= |g|p/q t,B,wαt/s . Combination of (2.3.6) and (2.3.7) yields p/q gp,B

≤ |g|

p/q

t,B,wαt/s

α(r−1)/s 1/(r−1) 1 · dx . w B

(2.3.8)

Using a similar method, we have p/q dup,B

≤ |du|

p/q

t,B,wαt/s

α(r−1)/s 1/(r−1) 1 · dx . w B

(2.3.9)

Combination of (2.3.2) and (2.3.3) gives α/s p/q p/q wdx . d vs,B,wα ≤ C1 hq,B + gp,B + dup,B B

(2.3.10)

64

2 Norm comparison theorems

Substituting (2.3.4), (2.3.8), and (2.3.9) into (2.3.10), we ﬁnd that

d vs,B,wα ≤ C1 ht,B,wαt/s + |g|p/q t,B,wαt/s + |du|p/q t,B,wαt/s

α/s 1 1/(r−1) α(r−1)/s dx . × B wdx B w (2.3.11) Since w(x) ∈ Ar (Ω), it follows that B

wdx

α/s 1 1/(r−1)

= |B|αr/s

w

B

1 |B|

α(r−1)/s dx

α/s wdx B

1 |B|

1 1/(r−1) B

w

α(r−1)/s dx

(2.3.12)

≤ C2 |B|αr/s . Finally, using (2.3.12) and (2.3.11), we obtain d vs,B,wα ≤ C3 |B|αr/s (ht,B,wαt/s + |g|p/q t,B,wαt/s + |du|p/q t,B,wαt/s ).

2.3.2 Ls-estimates for du Similar to the proof of Theorem 2.3.1, we have the following weighted Ls estimate for du. Theorem 2.3.2. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that dus,B,wα ≤ C|B|αr/s (gt,B,wαt/s + |h|q/p t,B,wαt/s + |d v|q/p t,B,wαt/s ) (2.3.13) for all balls B with B ⊂ Ω ⊂ Rn . Here α is any positive constant with 1 > αr, s = (1 − α)p, and t = s/(1 − αr) = ps/(s − αp(r − 1)). It is easy to see that inequality (2.3.13) is equivalent to |B|−1/s dus,B,wα ≤ C|B|−1/t (gt,B,wαt/s + |h|q/p t,B,wαt/s + |d v|q/p t,B,wαt/s ). (2.3.13)

2.3 The local weighted estimates

65

2.3.3 The norm comparison between d and d Theorem 2.3.3. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, for all balls B with B ⊂ Ω ⊂ Rn and any positive constant α with 1 > αr, there exist constants C1 , C2 , independent of u and v, such that (2.3.14) d vqs,B,wα ≤ C1 |B|αqr/s duppt/q,B,wαt/s for s = (1 − α)q, t = s/(1 − αr) = qs/(s − αq(r − 1)); and dups,B,wα ≤ C2 |B|αpr/s d vqqt/p,B,wαt/s

(2.3.15)

for s = (1 − α)p, t = s/(1 − αr) = ps/(s − αp(r − 1)). Proof. Applying Theorem 2.3.1 with g = 0 and h = 0, we obtain d vs,B,wα ≤ C1 |B|αr/s |du|p/q t,B,wαt/s . Note that

p/q

|du|p/q t,B,wαt/s = dupt/q,B,wαt/s .

(2.3.16) (2.3.17)

Combination of (2.3.16) and (2.3.17) yields d vqs,B,wα ≤ C1 |B|αqr/s duppt/q,B,wαt/s . Similarly, using Theorem 2.3.2 with g = 0 and h = 0, we have dups,B,wα ≤ C2 |B|αpr/s d vqqt/p,B,wαt/s . Theorem 2.3.4. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, for all balls B with B ⊂ Ω ⊂ Rn and any positive constant α with 1 > αr, there exist constants C1 , C2 , independent of u and v, such that u − uB s,B,wα ≤ C1 diam(B)|B|αr/s |d v|q/p t,σB,wαt/s

(2.3.18)

for s = (1 − α)p, t = s/(1 − αr) = ps/(s − αp(r − 1)); and d vs,B,wα ≤ C2 diam(B)−p/q |B|αr/s |u − c|p/q t,σB,wαt/s

(2.3.19)

for s = (1 − α)q, t = s/(1 − αr) = qs/(s − αq(r − 1)). Here σ > 1 is a constant. Proof. Applying the H¨ older inequality with p and s = (1 − α)p and Theorem 2.2.5, we ﬁnd that

66

2 Norm comparison theorems

u − uB s,B,wα = ≤

B

|u − uB |wα/s

|u − uB |p dx B

≤ u − uB p,B

s

1/s dx

1/p

B

wα/s

B

wdx

ps/(p−s)

(p−s)/ps dx

α/s

q/p

≤ C1 diam(B)d vq,σB

B

wdx

α/s

. (2.3.20)

Let

αqt(r − 1) sq + αqt(r − 1) =q+ , s s so that k > q. Using the H¨ older inequality again, it follows that k=

d vq,σB

1/q = σB |d v|q wαt/ks w−αt/ks dx

1/k 1 αqt/s(k−q) (k−q)/kq ≤ σB |d v|k wαt/s dx dx σB w ≤ d vk,σB,wαt/s

1 1/(r−1) B

w

(2.3.21)

(k−q)/kq dx

.

Note that α(r − 1) st αp(r − 1) k−q = · = . kq s q + αqt(r − 1) qs Thus, d

q/p vq,σB

≤ d

q/p vk,σB,wαt/s

·

1/(r−1) α(r−1)/s 1 dx . w σB

(2.3.22)

Also, we have q/p

d vk,σB,wαt/s = |d v|q/p t,σB,wαt/s .

(2.3.23)

Combination of (2.3.20), (2.3.22), and (2.3.23) gives u − uB s,B,wα ≤ C1 diam(B)|d v|q/p t,σB,wαt/s ·

α(r−1)/s 1/(r−1) × σB w1 dx .

B

wdx

Now, using the condition w(x) ∈ Ar (Ω), we ﬁnd that

α/s (2.3.24)

2.4 The global estimates

B

wdx

α/s

≤ |σB|αr/s

67

1 1/(r−1) σB

1 |σB|

w

σB

α(r−1)/s dx

α/s wdx

1 |σB|

1 1/(r−1)

σB

w

α(r−1)/s dx

≤ C2 |B|αr/s . (2.3.25) Finally, substituting (2.3.25) into (2.3.24) we obtain u − uB s,B,wα ≤ C3 diam(B)|B|αr/s |d v|q/p t,σB,wαt/s . This completes the proof of (2.3.18). The proof of (2.3.19) is similar to that of (2.3.18). Note. In this section, we have only proved the Ar (Ω)-weighted norm comparison theorems. Using a similar technique, we can obtain the local comparison theorems with other kinds of weights discussed in Section 1.4, including two-weight, etc.

2.4 The global estimates We have discussed the weighted local estimates for d vs,B,wα and dus,B,wα in the previous section. In this section, we prove some global norm comparison theorems using the local results.

2.4.1 Global estimates for dv Using the local weighted estimate developed above, we prove the following global estimate for d v. Theorem 2.4.1. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a bounded domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that d vs,Ω,wα ≤ C(ht,Ω,wαt/s +|g|p/q t,Ω,wαt/s +|du|p/q t,Ω,wαt/s ), (2.4.1) where α is any positive constant with 1 > αr, 1 ≤ s = (1 − α)q, and t = s/(1 − αr) = qs/(s − αq(r − 1)). Proof. Applying Theorem 2.3.1 and the covering lemma (Theorem 1.5.3), we have

68

2 Norm comparison theorems

d vs,Ω,wα = ≤

Ω

|d v|s wα dx

B∈V

≤ C1 ≤ C1

B

B∈V

B∈V

1/s

|d v|s wα dx

|B|

αr s

|Ω|

αr s

1/s p

p

p

p

ht,B,w αt + |g| q t,B,w αt + |du| q t,B,w αt s s s

ht,Ω,w αt + |g| q t,Ω,w αt + |du| q t,Ω,w αt s s s

·N + |du|p/q t,Ω,w αt ≤ C2 ht,Ω,wαt/s + |g|p/q t,Ω,w αt s s

≤ C3 ht,Ω,wαt/s + |g|p/q t,Ω,wαt/s + |du|p/q t,Ω,wαt/s since Ω is bounded. If we put g = 0 and h = 0 in Theorem 2.4.1, we obtain the following global norm comparison result for d v and du. Corollary 2.4.2. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a bounded domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that p/q

d vs,Ω,wα ≤ C|du|p/q t,Ω,wαt/s = Cdupt/q,Ω,wαt/s ,

(2.4.2)

where α is any positive constant with 1 > αr, s = (1 − α)q, and t = s/(1 − αr) = qs/(s − αq(r − 1)).

2.4.2 Global estimates for du Using Theorem 2.3.2 and the covering lemma, we obtain the following global Ls -estimate for du. Theorem 2.4.3. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a bounded domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that dus,Ω,wα ≤ C(gt,Ω,wαt/s +|h|q/p t,Ω,wαt/s +|d v|q/p t,Ω,wαt/s ). (2.4.3) Here α is any positive constant with 1 > αr, s = (1−α)p, and t = s/(1−αr) = ps/(s − αp(r − 1)).

2.4 The global estimates

69

Similarly, if we choose g = 0 and h = 0 in Theorem 2.4.3, we obtain the following global norm estimate for du in terms of d v. Corollary 2.4.4. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a bounded domain Ω ⊂ Rn . Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u and v, such that q/p

dus,Ω,wα ≤ C|d v|q/p t,Ω,wαt/s = d vqt/p,Ω,wαt/s .

(2.4.4)

Here α is any positive constant with 1 > αr, s = (1−α)p, and t = s/(1−αr) = ps/(s − αp(r − 1)).

2.4.3 Global Lp-estimates We denote the space of all l-forms that are Ls -integrable in Ω with respect to the measure μ by Ls (Ω, ∧, μ). Now we prove the following theorem that characterizes the integrability of a pair du and d v. Theorem 2.4.5. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a bounded domain Ω ⊂ Rn . If g ∈ Lp (B, ∧l , μ) and h ∈ Lq (B, ∧l , μ), then du ∈ Lp (B, ∧l , μ) if and only if d v ∈ Lq (B, ∧l , μ), where the measure μ is deﬁned by dμ = w(x)α dx, w(x) ∈ Ar (Ω) for some r > 1. Moreover, there exist constants C1 , C2 , independent of u and v, such that d vqq,Ω,wα ≤ C1 (hqq,Ω,wα + gpp,Ω,wα + dupp,Ω,wα ),

(2.4.5)

dupp,Ω,wα ≤ C2 (hqq,Ω,wα + gpp,Ω,wα + d vqq,Ω,wα )

(2.4.6)

for any real number α with α > 0. Proof. We only need to prove (2.4.5) and (2.4.6). Multiplying (2.2.6) by wα , we have (2.4.7) |d v|q wα ≤ C1 (|h|q wα + |g|p wα + |du|p wα ) since wα > 0. Integrating (2.4.7) over Ω we ﬁnd that (2.4.5) holds. Similarly, from (2.2.8), we obtain |du|p wα ≤ C2 (|g|p wα + |h|q wα + |d v|q wα ).

(2.4.8)

Hence, inequality (2.4.6) follows immediately. Setting g = 0 and h = 0 in Theorem 2.4.5, we have the following global norm comparison theorem that provides a powerful tool for the study of the global integrability of diﬀerential forms du and d v.

70

2 Norm comparison theorems

Corollary 2.4.6. Let u and v be a pair of solutions to the conjugate A-harmonic equation (2.1.1) in a bounded domain Ω ⊂ Rn . Then, du ∈ Lp (Ω, ∧l , μ) if and only if d v ∈ Lq (Ω, ∧l , μ), where the measure μ is deﬁned by dμ = w(x)α dx, w ∈ Ar (Ω) for some r > 1. Moreover, there exist constants C1 , C2 , independent of u and v, such that C1 dupp,Ω,wα ≤ d vqq,Ω,wα ≤ C2 dupp,Ω,wα

(2.4.9)

for any real number α with α > 0. Using Theorem 2.3.4 and the covering lemma, we ﬁnd the following norm comparison theorem for u − c and d v. Theorem 2.4.7. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a bounded domain Ω ⊂ Rn . Assume that w(x) ∈ Ar (Ω) for some r > 1. Then, for any positive constant α with 1 > αr, there exist constants C1 , C2 , independent of u and v, such that u − uB s,Ω,wα ≤ C1 |d v|q/p t,Ω,wαt/s

(2.4.10)

for s = (1 − α)p, t = s/(1 − αr) = ps/(s − αp(r − 1)); and d vs,Ω,wα ≤ C2 |u − c|p/q t,Ω,wαt/s

(2.4.11)

for s = (1 − α)q, t = s/(1 − αr) = qs/(s − αq(r − 1)).

2.4.4 Global Ls-estimates We have obtained the Lp -estimate for du and the Lq -estimate for d v with p−1 + q −1 = 1 in Theorem 2.4.5. However, in applications, we often need Ls -estimates for some s with 1 < s < ∞. Hence, we now state and prove the following Ls -norm comparison theorem. Theorem 2.4.8. Let u and v be a pair of solutions to the nonhomogeneous conjugate A-harmonic equation (2.1.2) in a bounded domain Ω ⊂ Rn and 1 < s < ∞. Assume that w ∈ Ar (Ω) for some r > 1. Then, p−1 d vs,Ω,wα ≤ C1 (hs,Ω,wα + gp−1 s(p−1),Ω,wα + dus(p−1),Ω,wα ), (2.4.12) q−1 dus,Ω,wα ≤ C2 (gs,Ω,wα + hq−1 s(q−1),Ω,wα + d vs(q−1),Ω,wα ), (2.4.13)

where C1 , C2 are constants. Proof. From (2.2.4), we know that |d v|s wα ≤ (|h| + a|g + du|p−1 )s wα

(2.4.14)

2.4 The global estimates

71

for any weight w(x) > 0. Integrating (2.4.14) over Ω and using the Minkowski inequality, we ﬁnd that d vs,Ω,wα ≤

Ω

(|h| + a|g + du|p−1 )s wα dx

1/s

≤ hs,Ω,wα + ag + dup−1 s(p−1),Ω,wα p−1 ≤ C1 (hs,Ω,wα + gp−1 s(p−1),Ω,wα + dus(p−1),Ω,wα ).

This completes the proof of (2.4.12). From (2.2.7), we ﬁnd that |g + du| ≤ |h + d v|1/(p−1) , and hence |du|s wα = |du + g − g|s wα ≤ (|du + g| + |g|)s wα ≤ (|g| + |h + d v|

(2.4.15)

1/(p−1) s

α

) w .

Integrating (2.4.15) over Ω and using (2.2.5) and Minkowski inequality again, we obtain dus,Ω,wα 1/s

s ≤ Ω |g| + (|h| + |d v|)1/(p−1) wα dx ≤ gs,Ω,wα +

Ω

(|h| + |d v|)

s/(p−1)

wα dx

1/s

1/s

s ≤ gs,Ω,wα + C2 Ω (h|1/(p−1) + |d v|1/(p−1) wα dx

1/s

1/s s/(p−1) α s/(p−1) α ≤ gs,Ω,wα + C3 |h| w dx + |d v| w dx Ω Ω 1/(p−1) 1/(p−1) ≤ C4 gs,Ω,wα + hs/(p−1),Ω,wα + d vs/(p−1),Ω,wα q−1 ≤ C4 (gs,Ω,wα + hq−1 s(q−1),Ω,wα + d vs(q−1),Ω,wα )

since 1/(p − 1) = q − 1. Remark. We have extended only few local norm inequalities into the global ones. Using similar methods, we can generalize other local results into the global ones. We can also obtain the global comparison theorems with other weights, including two-weight. Also, we notice that the global comparison norm inequalities contain a parameter α. Hence, by choosing diﬀerent value

72

2 Norm comparison theorems

of α, we have diﬀerent versions of the global norm inequalities. For example, choosing α = p > 1 in (2.4.5) and (2.4.6), we obtain d vqq,Ω,wp ≤ C1 (hqq,Ω,wp + gpp,Ω,wp + dupp,Ω,wp ) and dupp,Ω,wp ≤ C2 (hqq,Ω,wp + gpp,Ω,wp + d vqq,Ω,wp ), respectively.

2.5 Applications In this section, we discuss some applications of the global norm comparison theorems. We prove the global Sobolev–Poincar´e-type inequality, and obtain some versions of imbedding theorems for the homotopy operator T and the gradient operator ∇.

2.5.1 Imbedding theorems for diﬀerential forms From Corollary 4.1 in [99], for any u ∈ D (B, ∧l ) with du ∈ Lp (B, ∧l+1 ), we have (2.5.1) u − uB W 1,p (B,∧l ) ≤ C|B|dup,B . Combining (2.2.9) and (2.5.1), we obtain the following analog of Sobolev– Poincar´e-type imbedding theorem for diﬀerential forms satisfying the conjugate A-harmonic equation. Theorem 2.5.1. Let u and v be a pair of solutions to the conjugate Aharmonic equation (2.1.1) in a domain Ω ⊂ Rn . Then, u − uB pW 1,p (B,∧l ) ≤ Cd vqq,B

(2.5.2)

for all balls B with B ⊂ Ω, where C is a constant. Using (2.5.2) and the covering lemma, we prove the following global Sobolev–Poincar´e-type imbedding theorem. Theorem 2.5.2. Let u ∈ D (Ω, ∧0 ) and v ∈ D (Ω, ∧2 ) be a pair of solutions to the conjugate A-harmonic equation (2.1.1) in a δ-John domain Ω ⊂ Rn . Then, u − uB0 pW 1,p (Ω,∧l ) ≤ Cd vqq,Ω , where C is a constant and B0 ⊂ Ω is a ﬁxed ball.

2.5 Applications

73

Notes to Chapter 2. (i) We should note that all the results established in this chapter are about l-forms, l = 0, 1, . . . , n, and that the real functions in Rn are 0-forms. (ii) It is known that if f (x) = (f 1 , f 2 , . . . , f n ) is K-quasiregular in Rn , then u = f l df 1 ∧ df 2 ∧ · · · ∧ df l−1 , l = 1, 2, . . . , n − 1, and v = f l+1 df l+2 ∧ · · · ∧ df n , l = 1, 2, . . . , n − 1, are solutions of the conjugate A-harmonic equation (2.1.1). Thus, our results, such as Theorems 2.5.1 and 2.5.2, can be used to study the K-quasiregular mapping f .

Chapter 3

Poincar´ e-type inequalities

3.1 Introduction We begin this chapter with the following weak reverse H¨ older inequality, which is due to C. Nolder [71] and will be used repeatedly later. Lemma 3.1.1. Let u be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a domain Ω and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that us,B ≤ C|B|(t−s)/st ut,σB

(3.1.1)

for all balls or cubes B with σB ⊂ Ω for some σ > 1. Thus, for any s, t with 0 < s, t < ∞, the local Ls -norm and the local L -norm are comparable. t

3.2 Inequalities for diﬀerential forms We ﬁrst discuss the Poincar´e inequality for some diﬀerential forms. These forms are not necessarily be the solutions of any version of the A-harmonic equation.

3.2.1 Basic inequalities In 1989, Susan G. Staples [183] proved the following Poincar´e inequality for Sobolev functions in Ls -averaging domains. This inequality has been well studied and used in the development of the averaging domains; see [202]. R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 3,

75

76

3 Poincar´e-type inequalities

Theorem 3.2.1. If D is an Lp -averaging domain, p ≥ n, then there exists a constant C, such that

1 m(D)

1/p

|u − uD | dm

≤ C|D|

p

1/n

D

1/p

1 m(D)

|∇u| dm p

(3.2.1)

D

for each Sobolev function u deﬁned in D. In 1993, the following Poincar´e–Sobolev inequality was proved in [99], which can be used to generalize the theory of Sobolev functions to that of diﬀerential forms. Theorem 3.2.2. Let u ∈ D (Q, ∧l ) and du ∈ Lp (Q, ∧l+1 ). Then, u − uQ is in Lnp/(n−p) (Q, ∧l ) and (n−p)/np

|u − uQ |np/(n−p) dx

≤ Cp (n)

1/p |du|p dx

Q

(3.2.2)

Q

for Q a cube or a ball in Rn , l = 0, 1, . . . , n − 1, and 1 < p < n. From Corollary 4.1 in [99], we have the following version of Poincar´e inequality for diﬀerential forms. Theorem 3.2.3. Let u ∈ D (Q, ∧l ) and du ∈ Lp (Q, ∧l+1 ). Then, u − uQ is in W 1,p (Q, ∧l ) with 1 < p < ∞ and u − uQ p,Q ≤ C(n, p)|Q|1/n dup,Q

(3.2.3)

for Q a cube or a ball in Rn , l = 0, 1, . . . , n − 1.

3.2.2 Weighted inequalities In 1998, S. Ding and P. Shi [121] proved the following diﬀerent versions of the local Poincar´e inequalities for diﬀerential forms. We present these results in Theorems 3.2.4, 3.2.5, 3.2.6, 3.2.7, and 3.2.8. Also, using the local results, they have obtained the global Poincar´e inequalities in Ls (μ)-averaging domains. Theorem 3.2.4. Let u ∈ D (B, ∧l ) and du ∈ Ln (B, ∧l+1 ), l = 0, 1, . . . , n−1. If 1 < s < n and w ∈ An/s (n/s), then there exists a constant C, independent of u and du, such that

1/s |u − uB |s dμ

≤ C|B|1/s

B

for all balls B ⊂ Rn . Here dμ = w(x)dx.

1/n |du|n dμ

B

(3.2.4)

3.2 Inequalities for diﬀerential forms

77

Note that (3.2.4) can be written as u − uB s,B,w ≤ C|B|1/s dun,B,w or

1/s

1 |B|

|u − uB |s w(x)dx

1/n

≤C

|du|n w(x)dx

B

.

B

Theorem 3.2.5. Let u ∈ D (B, ∧l ) and du ∈ Lp (B, ∧l+1 ), l = 0, 1, . . . , n−1, and 1 < p < ∞. If w ∈ Ar (t/(t − s)), where r = t(p − s)/p(t − s) and 1 < s < t < p, then there exists a constant C, independent of u and du, such that 1/s 1/p 1 1 s 1/n p p(t−s)/st |u − uB | wdx ≤ C|B| |du| w dx |B| B |B| B for all balls B ⊂ Rn . Theorem 3.2.6. Let u ∈ D (Ω, ∧l ) and du ∈ Ln (Ω, ∧l+1 ), l = 0, 1, . . . , n−1. If 1 < s < n < βs and w ∈ Ar (1) ∩ An/s (n/s), then there exists a constant C, independent of u and du, such that

1 μ(Ω)

1/s

|u − uB0 |s wdx

1/κ

≤ C|Ω|1/βs

Ω

|du|κ wκ(s−n)/ns dx Ω

for any Ls (μ)-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here κ = βns/(βs − n) and β is the exponent in the reverse H¨ older inequality (see Lemma 1.4.7). Theorem 3.2.7. Let u ∈ D (Ω, ∧l ) and du ∈ Ln (Ω, ∧l+1 ), l = 0, 1, . . . , n−1. If 1 < s < n and w ∈ An/s (n/s) with w(x) ≥ α > 0, then there exists a constant C, independent of u and du, such that

1 μ(Ω)

1/s

|u − uB0 |s dμ Ω

1/n

≤C

|du|n dμ Ω

for any L (μ)-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here dμ = w(x)dx. s

Theorem 3.2.8. Let u ∈ D (Ω, ∧l ) and du ∈ Lp (Ω, ∧l+1 ), l = 0, 1, . . . , n−1. Let 1 < s < t < p < βs and 1/p = 1/n + 1/βs, where β is the exponent in the reverse H¨ older inequality (see Lemma 1.4.7). If w ∈ Ar (1) ∩ Ar (t/(t − s)) with r = t(p − s)/p(t − s), then there exists a constant C, independent of u and du, such that 1/s 1/n 1 s n −n/t |u − uB0 | wdx ≤C |du| w dx μ(Ω) Ω Ω for any Ls (μ)-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω.

78

3 Poincar´e-type inequalities

3.2.3 Inequalities for harmonic forms The above Poincar´e–Sobolev inequalities are about diﬀerential forms. We know that the A-harmonic tensors are diﬀerential forms that satisfy the A-harmonic equation. Then naturally, one would ask whether the Poincar´e– Sobolev inequalities for A-harmonic tensors are sharper than those for differential forms. The answer is “yes”. In [201], S. Ding and C. Nolder proved the following symmetric Poincar´e–Sobolev inequalities for solutions of the nonhomogeneous A-harmonic equation (1.2.10). Theorem 3.2.9. Let u ∈ D (Ω, ∧l ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that σ > 1, 0 < s < ∞ and w ∈ Ar for some r > 1. Then, (3.2.5) u − uB s,B,w ≤ C|B|1/n dus,σB,w for all balls B with σB ⊂ Ω. Here C is a constant independent of u and du. Note that (3.2.5) is equivalent to

1 μ(B)

1/s

|u − uB | dμ s

≤ C|B|

1/n

B

1 μ(B)

1/s

|du| dμ s

.

(3.2.5)

σB

Theorem 3.2.10. Let u ∈ D (Ω, ∧l ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that σ > 1, 0 < α ≤ 1, 1 + α(r − 1) < s < ∞, and w ∈ Ar for some r > 1. Then, u − uB s,B,wα ≤ C|B|1/n dus,σB,wα

(3.2.6)

for all balls B with σB ⊂ Ω. Here C is a constant independent of u and du. Note that (3.2.6) can be written as

1 |B|

1/s

|u − uB | w dx s

B

α

≤ C|B|

1/n

1 |B|

1/s

|du| w dx s

α

. (3.2.6)

σB

We have presented Theorems 3.2.4, 3.2.5, 3.2.6, 3.2.7, 3.2.8, and 3.2.9 without proof. Now, we present the proof of Theorem 3.2.10 as follows. Proof. We assume that 0 < α < 1 and choose t = s/(1 − α) so that 1 < s < t. Using the H¨ older inequality, we have

3.2 Inequalities for diﬀerential forms

B

|u − uB |s wα dx

1/s

79

1/s (|u − uB |wα/s )s dx

1/t

(t−s)/st ≤ B |u − uB |t dx wαt/(t−s) dx B

α/s = u − uB t,B B wdx . =

B

(3.2.7) Next, choose m=

s , α(r − 1) + 1

so that m < s. Since uB is a closed form, by Lemma 3.1.1 and Theorem 3.2.9 with w = 1, we ﬁnd that u − uB t,B ≤ C1 |B|(m−t)/mt u − uB m,σ1 B (3.2.8)

≤ C2 |B|(m−t)/mt |B|1/n dum,σB

for all balls B with σB ⊂ Ω. Here σ > σ1 > 1. Now since 1/m = 1/s + (s − m)/sm, by the H¨older inequality again, we obtain dum,σB = ≤ =

σB

|du|wα/s w−α/s

|du|s wα dx σB |du|s wα dx σB

m

1/m dx

1/s

1 αm/(s−m)

σB

1/s

w

1 1/(r−1)

σB

w

(s−m)/sm dx

(3.2.9)

α(r−1)/s dx

.

From (3.2.7), (3.2.8), and (3.2.9), we have B

|u − uB |s wα dx

1/s

≤ C2 |B|(m−t)/tm |B|1/n ×

|du|s wα dx

σB

wdx B

1/s

α/s

1 1/(r−1) σB

w

α(r−1)/s dx

. (3.2.10)

Since w ∈ Ar (Ω), it follows that

wdx B ≤

α/s

1 1/(r−1)

σB

σB

wdx

w

α(r−1)/s dx

1 1/(r−1) σB

w

1 = |σB|(r−1)+1 |σB| wdx σB

(r−1) α/s dx

80

3 Poincar´e-type inequalities

×

1 1/(r−1)

1 |σB|

w

σB

(r−1) α/s dx (3.2.11)

≤ C3 |σB|α(r−1)/s+α/s ≤ C4 |B|αr/s .

Substituting (3.2.11) into (3.2.10) and using (m−t)/mt = −α/s−α(r −1)/s, we obtain

1/s |u − uB |s wα dx

1/s

≤ C5 |B|1/n

|du|s wα dx

B

σB

which is equivalent to (3.2.6). For the case α = 1, by Lemma 1.4.7, there exist constants β > 1 and C6 > 0, such that w β,B ≤ C6 |B|(1−β)/β w 1,B

(3.2.12)

for any cube or ball B ⊂ Rn . Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by the H¨ older inequality and (3.2.12), we ﬁnd

1/s |u − uB |s wdx

s 1/s = B |u − uB |w1/s dx

B

≤

|u − uB |t dx B

1/t B

w1/s

st/(t−s)

(t−s)/st dx

(3.2.13)

1/s

= u − uB t,B · wβ,B 1/s

≤ C7 u − uB t,B · |B|(1−β)/βs w 1,B . Now, choose m = s/r so that m < s. By the weak reverse H¨older inequality and Theorem 3.2.3, we have u − uB t,B ≤ C8 |B|(m−t)/mt u − uB m,σB ≤ C9 |B|(m−t)/mt |B|1/n dum,σB . Now by the H¨ older inequality again, we obtain dum,σB =

σB

|du|w1/s w−1/s

m

1/m dx

(3.2.14)

3.2 Inequalities for diﬀerential forms

≤ =

σB

|du|s wdx

|du|s wdx σB

81

1/s

1 m/(s−m) σB

1/s

w

1 1/(r−1) σB

w

(s−m)/sm dx (r−1)/s

dx

(3.2.15) .

Combination of (3.2.14) and (3.2.15) yields u − uB t,B ≤ C10 |B|(m−t)/mt |B|1/n

σB

|du|s wdx

1/s σB

1 1/(r−1) w

(r−1)/s dx

. (3.2.16)

Inequality (3.2.11) with α = 1 is the same as 1/s

1/(r−1) (r−1)/s 1 dx ≤ C11 |B|r/s . w σB

wdx B

(3.2.17)

Substituting (3.2.16) into (3.2.13) and using (3.2.17), we conclude that u − uB s,B,w 1/s

≤ C11 |B|(1−β)/βs |B|(m−t)/mt |B|1/n dus,σB,w w1,B

(r−1)/s 1/(r−1) × σB w1 dx

(3.2.18)

≤ C12 |B|(1−β)/βs |B|(m−t)/mt |B|r/s |B|1/n dus,σB,w ≤ C12 |B|1/n dus,σB,w. Thus (3.2.6) holds for α = 1 also. Next, we shall prove the following global weighted Poincar´e–Sobolev inequality in Ls (μ)-averaging domains. Theorem 3.2.11. Let w ∈ Ar with w ≥ τ > 0, r > 1, where τ is a constant. Assume that u ∈ D (Ω, ∧0 ) is an A-harmonic tensor and du ∈ Ls (Ω, ∧1 ), then 1/s 1/s 1 |u − uB0 |s dμ ≤C |du|s dμ (3.2.19) μ(Ω) Ω Ω for any Ls (μ)-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx and C is a constant independent of u. Clearly, we can write (3.2.19) as u − uB0 s,Ω,w ≤ Cμ(Ω)1/s dus,Ω,w .

82

3 Poincar´e-type inequalities

Proof. Since Ω is an Ls (μ)-averaging domain, it follows that μ(Ω) ≤ M . For any ball B ∈ Ω, dμ = w(x)dx ≥ τ dx = τ |B|, μ(B) = B

B

so that

B

C1 1 ≤ . μ(B) |B|

(3.2.20)

Similarly, we have

μ(Ω) =

w(x)dx ≥

dμ = Ω

Ω

τ dx = τ |Ω| Ω

which implies |Ω| ≤ C2 μ(Ω) ≤ C3 .

(3.2.21)

By (3.2.20) and (3.2.21), Theorem 3.2.10 with α = 1, the deﬁnition of Ls (μ)averaging domains, and noticing s ≥ n, we ﬁnd that

1 μ(Ω)

≤

|u − uB0 |s dμ Ω

1 μ(B0 )

Ω

1/s

|u − uB0 |s dμ

≤ C4 sup2B⊂Ω

1 μ(B)

1/s

|u − uB |s dμ B

1/s

1/s 1 1/n s ≤ C5 sup2B⊂Ω |B| μ(B) σB |du| dμ 1/s C1 s ≤ C5 sup2B⊂Ω |B|1/n |B| |du| dμ σB

1/s ≤ C6 sup2B⊂Ω |B|1/n−1/s σB |du|s dμ

1/s ≤ C6 sup2B⊂Ω |Ω|1/n−1/s Ω |du|s dμ ≤ C7

Ω

|du|s dμ

1/s

.

In [203], we have obtained Poincar´e inequalities in which the integral on one side is about Lebesgue measure, but on the other side, the integral is about general measure induced by a weight w(x). We state these results in the following: Theorem 3.2.12. Let u ∈ D (Ω, ∧l ) be an A-harmonic tensor in a domain Ω ⊂ Rn and du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that σ > 1, 1 < s < ∞, and w ∈ Ar for some r > 1. Then, there exists a constant C, independent of u, such that

3.2 Inequalities for diﬀerential forms

1 μ(B)

83

1/s

|u − uB |s dμ

≤ C|B|1/n

B

1 |B|

1/s

|du|s dx

(3.2.22)

σB

for all balls B with σB ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx. Theorem 3.2.13. Let u ∈ D (Ω, ∧l ) be an A-harmonic tensor in a domain Ω ⊂ Rn and du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that σ > 1, 1 < s < ∞, and w ∈ Ar for some r > 1. Then,

1 |B|

1/s

|u − uB | dx

≤ C|B|

s

B

1/n

1/s

1 μ(σB)

|du| dμ s

(3.2.23)

σB

for all balls B with σB ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx and C is a constant independent of u and du.

3.2.4 Global inequalities in averaging domains Deﬁnition 3.2.14. We call a proper subdomain Ω ⊂ Rn an Ls (μ, 0)averaging domain, s ≥ 1, if μ(Ω) < ∞ and there exists a constant C such that 1/s 1/s 1 1 |u − uB0 |s dμ ≤ C sup |u − uB |s dμ μ(B0 ) Ω μ(B) B 2B⊂Ω (3.2.24) for some ball B0 ⊂ Ω and all u ∈ Lsloc (Ω; ∧0 , μ). Here the measure μ is deﬁned by dμ = w(x)dx, where w(x) is a weight and w(x) > 0 a.e., and the supremum is over all balls B with 2B ⊂ Ω. Theorem 3.2.15. Let w ∈ Ar for some r > 1, u ∈ D (Ω, ∧0 ), and du ∈ Ls (Ω, ∧1 ). If s ≥ n, then

1 μ(Ω)

1/s

|u − uB0 |s dμ

≤ C|Ω|1/n

Ω

1 |Ω|

1/s

|du|s dx

(3.2.25)

Ω

for any Ls (μ, 0)-averaging domain Ω with μ(Ω) < ∞ and some ball B0 with 2B0 ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx and C is a constant independent of u and du. Theorem 3.2.16. Let w ∈ Ar with w(x) ≥ α > 0, r > 1, u ∈ D (Ω, ∧0 ), and du ∈ Ls (Ω, ∧1 ). If s ≥ n, then

1 |Ω|

|u − uB0 |s dx Ω

1/s

≤ C|Ω|1/n

1 |Ω|

1/s

|du|s dμ

(3.2.26)

Ω

for any Ls -averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measure μ is deﬁned by dμ = w(x)dx and C is a constant independent of u.

84

3 Poincar´e-type inequalities

3.2.5 Aλ r -weighted inequalities We conclude this section with the following Aλr -weighted Poincar´e inequality for the solutions of the nonhomogeneous A-harmonic equation. Theorem 3.2.17. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that w ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1, σ > 1, and s > αλ(r − 1) + 1, then there exists a constant C, independent of u, such that

1/s

|u − uB |s wα dx

≤ C|B|1/n

B

1/s |du|s wαλ dx

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form. Proof. Choose t = s/(1 − α), so that 1 < s < t. Since 1/s = 1/t + (t − s)/st, by H¨ older’s inequality, we ﬁnd that B

= ≤

|u − uB |s wα dx

1/s

B

(|u − uB |wα/s )s dx

B

|u − uB |t dx

= u − uB t,B

1/s

1/t

B

B

wdx

Next, choose m=

wαt/(t−s) dx

α/s

(t−s)/st

(3.2.27)

.

s , αλ(r − 1) + 1

so that m > 1. Since uB is a closed form, by Lemma 3.1.1 and Theorem 3.2.3, we have u − uB t,B ≤ C1 |B|(m−t)/mt u − uB m,σB ≤ C2 |B|

(m−t)/mt

|B|

1/n

(3.2.28)

dum,σB

for all balls B with σB ⊂ Ω. Now since 1/m = 1/s + (s − m)/sm, by H¨ older’s inequality again, we obtain dum,σB

m 1/m = σB |du|wαλ/s w−αλ/s dx

3.2 Inequalities for diﬀerential forms

≤ =

|du|s wαλ dx

σB

|du|s wαλ dx σB

85

1/s

1 αλm/(s−m) w

σB

1/s

1 1/(r−1) w

σB

(s−m)/sm dx

αλ(r−1)/s dx

(3.2.29) .

From (3.2.27), (3.2.28), and (3.2.29), we have B

|u − uB |s wα dx

1/s

≤ C2 |B|(m−t)/tm |B|1/n ×

1 1/(r−1) w

σB

dx

B

wdx

α/s (3.2.30)

αλ(r−1)/s

|du|s wαλ dx

σB

1/s

.

Since w ∈ Aλr (Ω), it follows that B

wdx

α/s

1 1/(r−1)

≤

w

σB

σB

wdx

= |σB|λ(r−1)+1 ×

1 |σB|

1 1/(r−1) σB

w

1 |σB|

σB

λ(r−1) α/s dx

wdx σB

1 1/(r−1)

αλ(r−1)/s dx

w

(3.2.31)

λ(r−1)

α/s

dx

≤ C3 |σB|αλ(r−1)/s+α/s ≤ C4 |B|αλ(r−1)/s+α/s . Substituting (3.2.31) into (3.2.30) and noting (m−t)/mt = −α/s−αλ(r−1)/s, we obtain

1/s |u − uB | w dx s

B

α

≤ C|B|

1/s |du| w

1/n

s

αλ

dx

.

σB

If we choose λ = 1/α in Theorem 3.2.17, we get the following version of the Aλr -weighted Poincar´e inequality. Corollary 3.2.18. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), 1/α l = 0, 1, . . . , n − 1. Suppose that w ∈ Ar (Ω) for some r > 1 and 0 < α < 1. If s > r, then there exists a constant C, independent of u, such that

86

3 Poincar´e-type inequalities

1/s |u − uB |s wα dx

1/s

≤ C|B|1/n

B

|du|s wdx σB

for all balls B with σB ⊂ Ω. Here uB is a closed form. Selecting α = 1/s in Theorem 3.2.17, we obtain the following Aλr (Ω)weighted Poincar´e inequality. Corollary 3.2.19. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that w ∈ Aλr (Ω) for some r > 1 and λ > 0. If σ > 1 and s > λ(r − 1)/s + 1, then there exists a constant C, independent of u, such that

1/s

|u − uB | w s

B

1/s

dx

≤ C|B|

1/s |du| w

1/n

s

λ/s

dx

(3.2.32)

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form. It is interesting to note that (3.2.32) reduces to the following Ar (Ω)weighted inequality when λ = 1. Corollary 3.2.20. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that w ∈ Ar (Ω) for some r > 1. If σ > 1 and s > λ(r − 1)/s + 1, then there exists a constant C, independent of u, such that 1/s 1/s s 1/s 1/n s 1/s |u − uB | w dx ≤ C|B| |du| w dx B

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form.

3.3 Inequalities for Green’s operator Let ∧l Ω be the lth exterior power of the cotangent bundle and C ∞ (∧l Ω) be the space of smooth l-forms in a domain Ω. We use D (Ω, ∧l ) to denote the space of all diﬀerential l-forms on Ω and Lp (∧l Ω) to denote the l-forms ω(x) = ωI (x)dxI = ωi1 i2 ···il (x)dxi1 ∧ dxi2 ∧ · · · ∧ dxil I

on Ω satisfying Ω |ωI |p < ∞ for all ordered l-tuples I. The measure on the reference manifold is induced by the volume form *1. Integrals are deﬁned as usual using a partition of unity subordinate to atlas. We write

3.3 Inequalities for Green’s operator

87

1/s

us,Ω =

|u|s

.

Ω

The Laplace–Beltrami operator Δ is deﬁned by Δ = dd + d d. We say that u ∈ L1loc (∧l Ω) has a generalized gradient if, for each coordinate system, the pullbacks of the coordinate function of u have generalized gradient in the familiar sense, see [204]. We write (3.3.1) W(∧l Ω) = u ∈ L1loc (∧l Ω) : u has generalized gradient . As usual, the harmonic l-ﬁelds are deﬁned by H(∧l Ω) = u ∈ W(∧l Ω) : du = d u = 0, u ∈ Lp for some 1 = 0 for all h ∈ H . We deﬁne Green’s operator G : C ∞ (∧l Ω) → H⊥ ∩ C ∞ (∧l Ω) by setting G(u) equal to the unique element of H⊥ ∩ C ∞ (∧l Ω) satisfying Poisson’s equation ΔG(ω) = ω − H(ω), where H is the harmonic projection or the harmonic part of the Hodge decomposition ω = dα + d∗ β + H. We also have for 1 < s < ∞ dαs,Ω + d∗ βs,Ω + Hs,Ω ≤ Cp (Ω)ωs,Ω .

(3.3.2)

We know that G is a bounded self-adjoint linear operator. It has been proved in [21] that for 1 < p < ∞ a solution exists ω ∈ Lp (∧l Ω). We will need the following lemma about Ls -estimates for Green’s operator which appeared in [21]. The proof of this lemma will be given in Chapter 7, where we will study various operators systematically. Lemma 3.3.1. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n. For 1 < s < ∞, there exists a constant C, independent of u, such that dd G(u)s,Ω + d dG(u)s,Ω + dG(u)s,Ω

88

3 Poincar´e-type inequalities

+d G(u)s,Ω + G(u)s,Ω ≤ Cus,Ω .

(3.3.3)

3.3.1 Basic estimates for operators In the following results we estimate the composition of the Laplace–Beltrami operator and Green’s operator. Theorem 3.3.2. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, and 1 < s < ∞. Then, there exists a constant C, independent of u, such that Δ(G(u))s,Ω ≤ Cus,Ω . Proof. From the deﬁnition of the Laplace–Beltrami operator Δ, (3.3.3), and Minkowski’s inequality, we have Δ(G(u))s,Ω = u − H(u)s,Ω ≤ C||u||s,Ω . Theorem 3.3.3. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n. Assume that 1 < s < ∞. Then, there exists a constant C, independent of u, such that G(Δu)s,Ω ≤ Cus,Ω . Proof. We know that Green’s operator commutes with d and d (see [23]), that is, for any diﬀerential form u ∈ C ∞ (∧l Ω), we have dG(u) = Gd(u), d G(u) = Gd (u).

(3.3.4)

Hence G(Δu)s,Ω = Δ(G(u))s,Ω . Using Minkowski’s inequality and combining Theorems 3.3.2 and 3.3.3, we obtain the following corollary immediately. Corollary 3.3.4. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n. For 1 < s < ∞, there exists a constant C, independent of u, such that (GΔ + ΔG)us,Ω ≤ Cus,Ω .

(3.3.5)

Theorem 3.3.5. Let u ∈ C ∞ (Ω, ∧l ), l = 1, 2, . . . , n. If 1 < s < ∞, then there exists a constant C, independent of u, such that (G(u))D s,D ≤ C|D|us,D for any convex and bounded D with D ⊂ Ω.

(3.3.6)

3.3 Inequalities for Green’s operator

89

Proof. First, we assume that 1 ≤ l ≤ n. We recall the deﬁnition of the l-form ωD ∈ D (D, ∧l ), ωD = |D|−1 ω(y)dy, l = 0, and ωD = d(T ω), l = 1, 2, . . . , n D

for all ω ∈ Ls (D, ∧l ), and the estimate T uW 1,s (D) ≤ C1 |D|us,D .

(3.3.7)

We ﬁnd that, using (3.3.3), (G(u))D s,D = d(T (G(u)))s,D ≤ ∇T (G(u))s,D (3.3.8)

≤ C|D|G(u)s,D ≤ C|D|us,D .

Next, if l = 0, using (3.3.3) and H¨ older’s inequality with 1 = 1/s + 1/q, we obtain

1/s (G(u))D s,D = D |(G(u))D |s dx s 1/s 1 = D |D| G(u(y))dy dx D ≤ =

1 |D|

|G(u(y))|dy D

1 1/s |D| |D| D

≤ |D|1/s−1

D

s

1/s 1dx D

|G(u(y))|dy |G(u(y))|s dy

(3.3.9)

1/s D

1q dy

1/q

= G(u)s,D ≤ C6 us,D .

Corollary 3.3.6. Let u ∈ C ∞ (Ω, ∧l ), l = 1, 2, . . . , n. Assume that 1 < s < ∞. Then, for any convex and bounded D with D ⊂ Ω, there exists a constant C, independent of u, such that G(u) − (G(u))D s,D ≤ CG(u) − cs,D

(3.3.10)

for any closed form c, and G(u) − (G(u))D s,D ≤ Cus,D .

(3.3.11)

90

3 Poincar´e-type inequalities

Proof. We know that cD = c if c is a closed form. Hence, we have G(u) − (G(u))D s,D ≤ (G(u) − c) − ((G(u))D − cD )s,D ≤ G(u) − cs,D + (G(u) − c)D s,D

(3.3.12)

≤ G(u) − cs,D + C1 G(u) − cs,D ≤ C2 G(u) − cs,D . Thus, (3.3.10) holds. From (3.3.6) and (3.3.3), we ﬁnd that G(u) − (G(u))D s,D ≤ G(u)s,D + (G(u))D s,D ≤ C3 us,D + C4 us,D

(3.3.13)

≤ C5 us,D .

3.3.2 Weighted inequality for Green’s operator We have made necessary preparation in the previous section to prove the following Poincar´e-type inequality for Green’s operator. Theorem 3.3.7. Let u ∈ C ∞ (Ω, ∧l ), l = 1, 2, . . . , n. Assume that 1 < s < ∞. Then, there exists a constant C, independent of u, such that G(u) − (G(u))Q W 1,s (Q) ≤ Cdus,Q

(3.3.14)

for all cubes Q with Q ⊂ Ω. Proof. For any diﬀerential form u, we have decomposition u = d(T u) + T (du).

(3.3.15)

Noticing that uQ = d(T u) and replacing u by G(u) in (3.3.15), we obtain G(u) − (G(u))Q W 1,s (Q) = T d(G(u))W 1,s (Q) ≤ Cμ(Q)d(G(u))s,Q = CG(d(u))s,Q ≤ Cdus,Q .

3.3 Inequalities for Green’s operator

91

A weighted result follows from the following general inequality. Theorem 3.3.8. Suppose that f, g ≥ 0 a.e. in Ω, β > 1, p > 0, f ∈ WRH pβ (Ω), w ∈ Ar (Ω)∩RHβ (Ω), and β−1

1 |Q|

f

p r

pr

≤ C1

σQ

1 |Q|

g

p r

pr

σ2 Q

for all cubes Q with σ 2 Q ⊂ Ω. Then, there exists a constant C2 , depending only on p, β, n, r, σ, and C1 such that

1 |Q|

p1

p

f w

≤ C2

Q

1 |Q|

p1

p

g w σ2 Q

for all cubes Q with σ 2 Q ⊂ Ω. Proof. Since f ∈ W RH pβ (Ω), there exist constants C1 and 0 < γ0 < β−1 such that

1 |Q|

f

pβ β−1

β−1 pβ

≤ C1

dx

Q

1 |Q|

pβ β−1

1/γ0

γ0

f dx

.

σQ

Thus, from (1.4.1) , we obtain

1 |Q|

f

pβ β−1

β−1 pβ

≤ C3

Q

1 |Q|

f

p r

pr .

(3.3.16)

σQ

We have using (3.3.16), Holder’s inequality, and the assumptions that

1 |Q|

≤ ≤

f pw Q

1 |Q| 1 |Q|

≤ C1 ≤

1 |Q|

≤ C2

p1

Q

w

w Q

1 |Q|

1 βp

p1

w Q

β

σ2 Q

w

1 |Q|

1 |Q|

p1

p1

gp w σ2 Q

p1

Q

f p

fr σQ

1 |Q|

1 |Q|

.

pβ β−1

β−1 pβ

pr

p

σ2 Q

σ2 Q

gr

w

pr

1 1−r

r−1 p

1 |Q|

σ2 Q

gp w

p1

92

3 Poincar´e-type inequalities

Corollary 3.3.9. Suppose that p > r, w ∈ Ar (Ω)∩RHβ , and |T u|, |∇T u| ∈ WRH pβ (Ω). Then β−1

G(u) − (G(u))Q W 1,p (Q),w ≤ Cdup,Q,w . Proof. Since p > r, (3.3.14) holds with s = pr . The result follows then from Theorem 3.3.8.

3.3.3 Global inequality for Green’s operator In [103], Y. Wang and C. Wu proved the following global-weighted Poincar´etype inequality for Green’s operator that is applied to the solutions of the nonhomogeneous A-harmonic equation (1.2.10). We will use the following result from [119]. Theorem 3.3.10. Suppose that Ω is a John domain, s > 0, σ > 1, w ∈ Ar (Ω), and f is a measurable function. If there exists a constant C1 and constants bQ such that f − fQ s,Q,w ≤ C1 bQ for all cubes Q with σQ ⊂ Ω, then for some cube Q0 there exists a constant C2 , independent of f , such that f − fQ0 s,Ω,w ≤ C2 ΣQ bQ . Combining this result with Theorem 3.3.8, we obtain the following result.

Theorem 3.3.11. If u ∈ D (Ω, ∧0 ), w ∈ Ar (Ω) and Ω is a John domain, then for some cube Q0 there exists a constant C, independent of u, such that G(u) − (G(u))Q0 p,Ω,w ≤ C∇up,Ω,w .

3.4 Inequalities with Orlicz norms In this section, we will develop the local and global Poincar´e inequalities with Orlicz norms. We will assume that u is a solution of the nonhomogeneous A-harmonic equation (3.4.1) d A(x, dω) = B(x, dω)

3.4 Inequalities with Orlicz norms

93

which has been introduced in Section 1.2. We will need the following deﬁnition of Lϕ (μ)-domains. Deﬁnition 3.4.1. Let ϕ be a Young function on [0, ∞) with ϕ(0) = 0. We call a proper subdomain Ω ⊂Rn an Lϕ (μ)-domain, if μ(Ω) < ∞ and there exists a constant C such that ϕ (σ|u − uΩ |) dμ ≤ C sup ϕ (στ |u − uB |) dμ Ω

B⊂Ω

B

for all u such that ϕ(|u|) ∈ L1loc (Ω; μ), where the measure μ is deﬁned by dμ = w(x)dx, w(x) is a weight, and τ, σ are constants with 0 < τ ≤ 1, 0 < σ ≤ 1, and the supremum is over all balls B ⊂ Ω. One of the main results of this section is the following global-weighted Poincar´e inequality for a solution of the nonhomogeneous A-harmonic equation in an Lϕ (μ)-domain Ω. Theorem 3.4.2. Assume that Ω ⊂Rn is an Lϕ (μ)-domain with ϕ(t) = tp logα (e + t/k), where k = u − uB0 p,Ω , 1 1. Then, there is a constant C, independent of u, such that u − uΩ Lp (log L)α (Ω,w) ≤ C|Ω|1/n duLp (log L)α (Ω,w) for any constant α > 0.

3.4.1 Local inequality To prove Theorem 3.4.2, we need the following local Poincar´e inequalities, Theorems 3.4.3 and 3.4.4, with Orlicz norms. Theorem 3.4.3. Let u ∈ D (Ω, ∧l ) be a solution of the nonhomogeneous A-harmonic equation in a domain Ω ⊂Rn and du ∈ Lp (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that 1 0 is any constant, ρ > 1 and d0 > 0 are some constants. Proof. Let B ⊂ Ω be a ball with diam(B) ≥ d0 > 0. Choose ε > 0 small enough and a constant C1 such that

94

3 Poincar´e-type inequalities

|B|−ε/p ≤ C1 . 2

(3.4.2)

From Lemma 3.1.1, we ﬁnd that u − uB p+ε,B ≤ C2 |B|(p−(p+ε))/p(p+ε) u − uB p,σ1 B

(3.4.3)

for some σ1 > 1. We assume that |u − uB | ≥ 1. u − uB p,B Then, for above ε > 0, there exists C3 > 0 such that ε |u − uB | |u − uB | ≤ C3 . logα e + u − uB p,B ||u − uB ||p,σ1 B

(3.4.4)

Otherwise, setting B1 = x ∈ B :

|u−uB |

u−uB p,B

B2 = x ∈ B :

|u−uB |

u−uB p,B

! ≥1 , ! <1

and using the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ), where s > 0 is any constant, we ﬁnd u − uB Lp (log L)α (B) 1/p |u−uB | dx = B |u − uB |p logα e + u−u B p,B |u−uB | dx = B1 |u − uB |p logα e + u−u B p,B +

α p e+ |u − u | log B B2

≤ 21/p

+21/p

|u−uB |

u−uB p,B

α p e+ |u − u | log B B1

dx

|u−uB |

u−uB p,B

α p e+ |u − u | log B B2

p1

(3.4.5)

|u−uB |

u−uB p,B

dx

p1

dx

p1

.

Now, we shall estimate the ﬁrst term of the right side. Since |u − uB | ≥1 u − uB p,B on B1 , for ε > 0 in (3.4.2), there exists C4 > 0 such that ε |u − uB | |u − uB | α ≤ C4 . log e + u − uB p,B u − uB p,σ1 B

(3.4.6)

3.4 Inequalities with Orlicz norms

95

Combining (3.4.3), (3.4.4), and (3.4.6), we obtain

p α e+ |u − u | log B B1

≤ C5

1

u−uB εp,σ

≤ C5 =

C5 ε/p

u−uB p,σ

≤

C6 ε/p

u−uB p,σ

dx

dx

|u − uB |p+ε dx

B

1B

|u − uB |

p+ε

B

|u − uB |

1/p

1B

1/p dx

p+ε

B1

1B

1/p

1

u−uB εp,σ

|u−uB | ||u−uB ||p,B

(3.4.7)

1 (p+ε)/p

p+ε

|B|(p−(p+ε))/p(p+ε) u − uB p,σ1 B

(p+ε)/p

1B

≤ C7 u − uB p,σ1 B , where σ1 > 1 is a constant. For the second term of (3.4.5), since |u − uB | α ≤ M1 logα (e + 1) ≤ M2 , x ∈ B2 , log e + ||u − uB ||p,B we obtain the similar estimate 1/p |u − uB | α p dx |u − uB | log e + ≤ C8 u − uB p,σ2 B , ||u − uB ||p,B B2 (3.4.8) where σ2 > 1 is a constant. From (3.4.5), (3.4.7), and (3.4.8), we have u − uB Lp (log L)α (B) ≤ C9 u − uB p,σ3 B ,

(3.4.9)

where σ3 = max{σ1 , σ2 }. Applying Theorem 3.2.9 with w = 1, we obtain u − uB p,σ3 B ≤ C10 |B|1/n dup,σ4 B for some σ4 > σ3 . Note that logα e +

|u − uB | ||u − uB ||p,σ2 B

≥ 1,

α > 0.

Now the combination of the last three inequalities yields u − uB Lp (log L)α (B) ≤ C11 |B|1/n dup,σ4 B ≤ C12 |B|1/n duLp (log L)α (σ4 B) .

(3.4.10)

96

3 Poincar´e-type inequalities

3.4.2 Weighted inequalities It is easy to see that for any constant k, there exist constants m > 0 and M > 0, such that t ≤ M log(e + t), t > 0. (3.4.11) m log(e + t) ≤ log e + k From the weak reverse H¨older inequality (Lemma 3.1.1), we know that the norms us,B and ut,B are comparable when 0 < d1 ≤ diam(B) ≤ d2 < ∞. Hence, we may assume that 0 < m1 ≤ us,B ≤ M1 < ∞ and 0 < m2 ≤ ut,B ≤ M2 < ∞ for some constants mi and Mi , i = 1, 2. Thus, we have |u| ≤ C2 log (e + |u|) (3.4.12) C1 log (e + |u|) ≤ log e + us,B and

C3 log(e + |u|) ≤ log e +

|u| ut,B

≤ C4 log (e + |u|)

(3.4.13)

for any s > 0 and t > 0, where Ci are constants, i = 1, 2, 3, 4. Using (3.4.12) and (3.4.13), we obtain C5

α s e+ |u| log B

|u| ||u||t,B

1/s dx

≤ uLs (log L)α (B) ≤ C6

(3.4.14)

B

|u|s logα e +

|u| ||u||t,B

1/s dx

and

C7 uLt (log L)α (B) ≤

|u| log t

B

α

|u| e+ ||u||s,B

1t dx

≤ C8 uLt (log L)α (B)

(3.4.15) for any ball B and any s > 0, t > 0, and α > 0. Consequently, we ﬁnd that uLs (log L)α (B) < ∞ if and only if

|u| log s

B

α

|u| e+ ||u||t,B

1/s dx

< ∞.

Theorem 3.4.4. Let u ∈ D (Ω, ∧l ) be a solution of the nonhomogeneous A-harmonic equation in a domain Ω ⊂Rn and du ∈ Lp (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Assume that 1 1.

3.4 Inequalities with Orlicz norms

97

Then, there is a constant C, independent of u, such that u − uB Lp (log L)α (B,w) ≤ C|B|1/n duLp (log L)α (σB,w) for all balls B with σB ⊂ Ω and diam(B) ≥ d0 . Here α > 0 is any constant, σ > 1 and d0 > 0 are some constants. Proof. In view of Lemma 1.4.7, there exist constants k > 1 and C0 > 0, such that (3.4.16) w k,B ≤ C0 |B|(1−k)/k w 1,B . Choose s = kp/(k − 1), so that p < s. Using the H¨ older inequality with 1/p = 1/s + (s − p)/sp, (3.4.16), and (3.4.14), we obtain u − uB Lp (log L)α (B,w) α = B |u − uB | log p e + ≤

|u − uB |s log B

≤ C1

B

αs p

|u−uB | ||u−uB ||p,B

e+

|u − uB |s log

αs p

1/p

≤ C2 |B|(1−k)/kp w 1,B

|u−uB | ||u−uB ||p,B

e+

w1/p dx

|u−uB | ||u−uB ||s,B

p

1/p dx

1/s B

|u − uB |s log B

dx αs p

ws/(s−p) dx

1/s B

e+

wk dx

(s−p)/ps

1/k 1/p

|u−uB | ||u−uB ||s,B

1/s dx

.

(3.4.17) Applying Theorem 3.4.3, we ﬁnd that B

|u − uB |s log

αs p

e+

|u−uB | ||u−uB ||s,B

1/s dx (3.4.18)

≤ C3 |B|1/n duLs (log L)αs/p (σ1 B) , where σ1 > 1 is some constant. Let t = p/r. From Lemma 2.6 in [205] with β = α/r, we have

|du|s log σ1 B

αs p

e+

|du| ||du||s,B

1/s dx (3.4.19)

≤ C4 |B|(t−s)/st duLt (log L)β (σ2 B) for some σ2 > σ1 . Using the H¨ older inequality again with 1/t = 1/p + (p − t)/pt, we obtain duLt (log L)β (σ2 B) =

β t e+ |du| log σ2 B

|du| ||du||t,σ2 B

1/t dx

98

3 Poincar´e-type inequalities

≤

σ2 B

|du| logβ/t e +

σ2 B

|du|p logβp/t e +

=

≤ duLp (log L)α (σ2 B,w)

|du| ||du||t,σ2 B |du| ||du||t,σ2 B

σ2 B

w1/p w−1/p

wdx

1 1 r−1

w

1/t

t dx

1/p

t 1 p−t

σ2 B

w

(p−t)/pt dx

(r−1)/p dx

. (3.4.20)

Combining (3.4.17), (3.4.18), (3.4.19), and (3.4.20), we have u − uB Lp (log L)α (B,w) ≤ C5 |B| n − p duLp (log L)α (σ2 B,w) 1

r

wdx B

1 1 r−1

σ2 B w

(r−1) 1/p dx

1/p 1 r ≤ C5 |B| n − p duLp (log L)α (σ2 B,w) w1,σ2 B · 1/w1/(r−1),σ2 B . (3.4.21) Now since w ∈ Ar (Ω), it follows that

1/p w1,σ2 B · 1/w1/(r−1),σ2 B r−1 1/p 1/(r−1) ≤ wdx (1/w) dx σ2 B σ2 B =

1/(r−1) r−1 1/p |σ2 B|r |σ21B| σ2 B wdx |σ21B| σ2 B w1 dx

≤ C6 |B|r/p . (3.4.22) Substituting (3.4.22) into (3.4.21), we ﬁnd u − uB Lp (log L)α (B,w) ≤ C7 |B|1/n duLp (log L)α (σ2 B,w) .

(3.4.23)

3.4.3 The proof of the global inequality Now, we are ready to prove our global inequality given in Theorem 3.4.2. Proof of Theorem 3.4.2. For any constants ki > 0, i = 1, 2, 3, there are constants C1 > 0 and C2 > 0 such that t t t C1 log e + ≤ log e + ≤ C2 log e + (3.4.24) k1 k2 k3 for any t > 0. Therefore, we have

3.4 Inequalities with Orlicz norms

C1

≤

B

99

|u|t logα e +

|u| k1

|u|t logα e + B

≤ C2

B

dx

|u| k2

|u|t logα e +

1t

dx |u| k3

1t

(3.4.25)

dx

1 t

.

By properly selecting constants ki in (3.4.25), we obtain inequalities that we will need. Using the deﬁnition of Lϕ (μ)-domains with τ = 1, σ = 1, and ϕ(t) = tp logα (e + t/k), where k = ||u − uB0 ||p,Ω , Theorem 3.4.4, and (3.4.25), we obtain u − uΩ pLp (log L)α (Ω,w) = Ω |u − uΩ |p logα e + ≤ C1 supB⊂Ω ≤ C2 supB⊂Ω

|u−uΩ | ||u−uB0 ||p,Ω

α p e+ |u − u | log B B

B

|u − uB |p logα e +

wdx |u−uB | ||u−uB0 ||p,Ω |u−uB | ||u−uB ||p,B

wdx

wdx

≤ C3 supB⊂Ω |B|p/n dupLp (log L)α (σB,w) ≤ C3 supB⊂Ω |Ω|p/n dupLp (log L)α (Ω,w) ≤ C3 |Ω|p/n dupLp (log L)α (Ω,w) which is equivalent to u − uΩ Lp (log L)α (Ω,w) ≤ C|Ω|1/n duLp (log L)α (Ω,w) . We can prove that the global inequality also holds in δ-John domains. Speciﬁcally, we have the following result. Theorem 3.4.5. Let u ∈ D (Ω, ∧0 ) be a solution of the nonhomogeneous A-harmonic equation in a δ-John domain Ω ⊂Rn and du ∈ Lp (Ω, ∧1 ). Assume that 1 1. Then, there is a constant C, independent of u, such that u − uΩ Lp (log L)α (Ω,w) ≤ C|Ω|1/n duLp (log L)α (Ω,w) for any constant α > 0. Note. In this section, we have only included the global inequalities with Orlicz norms in two kinds of domains, Lϕ (μ)-domains and δ-John domains.

100

3 Poincar´e-type inequalities

However, the local results can be extended in some other types of domains also, such as uniform domains and Ls -averaging domains.

3.5 Two-weight inequalities In recent years, several versions of the two-weight Poincar´e inequalities have been developed, see [206, 82, 207], for example. Here we state and prove the results of Y. Wang [206].

3.5.1 Statements of two-weight inequalities Theorem 3.5.1. Let u ∈ D (B, ∧l ) and du ∈ Lt (B, ∧l+1 ), l = 0, 1, . . . , n−1. s/t Then, there exists a constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ At/s , where 1 < s < n, t = sβ, and r > 1, we have

1 |B|

1/s

|u − uB | w1 dx

1/t

≤C

s

B

|du| w2 dx t

(3.5.1)

B

for all balls B ⊂ Rn . Here C is a constant independent of u and du. Theorem 3.5.2. Let u ∈ D (B, ∧l ) and du ∈ Ln (B, ∧l+1 ), l = 0, 1, . . . , n−1. If 1 < s < n and (w1 , w2 ) ∈ A1n/s , then there exists a constant C, independent of u and du, such that

1 |B|

1/s

s/n

|u − uB |s w1 dx B

1/n

≤C

|du|n w2 dx

(3.5.2)

B

for any ball or cube B ⊂ Rn . We remark that the exponents t and n on the right-hand sides of (3.5.1) and (3.5.2) can be improved. In fact, the following result is with the sharper right-hand side. Theorem 3.5.3. Let u ∈ D (B, ∧l ) and du ∈ Lt (B, ∧l+1 ), l = 0, 1, . . . , n−1. Then, there exists a constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ αs/t Ar , where 1 < s < n, t = s + αs(r − 1), and r > 1, it follows that

1 |B|β

|u − uB |s w1 dx B

1/s

≤ C|B|α(r−1)/t

1/t |du|t w2α dx

(3.5.3)

B

for all balls B ⊂ Rn and any constant α > 0. Here C is a constant independent of u and du. Clearly, in this result if α → 0, then t → s.

3.5 Two-weight inequalities

101

Theorem 3.5.4. Let u ∈ D (Ω, ∧0 ) and du ∈ Lt (Ω, ∧1 ). Then, there exists s/t a constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ At/s , where 1 < s < n, t = sβ, r > 1, and w1 > w2 ≥ η > 0, we have

1 μ1 (Ω)

1/s

|u − uB0 | dμ1

1/t

≤C

s

|du| dμ2 t

Ω

(3.5.4)

Ω

for any Ls (μ1 )-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measures μ1 and μ2 are deﬁned by dμ1 = w1 (x)dx, dμ2 = w2 (x)dx, and C is a constant independent of u and du.

3.5.2 Proofs of the main theorems Proof of Theorem 3.5.1. Since w1 ∈ A1r , r > 1, by Lemma 1.4.7, there exist constants β > 1 and C1 > 0, such that w1 β,B ≤ C1 |B|(1−β)/β w1 1,B

(3.5.5)

for any cube or ball B ⊂ Rn . Let s = n/β, so that β = n/s. Applying H¨ older’s inequality, Theorem 3.2.2, and (3.5.5), we ﬁnd that B

≤

|u − uB |s w1 dx

1/s

n

w1

B

≤ C2 dus,B

1/s

dx

1/n

B

|u − uB |ns/(n−s) dx

(n−s)/ns

1/n

n/s

(3.5.6)

w dx B 1 1/s

= C2 dus,B · w1 β,B 1/s

≤ C3 |B|(1−β)/sβ w1 1,B · dus,B . Setting t = sβ and using H¨ older’s inequality again, we obtain dus,B =

B

−1/t

1/t

|du|w2 w2

≤ =

B

1/t |du|w2

|du|t w2 dx B

s

1/s dx

1/t

t dx

B

1 w2

"

1/t " " 1/t " · "(1/w2 ) "

Combination of (3.5.6) and (3.5.7) gives

(t−s)/ts

s/(t−s)

ts/(t−s),B

dx .

(3.5.7)

102

3 Poincar´e-type inequalities

B

|u − uB |s w1 dx

1/s

" " 1/s " 1/t " ≤ C3 |B|(1−β)/sβ w1 1,B · "(1/w2 ) "

ts/(t−s),B

·

B

|du|t w2 dx

1/t

.

(3.5.8) s/t

Using the condition (w1 , w2 ) ∈ At/s , we have " " 1/s " 1/t " w1 1,B · "(1/w2 ) "

ts/(t−s),B

=

=

B

w1 dx

B

1/s B

w1 dx

B

1 w2

1 w2

(t−s)/ts

s/(t−s) dx

(s/t)(t/s−1) 1/s

1/(t/s−1) dx

1 1/(t/s−1) st ( st −1) 1/s 1 1 = |B|1+(s/t)(t/s−1) |B| w dx dx 1 |B| B w2 B ≤ C4 |B|2/s−1/t . (3.5.9) Now substituting (3.5.9) into (3.5.8) and using t = sβ, we ﬁnd that

1/s

|u − uB |s w1 dx

≤ C5 |B|1/s

B

1/t |du|t w2 dx

,

B

that is

1 |B|

1/s

|u − uB | w1 dx

≤ C5

s

B

1/t |du| w2 dx t

.

B

The proof of Theorem 3.5.2 is similar to that of Theorem 3.5.1, and therefore not included here. Proof of Theorem 3.5.3. Let s = n/β, where β is the constant appearing in Lemma 1.4.7. Applying H¨ older’s inequality, Theorem 3.2.2, and (3.5.5), we ﬁnd that

1/s |u − uB |s w1 dx B ≤

B

1/s

n

w1

≤ C2 dus,B

dx

1/n B n/s

w dx B 1

|u − uB |ns/(n−s) dx

1/n

(n−s)/ns

3.5 Two-weight inequalities

103 1/s

= C2 dus,B · w1 β,B (3.5.10)

1/s

≤ C3 |B|(1−β)/sβ w1 1,B · dus,B .

Let t = s + αs(r − 1). Applying H¨ older’s inequality again, we obtain dus,B =

≤

=

B

α/t |du|w2

|du|

t

B

−α/t

α/t

|du|w2 w2

B

s

1/s dx

1/t

t

dx

1/t w2α dx

·

B

B

1 w2

1 w2

(t−s)/ts

αs/(t−s) dx

(3.5.11)

α(r−1)/t

1/(r−1) dx

.

Combination of (3.5.10) and (3.5.11) gives B

|u − uB |s w1 dx

1/s 1/s

≤ C3 |B|(1−β)/sβ w1 1,B ·

B

1 w2

1 r−1

dx

α(r−1)

1/t t · B |du|n w2α dx . (3.5.12)

Using the condition (w1 , w2 ) ∈ 1/s w1 1,B

·

B

1 w2

|B|

1+αs(r−1)/t

we have

α(r−1)/t

1/(r−1)

=

αs/t Ar ,

dx

1 |B|

B

w1 dx

1 |B|

B

1 w2

1 r−1

(αs/t)(r−1) 1/s dx

≤ C4 |B|1/s+α(r−1)/t . (3.5.13) Substituting (3.5.13) into (3.5.12), we ﬁnd that

1/s

|u − uB |s w1 dx

≤ C5 |B|1/βs+α(r−1)/t

B

1/t |du|t w2α dx

B

which is equivalent to (3.5.3). Proof of Theorem 3.5.4. Since dμ2 = w2 (x)dx and w2 ≥ η > 0, we have μ2 (B) = w2 dx ≥ ηdx = η|B|, B

B

104

3 Poincar´e-type inequalities

that is

|B| ≤ C1 , μ2 (B)

(3.5.14)

where C1 = 1/η. From Theorem 3.5.1, deﬁnition of Ls (μ)-averaging domains, and inequality (3.5.14), we obtain

1 μ1 (Ω)

≤

|u − uB0 |s dμ1 Ω

1 μ1 (B0 )

Ω

|u − uB0 |s dμ1

≤ C2 sup2B⊂Ω

1 μ1 (B)

= C2 sup2B⊂Ω ≤ C2 sup2B⊂Ω ≤ C4 sup2B⊂Ω = C4

Ω

1/s

|B| μ1 (B) |B| μ2 (B)

Ω

|du|t dμ2

|u − uB |s dμ1 B 1/s

1 |B|

1/s C3

|du|t dμ2

1/t

1/s 1/s

|u − uB |s dμ1 B

B

|du| dμ2 t

1/t

1/s

1/t

.

Remark. (1) Theorems 3.5.2 and 3.5.3 can be extended to the global versions. (2) From [184], we know that John domains are Ls (μ)-averaging domains. Thus, the global results and Theorem 3.5.4 also hold if Ω is a John domain.

3.5.3 Aλ r (Ω)-weighted inequalities Next, we discuss the following version of two-weight Poincar´e inequality for diﬀerential forms. Theorem 3.5.5. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1, σ > 1, and s > αλ(r − 1) + 1, then there exists a constant C, independent of u, such that |u − uB |

s

B

1/s

w1α dx

≤ C|B|

1/s |du|

1/n

s

w2αλ dx

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form.

(3.5.15)

3.5 Two-weight inequalities

105

Proof. Choose t = s/(1 − α), so that 1 < s < t. Since 1/s = 1/t + (t − s)/st, by H¨ older’s inequality, we ﬁnd that

|u − uB |s w1α dx

B

= ≤

1/s α/s

(|u − uB |w1 )s dx B

|u − uB |t dx B

= u − uB t,B

1/t

B

w1 dx

Next, choose m=

1/s

αt/(t−s)

w B 1

α/s

(t−s)/st

(3.5.16)

dx

.

s , αλ(r − 1) + 1

so that m > 1. Since uB is a closed form, using Lemma 3.1.1 and Theorem 3.2.3, we have u − uB t,B ≤ C1 |B|(m−t)/mt u − uB m,σB

(3.5.17)

≤ C2 |B|(m−t)/mt |B|1/n dum,σB for all balls B with σB ⊂ Ω. Now since 1/m = 1/s + (s − m)/sm, by H¨ older’s inequality again, we obtain dum,σB = ≤

=

σB

αλ/s

|du|w2

|du|s w2αλ dx σB

|du|s w2αλ dx σB

−αλ/s

m

w2

1/m dx

1/s

σB

1/s

σB

1 w2

1 w2

(s−m)/sm

αλm/(s−m) dx

(3.5.18)

αλ(r−1)/s

1/(r−1) dx

.

From (3.5.16), (3.5.17), and (3.5.18), we have B

|u − uB |s w1α dx

1/s

≤ C2 |B|(m−t)/tm |B|1/n ×

σB

1 w2

B

w1 dx

α/s

αλ(r−1)/s

1/(r−1) dx

(3.5.19) σB

|du|s w2αλ dx

1/s

.

106

3 Poincar´e-type inequalities

Using the condition (w1 , w2 ) ∈ Aλr (Ω), we ﬁnd B

w1 dx

≤

α/s

B

σB

w1 dx

= |σB|λ(r−1)+1

×

1 |σB|

σB

dx

1 w2

1 |σB|

1 w2

αλ(r−1)/s

1/(r−1)

σB

1 w2

λ(r−1) α/s

1/(r−1) dx

B

w1 dx

(3.5.20)

λ(r−1) α/s

1/(r−1) dx

≤ C3 |σB|αλ(r−1)/s+α/s ≤ C4 |B|αλ(r−1)/s+α/s . Now substituting (3.5.20) in (3.5.19) and using −α αλ(r − 1) m−t = − , mt s s we obtain

1/s |u − uB |s w1α dx

B

≤ C|B|1/n

1/s |du|s w2αλ dx

.

σB

If we choose λ = 1/α in Theorem 3.5.5, we obtain the following version of the Aλr -weighted Poincar´e inequality. Corollary 3.5.6. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), 1/α l = 0, 1, . . . , n − 1. Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and 0 < α < 1. If s > r, then there exists a constant C, independent of u, such that 1/s 1/s |u − uB |s w1α dx ≤ C|B|1/n |du|s w2 dx B

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form. Selecting α = 1/s in Theorem 3.5.5, we have the following two-weighted Poincar´e inequality.

3.6 Inequalities for Jacobians

107

Corollary 3.5.7. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If σ > 1 and s > λ(r − 1)/s + 1, then there exists a constant C, independent of u, such that

1/s

|u − uB |

s

1/s w1 dx

≤ C|B|

B

1/s |du|

1/n

s

λ/s w2 dx

(3.5.21)

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form. When λ = 1 in Corollary 3.5.7, we obtain the following symmetric twoweighted inequality. Corollary 3.5.8. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.10) in a domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1. Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If σ > 1 and s > (r − 1)/s + 1, then there exists a constant C, independent of u, such that 1/s 1/s s 1/s 1/n s 1/s |u − uB | w1 dx ≤ C|B| |du| w2 dx B

σB

for all balls B with σB ⊂ Ω. Here uB is a closed form.

3.6 Inequalities for Jacobians The Jacobian (determinant) has played a critical role in multidimensional analysis and related ﬁelds from the very beginning, mainly pertaining to the change of variables in multiple integrals. The recent developments in nonlinear elasticity, weakly diﬀerentiable mappings, continuum mechanics, nonlinear PDEs, calculus of variations, and so on have rather clearly shown that the Jacobian is a powerful tool in many branches of science and engineering. For example, Jacobian and the related diﬀerential matrix satisfy the Beltrami equation which builds a bridge between mappings of ﬁnite distortion and PDEs; in quasiconformal analysis and nonlinear elasticity theory, the socalled stored energy integral of the div-curl ﬁeld the integrand is a Jacobian. The recent progress in these theories relies basically on estimates of the Jacobians in terms of the cofactor matrix. The present knowledge of Jacobians tells us something more, such as the regularity and topological behavior of the related mappings. Since one of the major applications of Jacobians is to evaluate multiple integrals, the integrability of Jacobians has become a rather important topic.

108

3 Poincar´e-type inequalities

3.6.1 Some notations In this section, we establish some Poincar´e-type inequalities for Jacobians in some domains, such as Ls (μ)-averaging domain. For this, we recall that earlier in this chapter we have developed some versions of Poincar´e-type inequalities for diﬀerential forms which do not satisfy any version of the A-harmonic equation. Then, as applications, we estimate subdeterminants of the Jacobian J(x, f ) of a mapping f : Ω → Rn , f = (f 1 , . . . , f n ). It is well known that Jacobian J(x, f ) is an n-form, speciﬁcally, J(x, f )dx = df 1 ∧ · · · ∧ df n , where dx = dx1 ∧dx2 ∧· · ·∧dxn . For example, let f = (f 1 , f 2 ) be a diﬀerential mapping in R2 . Then, 1 fx fy1 J(x, f )dx ∧ dy = 2 2 dx ∧ dy fx fy

= fx1 fy2 − fy1 fx2 dx ∧ dy and

df 1 ∧ df 2 = fx1 dx + fy1 dy ∧ fx2 dx + fy2 dy = fy1 fx2 dy ∧ dx + fx1 fy2 dx ∧ dy

= fx1 fy2 − fy1 fx2 dx ∧ dy,

where we have used the property dxi ∧ dxj =

0, i=j −dxj ∧ dxi , i = j.

Clearly, J(x, f )dx ∧ dy = df 1 ∧ df 2 . Recall the notations we have introduced in Section 1.2, where we assumed that f : Ω → Rn , f = (f 1 , . . . , f n ) is a mapping of Sobolev class 1,p (Ω, Rn ), 1 ≤ p < ∞, whose distributional diﬀerential Df = [∂f i /∂xj ] : Wloc Ω → GL(n) is a locally integrable function on Ω with values in the space GL(n) of all n × n-matrices. A homeomorphism f : Ω → Rn of Sobolev class 1,n (Ω, Rn ) is said to be K-quasiconformal, 1 ≤ K < ∞, if its diﬀerential Wloc matrix Df (x) and the Jacobian determinant

3.6 Inequalities for Jacobians

109

J(x, f ) = detDf (x) 1 fx1 fx12 f2 f2 x2 x1 = .. .. . . n fx fxn 2 1

fx13

···

fx23

···

.. .

..

fxn3

···

.

fx1n 2 fxn .. . n fxn

satisfy |Df (x)|n ≤ KJ(x, f ), where |Df (x)| = max{|Df (x)h| : |h| = 1} denotes the norm of the Jacobi matrix Df (x). Let u be the subdeterminant of Jacobian J(x, f ) which is obtained by deleting the k rows and k columns, k = 0, 1, . . . , n − 1, say,

J xj1 , xj2 , · · · , xjn−k ; f i1 , f i2 , · · · , f in−k i1 fxj1 fxi1j2 fxi1j3 · · · fxi1j n−k f i2 fxi2j2 fxi2j3 · · · fxi2j xj1 n−k = . .. .. .. , .. .. . . . . f in−k f in−k f in−k · · · f in−k xj2 xj3 xjn−k xj1 which is an (n − k) × (n − k) subdeterminant of J(x, f ), {i1 , i2 , . . . , in−k } ⊂ {1, 2, . . . , n}, and {j1 , j2 , . . . , jn−k } ⊂ {1, 2, . . . , n}. Also, it is easy to see that J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k )dxj1 ∧ dxj2 ∧ · · · ∧ dxjn−k is an (n − k)-form. Thus, all estimates for diﬀerential forms

are applicable to the (n−k)-form J xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k dxj1 ∧dxj2 ∧· · ·∧dxjn−k . For example, if we choose

u = J xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k dxj1 ∧ dxj2 ∧ · · · ∧ dxjn−k and use the same method which was developed in the proof of Theorem 3.5.4, we can prove the following Theorems 3.6.1 and 3.6.2.

3.6.2 Two-weight estimates Theorem 3.6.1. Let

u = J xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . f in−k dxj1 ∧ dxj2 ∧ · · · ∧ dxjn−k

110

3 Poincar´e-type inequalities

and du =

n ∂J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . f in−k ) dxm ∧ dxj1 ∧ · · · ∧ dxjn−k , ∂xm m=1

where k = 0, 1, . . . , n − 1. Assume that s > 1 and p > max{s, n}. Then, there s/p exists a constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ Ap/γ , where r > 1, sβ/(β − 1) ≤ γ < p, and w1 > w2 ≥ η > 0, we have

1 μ1 (Ω)

1/s

|u − uB0 | w1 dx

≤ Cμ2 (Ω)

s

1/n

Ω

1 μ2 (Ω)

1/p

|du| w2 dx p

Ω

for any Ls (μ1 )-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measures μ1 and μ2 are deﬁned by dμ1 = w1 (x)dx, dμ2 = w2 (x)dx, and C is a constant independent of u and du. Theorem 3.6.2. Let u and du be as Theorem 3.6.1. Then, there exists a s/n constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ An/s , where s = n/β, r > 1, and w1 > w2 ≥ η > 0, we have

1 μ1 (Ω)

1/s

|u − uB0 |s w1 dx

≤ Cμ2 (Ω)1/n

Ω

1 μ2 (Ω)

1/n

|du|n w2 dx Ω

for any Ls (μ1 )-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measures μ1 and μ2 are deﬁned by dμ1 = w1 (x)dx, dμ2 = w2 (x)dx, and C is a constant independent of u and du. Now recall that u = df i1 ∧ df i2 ∧ · · · ∧ df im is an m-form, where {i1 , i2 , . . . , im } ⊂ {1, 2, . . . , n}, m = 1, 2, . . . , n. Thus, any estimate for m-forms can be used to estimate the products of the components of a K-quasiconformal mapping. In fact, similar to Theorems 3.6.1 and 3.6.2, we can prove the following Theorems 3.6.3 and 3.6.4. Theorem 3.6.3. Let u = df i1 ∧ df i2 ∧ · · · ∧ df im and du ∈ Lp (B, ∧m+1 ), m = 1, 2, . . . , n − 1. Assume that 1 < s 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ Ap/k for some r > 1 and k with sβ/(β − 1) ≤ k < p, we have

1 |B|

|u − uB |s w1 dx B

1/s

≤ C|B|1/n

1 |B|

1/p

|du|p w2 dx B

for all balls B ⊂ Rn . Here C is a constant independent of u and du.

3.7 Inequalities for the projection operator

111

Theorem 3.6.4. Let u = df i1 ∧ df i2 ∧ · · · ∧ df im and du ∈ Lp (B, ∧m+1 ), m = 1, 2, . . . , n − 1. Assume that s > 1 and p > max{s, n}. Then, there exists s/p a constant β > 1 such that if w1 ∈ A1r and (w1 , w2 ) ∈ Ap/k , where r > 1, sβ/(β − 1) ≤ k < p, and w1 > w2 ≥ η > 0, we have

1 μ1 (Ω)

1/s

|u − uB0 | w1 dx s

≤ Cμ2 (Ω)

1/n

Ω

1 μ2 (Ω)

1/p

|du| w2 dx p

Ω

for any Ls (μ1 )-averaging domain Ω and some ball B0 with 2B0 ⊂ Ω. Here the measures μ1 and μ2 are deﬁned by dμ1 = w1 (x)dx, dμ2 = w2 (x)dx, and C is a constant independent of u and du. Remark. (1) Since Ls (μ)-averaging domains reduce to Ls -averaging domains if w = 1 (so dμ = w(x)dx = dx), Theorems 3.6.1, 3.6.2, and 3.6.4 also hold if Ω ⊂ Rn is an Ls -averaging domain. (2) From [184], we know that John domains are Ls (μ)-averaging domains. Thus, our global results, Theorems 3.6.1, 3.6.2, and 3.6.4 also hold if Ω is a John domain.

3.7 Inequalities for the projection operator In this section, we will prove Poincar´e inequalities for the projection operator applied to A-harmonic tensors on Ω. We continue to use the notation and terms that were introduced in Section 3.3.

3.7.1 Statement of the main theorem As usual, for u ∈ L1 (Ω, ∧l ), the projection operator H is deﬁned by setting H(u) to be the unique element of H such that = 0

(3.7.1)

for all h ∈ H. See [21] for the deﬁnition of the projection operator H and its properties. From [23, Chapter 6] we recall that the projection operator, Green’s operator, and the Laplace–Beltrami operator satisfy Poisson’s equation H(u) = u − ΔG(u). (3.7.2) We ﬁrst present the following local Poincar´e inequality for the projection operator applied to A-harmonic tensors on Ω.

112

3 Poincar´e-type inequalities

Theorem 3.7.1. Let u ∈ D (Ω, ∧l ) be an A-harmonic tensor on Ω and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n − 1, and H be the projection operator. Assume that ρ > 1, 0 < α ≤ 1, 1 + α(r − 1) < s < ∞, and w ∈ Ar for some r > 1. Then, H(u) − (H(u))B s,B,wα ≤ C|B|1/n dus,ρB,wα

(3.7.3)

for all balls B with ρB ⊂ Ω. Here C is a constant independent of u. Note that (3.7.3) can be written in the following symmetric form:

1 |B|

1/s

|H(u) − (H(u))B | w dx s

α

≤ C|B|

1/n

B

1 |B|

1/s

|du| w dx s

α

ρB

.

(3.7.3)

3.7.2 Inequality for Δ and G In order to prove Theorem 3.7.1, we need the following Poincar´e inequality for the composition of Δ and G. Lemma 3.7.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be an A-harmonic tensor on Ω. Assume that ρ > 1, 1 < s < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that ΔG(u) − (ΔG(u))B s,B,wα ≤ Cdiam(B)dus,ρB,wα

(3.7.4)

for any ball B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. Since G commutes with d (see [23, P225]) and d(du) = 0 for any diﬀerential form u, we ﬁnd that dΔG(u) = d(dd + d d)G(u) = dd dG(u) = (dd d + d dd)G(u) = (dd + d d)dG(u)

(3.7.5)

= ΔdG(u) = ΔG(du). From the properties of the homotopy operator T (see Section 1.5), we have u = d(T u) + T (du)

(3.7.6)

3.7 Inequalities for the projection operator

113

and T us,B ≤ C1 diam(B)us,B .

(3.7.7)

From [21], we know that for any smooth l-form ω, dd G(ω)s,B + d dG(ω)s,B ≤ C2 ωs,B ,

(3.7.8)

where 1 < s < ∞ and C2 is a constant. Choosing ω = du, we ﬁnd that dd G(du)s,B + d dG(du)s,B ≤ C2 dus,B .

(3.7.9)

By the deﬁnition of the Laplace–Beltrami operator Δ, Minkowski’s inequality, and (3.7.9), we obtain ΔG(du)s,B = (d d + dd )G(du)s,B ≤ d dG(du)s,B + dd G(du)s,B

(3.7.10)

≤ C2 dus,B . Thus, it follows that ΔG(u) − (ΔG(u))B s,B = T d(ΔG(u))s,B = T (ΔG(du))s,B

(3.7.11)

≤ C1 diam(B)ΔG(du)s,B ≤ C3 diam(B)dus,B . We shall ﬁrst show that (3.7.4) holds for 0 < α < 1. Let t = s/(1 − α). Using H¨ older’s inequality and (3.7.11), we have ΔG(u) − (ΔG(u))B s,B,wα =

B

|ΔG(u) − (ΔG(u))B |wα/s

≤ ΔG(u) − (ΔG(u))B t,B = ΔG(u) − (ΔG(u))B t,B ≤ C3 diam(B)dut,B

B

s

1/s dx

B

wtα/(t−s) dx

B

wdx

wdx

α/s

(t−s)/st

(3.7.12)

α/s

.

Choose m = s/(1 + α(r − 1)), so that m < s. Applying the weak reverse H¨older inequality, we obtain

114

3 Poincar´e-type inequalities

dut,B ≤ C4 |B|(m−t)/mt dum,ρB .

(3.7.13)

Substituting (3.7.13) into (3.7.12), we have ΔG(u) − (ΔG(u))B s,B,wα ≤ C5 |B|1/n+(m−t)/mt dum,ρB

B

wdx

α/s

(3.7.14)

.

Using H¨ older’s inequality again with 1/m = 1/s + (s − m)/sm, we ﬁnd that

dum,ρB =

|du|m dx ρB

=

1/m

|du|wα/s w−α/s

ρB

≤ dus,ρB,wα

m

1/m dx

(3.7.15)

1 1/(r−1) w

ρB

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Substitution of (3.7.15) into (3.7.14) gives ΔG(u) − (ΔG(u))B s,B,wα 1

≤ C6 |B| n +

m−t mt

dus,ρB,wα

wdx B

α/s

1 1/(r−1) ρB

w

α(r−1)/s dx

. (3.7.16)

Since w ∈ Ar (Ω), it follows that α/s

α/s

w1,B · 1/w1/(r−1),ρB ≤

wdx ρB

=

|ρB|

r

1 |ρB|

1 1/(r−1) ρB

ρB

w

wdx

1 |ρB|

r−1 α/s dx ρB

1 1/(r−1) w

r−1 α/s

(3.7.17)

dx

≤ C7 |B|αr/s . Now combination of (3.7.17) and (3.7.16) leads to ΔG(u) − (ΔG(u))B s,B,wα ≤ C8 |B|1/n dus,ρB,wα

(3.7.18)

for all balls B with ρB ⊂ Ω. Thus, we have proved (3.7.4) for 0 < α < 1. Next, we shall show that (3.7.4) is also true for α = 1, that is, we need to prove that ΔG(u) − (ΔG(u))B s,B,w ≤ C|B|1/n dus,ρB,w .

(3.7.19)

3.7 Inequalities for the projection operator

115

By Lemma 1.4.7, there exist constants β > 1 and C9 > 0, such that w β,B ≤ C9 |B|(1−β)/β w 1,B

(3.7.20)

for any cube or ball B ⊂ Rn . Set t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Note that 1/s = 1/t + (t − s)/st. Using H¨ older’s inequality, (3.7.11), and (3.7.20), we ﬁnd

1/s |ΔG(u) − (ΔG(u))B |s wdx

s 1/s = B |ΔG(u) − (ΔG(u))B |w1/s dx

1/t 1/s st/(t−s) (t−s)/st w ≤ B |ΔG(u) − (ΔG(u))B |t dx dx B

B

1/s

≤ C10 ΔG(u) − (ΔG(u))B t,B · wβ,B 1/s

≤ C10 |B|1/n dut,B · wβ,B 1/s

≤ C11 |B|1/n+(1−β)/βs w1,B · dut,B 1/s

≤ C11 |B|1/n−1/t w1,B · dut,B . (3.7.21) Next, let m = s/r. From Lemma 3.1.1, we obtain dut,B ≤ C12 |B|(m−t)/mt dum,ρB .

(3.7.22)

Using Lemma 1.1.4, we ﬁnd that dum,ρB = ≤

ρB

|du|w1/s w−1/s

|du|s wdx ρB

m

1/s

1/m dx 1 1/(r−1)

ρB

w

(r−1)/s

(3.7.23)

dx

for all balls B with ρB ⊂ Ω. Now since w ∈ Ar (Ω), similar to (3.7.17), we have 1/s 1/s (3.7.24) w1,B · 1/w1/(r−1),ρB ≤ C13 |B|r/s . Finally, combining (3.7.21), (3.7.22), (3.7.23), and (3.7.24), it follows that ΔG(u) − (ΔG(u))B s,B,w 1/s

≤ C14 |B|1/n−1/t w1,B |B|(m−t)/mt dum,ρB 1/s

1/s

≤ C14 |B|1/n−1/m w1,B · 1/w1/(r−1),ρB dus,ρB,w ≤ C15 |B|1/n dus,ρB,w for all balls B with ρB ⊂ Ω. Hence, (3.7.19) follows.

116

3 Poincar´e-type inequalities

3.7.3 Proof of the main theorem Now, we are ready to prove Theorem 3.7.1, the local Poincar´e inequality for the projection operator applied to A-harmonic tensors on Ω. Proof of Theorem 3.7.1. using (3.7.2), we obtain

Applying Theorem 3.2.3, Lemma 3.7.2, and

H(u) − (H(u))B s,B,wα = u − ΔG(u) − (u − ΔG(u))B s,B,wα = u − ΔG(u) − (uB − (ΔG(u))B )s,B,wα ≤ u − uB s,B,wα + ΔG(u) − (ΔG(u))B s,B,wα ≤ C1 |B|1/n dus,ρ2 B,wα + C2 |B|1/n dus,ρ1 B,wα ≤ C3 |B|1/n dus,ρB,wα , where ρ = max{ρ1 , ρ2 }. Using Theorems 3.7.1 and 1.5.3, and employing the method similar to the proof of Theorem 1.5.6, we can prove the following global Poincar´e inequality for the projection operator. Theorem 3.7.3. Let u ∈ D (Ω, ∧0 ) be a solution of the A-harmonic equation (1.2.4) in a δ-John domain Ω, du ∈ Ls (Ω, ∧1 ), and H be the projection operator. Assume that 0 < α ≤ 1, 1 + α(r − 1) < s < ∞, and w ∈ Ar (Ω) for some r > 1. Then, H(u) − (H(u))B0 s,Ω,wα ≤ Cdiam(Ω)dus,Ω,wα ,

(3.7.25)

where B0 ⊂ Ω is a ﬁxed ball and C is a constant independent of u.

3.8 Other Poincar´ e-type inequalities In Sections 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7, we have discussed several versions of the Poincar´e inequality for diﬀerential forms and the related operators applied to diﬀerential forms. There are many other versions of the Poincar´e inequality which have been widely investigated and used in many ﬁelds of mathematics, such as potential theory, partial diﬀerential equations. For example, Sobolev and Poincar´e inequalities and quasiconformal mappings were discussed in [208]; weighted Poincar´e inequalities and Minkowski contents were investigated in [209]; and Poincar´e-type inequalities on stratiﬁed sets

3.8 Other Poincar´e-type inequalities

117

and their applications were studied in [210]. S.M. Buckley and P. Koskela proved the following Sobolev–Poincar´e inequality in [211]: if Ω is a John domain and ∇u satisﬁes a certain mild condition, then 1/q

|u − uB0 |

1/p

≤C

q

|∇u|

p

Ω

Ω

1,1 for any function u ∈ Wloc (Ω), any p with 0 0.

Now let (M, g) be a smooth compact Riemannian n-dimensional manifold, n ≥ 3, and H12 (M ) be the Sobolev space deﬁned as the completion of C ∞ (M ) with respect to u2H 2 = 1

|∇u|2 dvg + M

u2 dvg . M

In 2002, E. Hebey [212] investigated the following sharp Sobolev–Poincar´e inequality on compact Riemannian manifold M ,

(n−2)/n |u|

2n/(n−2)

M

dvg

≤

|∇u| dvg + B

|u|dvg

2

M

2

Kn2

, (3.8.1)

M

where u ∈ H12 (M ) and B > 0 is independent of u. We say that (3.8.1) is valid if there exists a B such that (3.8.1) holds for all u ∈ H12 (M ). E. Hebey provided the validity criterion of inequality (3.8.1). Recently, in [213–215], P. Koskela and his colleagues investigated the Poincar´e inequality on metric spaces, Dirichlet forms, and Steiner symmetrization, respectively. In [216], O. Martio studied John domains, biLipschitz balls, and Poincar´e inequality. R. Nibbi discussed Poincar´e inequalities and applications to electromagnetism in [151]. M. Pearson investigated Poincar´e inequality and entire functions in [160]. Recently, C. Wang studied Poincar´e inequality and removable singularities for harmonic maps in [217]. In [218–221, 159], several versions of the Poincar´e inequalities for convex domains and planar domains were developed. In [222, 198, 223, 175, 176], Poincar´e inequalities on unbounded domains, Lipschitz domains, punctured domains, and trees were treated, respectively. In [224–227], Poincar´e inequalities for a class of Ap -weights, with a doubling weights, and with reverse doubling weights were established, respectively. In [228–234], Poincar´e-type inequalities for p-Laplacian, pointwise estimates for a class of degenerate elliptic equations, and the Poincar´e-type inequalities for Crushin-type operator, etc., were studied. In [235, 214], Poincar´e inequalities associated with Dirichlet forms were explored. For several other diﬀerent versions of Poincar´e inequalities, see [236–254, 34, 212, 150, 152, 162, 166, 172]. For some work related to Poincar´e inequalities, see [255–263, 238].

Chapter 4

Caccioppoli inequalities

In Chapter 3, we have discussed various versions of the Poincar´e-type inequalities in which we have estimated the norm of u − uB in terms of the corresponding norm of du. In this chapter, we develop a series of estimates which provide upper bounds for the norms of ∇u (if u is a function) or du (if u is a form) in terms of the corresponding norm u − c, where c is any closed form. These kinds of estimates are called the Caccioppoli-type estimates or the Caccioppoli inequalities. In Section 4.2, we study the local Ar (Ω)-weighted Caccioppoli inequalities. The local Caccioppoli inequalities with two-weights are discussed in Section 4.3. The global versions of Caccioppoli inequalities on Riemannian manifolds and bounded domains are developed in Sections 4.4 and 4.5, respectively. In Section 4.6, we present Caccioppoli inequalities with Orlicz norms. Finally, in Section 4.7, we address few versions of Caccioppoli inequalities related to the codiﬀerential operator d .

4.1 Preliminary results We assume that Ω is bounded and ψ is any function deﬁned in Ω with values in the extended reals [−∞, ∞], and ϑ is a function in the Sobolev space H 1,p (Ω; μ). Let χψ,ϑ = χψ,ϑ (Ω) = {v ∈ H 1,p (Ω; μ) : v ≥ ψ a.e. in Ω, v − ϑ ∈ H01,p (Ω; μ)}, where H01,p (Ω; μ) is the closure of C0∞ (Ω) in H 1,p (Ω; μ). For ψ = ϑ, we write χψ,ψ (Ω) = χψ (Ω). The problem is to ﬁnd a u in χψ,ϑ such that A(x, ∇u) · ∇(v − u)dx ≥ 0 (4.1.1) Ω

whenever v ∈ χψ,ϑ . The function ψ is called an obstacle. R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 4,

119

120

4 Caccioppoli inequalities

A function u ∈ χψ,ϑ (Ω) which satisﬁes (4.1.1) for all v ∈ χψ,ϑ (Ω) is called a solution to the obstacle problem with obstacle ψ and boundary values ϑ, or a solution to the obstacle problem in χψ,ϑ (Ω). See [264–266, 135] for results 1,p on obstacle problems. A function u ∈ Hloc (Ω; μ) is called a supersolution of the equation − div A(x, ∇u) = 0 (4.1.2) in Ω if − div A(x, ∇u) ≥ 0 weakly in Ω, that is

A(x, ∇u) · ∇ϕdx ≥ 0 Ω

whenever ϕ ∈ C0∞ (Ω) is nonnegative. Here A is an operator satisfying certain conditions. A function v is a subsolution of (4.1.2) if −v is a supersolution of (4.1.2). We begin this chapter with the following Caccioppoli inequality for a solution u to the obstacle problem in χψ,ϑ (Ω) with nonpositive obstacle ψ. The proof of Theorem 4.1.1 can be found in [59]. Theorem 4.1.1. Let η ∈ C0∞ (Ω) is nonnegative and q ≥ 0. i) If u is a solution to the obstacle problem in χψ,ϑ (Ω) with nonpositive obstacle ψ, then |u+ |q |∇u+ |p η p dμ ≥ C |u+ |p+q |∇η|p dμ. (4.1.3) Ω

Ω

ii) If u is a supersolution of (4.1.2) in Ω, then − q − p p |u | |∇u | η dμ ≥ C |u− |p+q |∇η|p dμ. Ω

(4.1.4)

Ω

Here u+ = max(u, 0), u− = min(u, 0), and C is a constant.

4.2 Local and global weighted cases In this section, we develop the local Ar (Ω)-weighted Caccioppoli inequalities which will be used to establish the global weighted estimates in later sections. We begin with an elementary Caccioppoli inequality without weights, and then use it to obtain some weighted inequalities.

4.2.1 Ar (Ω)-weighted inequality C. Nolder [71] obtained the following local Caccioppoli inequality for diﬀerential forms satisfying the homogeneous A-harmonic equation

4.2 Local and global weighted cases

121

d A(x, dω) = 0

(4.2.1)

in a ball or cube B ⊂ Rn (see Section 1.2 for the properties of operator A). Theorem 4.2.1. Let u be a diﬀerential form satisfying the homogeneous A-harmonic equation in Ω and let σ > 1. Then, there exists a constant C, independent of u and du, such that dus,B ≤

C u − cs,σB diam(B)

for all balls or cubes B with σB ⊂ Ω and all closed forms c. Here 1 < s < ∞ is the exponent associated with the A-harmonic equation (4.2.1). The above local Caccioppoli inequality was extended to the Ar (Ω)weighted version in [267]. Theorem 4.2.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be an A-harmonic tensor in a domain Ω ⊂ Rn and ρ > 1. Assume that 1 < s < ∞ is the exponent associated with the A-harmonic equation and w ∈ Ar for some r > 1. Then, there exists a constant C, independent of u and du, such that dus,B,w ≤

C u − cs,ρB,w diam(B)

(4.2.2)

for all balls B with ρB ⊂ Ω and all closed forms c. Note that (4.2.2) can be written as

1/s |du|s wdx

≤

B

C diam(B)

1/s |u − c|s wdx

(4.2.3)

ρB

or

1/s |du|s dμ B

≤

C diam(B)

1/s |u − c|s dμ

,

(4.2.4)

ρB

where the measure μ is deﬁned by dμ = w(x)dx and w ∈ Ar . Proof. Since w ∈ Ar for some r > 1, by Lemma 1.4.7, there exist constants β > 1 and C1 > 0, such that w β,B ≤ C1 |B|(1−β)/β w 1,B

(4.2.5)

for any cube or ball B ⊂ Rn . Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by H¨older’s inequality, Theorem 4.2.1, and (4.2.5), it follows that

122

4 Caccioppoli inequalities

dus,B,w = ≤

B

|du|w1/s

|du|t dx B

s

1/s dx

1/t B

w1/s

st/(t−s)

(t−s)/st dx

1/s wβ,B

≤ C2 dut,B ·

−1

≤ C3 diam(B)

(4.2.6)

u − ct,σB ·

1/s wβ,B

≤ C4 diam(B)−1 |B|(1−β)/βs w1,B · u − ct,σB 1/s

= C4 diam(B)−1 |B|−1/t · w1,B · u − ct,σB 1/s

for all balls B with σB ⊂ Ω and all closed forms c. Since c is a closed form and u is an A-harmonic tensor, u − c is also an A-harmonic tensor. Taking m = s/r, we ﬁnd that m < s < t. Applying Lemma 3.1.1, we ﬁnd that u − ct,σB ≤ C5 |B|(m−t)/mt u − cm,σ2 B ≤ C5 |B|(m−t)/mt u − cm,ρB ,

(4.2.7)

where ρ = σ 2 . Substituting (4.2.7) into (4.2.6), we ﬁnd dus,B,w ≤ C6 diam(B)−1 |B|−1/m · w1,B · u − cm,ρB . 1/s

(4.2.8)

Now since 1/m = 1/s + (s − m)/sm, by H¨ older’s inequality again, we obtain u − cm,ρB 1/m = ρB |u − c|m dx = ≤

ρB

ρB

|u − c|w1/s w−1/s

|u − c|s wdx

m

1/s

1/m dx

(4.2.9)

1 m/(s−m) ρB

w

(s−m)/sm dx

1/s

≤ u − cs,ρB,w · 1/wm/(s−m),ρB for all balls B with ρB ⊂ Ω and all closed forms c. Combining (4.2.8) and (4.2.9), we have dus,B,w ≤ C6 diam(B)−1 |B|−1/m · w1,B · 1/wm/(s−m),ρB · u − cs,ρB,w . (4.2.10) Since w ∈ Ar , it follows that 1/s

1/s

4.2 Local and global weighted cases 1/s

123

1/s

w1,B · 1/wm/(s−m),ρB

1/s 1 m/(s−m) (s−m)/sm = B wdx dx ρB w ≤

wdx ρB

=

|ρB|

s/m

1 1/(s/m−1) ρB

1 |ρB|

w

ρB

wdx

1 |ρB|

s/m−1 1/s dx

(4.2.11) 1 1/(r−1)

ρB

w

r−1 1/s dx

≤ C7 |B|1/m . Substituting (4.2.11) into (4.2.10), we ﬁnd that dus,B,w ≤ Cdiam(B)−1 u − cs,ρB,w for all balls B with ρB ⊂ Ω and all closed forms c.

4.2.2 Ar (λ, Ω)-weighted inequality The following Ar (λ, Ω)-weighted Caccioppoli-type estimate for diﬀerential forms satisfying the homogeneous A-harmonic equation (4.2.1) was proved by G. Bao in [18] (see Section 1.4 for the deﬁnition of Ar (λ, Ω)-weights). Theorem 4.2.3. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be an A-harmonic tensor in a domain Ω ⊂ Rn and ρ > 1. Assume that 1 < s < ∞ is a ﬁxed exponent associated with the A-harmonic equation and w ∈ Ar (λ, Ω) for some r > 1 and λ > 1. Then, there exists a constant C, independent of u, such that 1/s 1/s 2 C |du|s w1/λ dx ≤ |u − c|s w1/λ dx (4.2.12) diam(B) B ρB for all balls B with ρB ⊂ Ω and all closed forms c. Proof. Choose t = sλ2 /(λ2 − 1), so that 1 < s < t. Since 1/s = 1/t + (t − s)/st, by H¨ older’s inequality and Theorem 4.2.1, we have

1/s |du|s w1/λ dx

s 1/s = B |du|w1/λs dx

B

≤

|du|t dx B

≤ dut,B ·

1/t B

B

wλ dx

w1/λs

st/(t−s)

(t−s)/st dx

1/λ2 s

= C1 diam(B)−1 u − ct,σB

B

wλ dx

1/λ2 s

(4.2.13)

124

4 Caccioppoli inequalities

for all balls B with σB ⊂ Ω and all closed forms c. Since c is a closed form and u is an A-harmonic tensor, u − c is also an A-harmonic tensor. Taking m=

λ2 s , +r−1

λ2

we ﬁnd m < s < t. Applying Lemma 3.1.1, we have u − ct,σB ≤ C2 |B|(m−t)/mt u − cm,σ2 B

(4.2.14)

= C2 |B|(m−t)/mt u − cm,ρB , where ρ = σ 2 . Substituting (4.2.14) into (4.2.13), we have B

|du|s w1/λ dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt u − cm,ρB

wλ dx B

1/λ2 s

(4.2.15) .

Now since 1/m = 1/s + (s − m)/sm, by H¨ older’s inequality again, we obtain u − cm,ρB 1/m = ρB |u − c|m dx

=

|u − c|w1/λ s w−1/λ

ρB

≤

2

2

|u − c|s w1/λ dx ρB

2

s

m

1/m

(4.2.16)

dx

1/s

1 1/(r−1)

ρB

w

(r−1)/λ2 s dx

for all balls B with ρB ⊂ Ω and all closed forms c. Combining (4.2.15) and (4.2.16), we obtain

1/s |du|s w1/λ dx B 1s m−t 2 1/λs 1/λ2 s C3 ≤ diam(B) |B| mt wλ,B w1 1/(r−1),ρB ρB |u − c|s w1/λ dx . (4.2.17) Since w ∈ Ar (λ), it follows that 1/λ2 s

1/λs

wλ,B · 1/w1/(r−1),ρB ≤

λ

ρB

=

|ρB|r

w dx

1 |ρB|

2

≤ C4 |B|r/λ s .

1/(r−1)

ρB

ρB

(1/w)

wλ dx

1 |ρB|

r−1 1/λ2 s dx 1 1/(r−1)

ρB

w

r−1 1/λ2 s dx

(4.2.18)

4.2 Local and global weighted cases

125

Substituting (4.2.18) into (4.2.17), we ﬁnd that 1/s

|du| w s

1/λ

dx

B

C ≤ diam(B)

|u − c| w s

1/λ2

1/s dx

ρB

for all balls B with ρB ⊂ Ω and all closed forms c.

4.2.3 Aλ r (Ω)-weighted inequality We have presented the Ar (λ, Ω)-weighted Caccioppoli-type estimate in the previous section. A similar result holds for Aλr (Ω)-weights. Theorem 4.2.4. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (4.2.1) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n − 1 and ρ > 1. Assume that w ∈ Aλr (Ω) for some r > 1 and λ > 1. If 1 < s < ∞ is a ﬁxed exponent associated with the A-harmonic equation, then there exists a constant C, independent of u, such that

1/s |du|s w1/λ dx

≤ Cdiam(B)−1

B

1/s |u − c|s wdx

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Proof. Choose t = λs/(λ − 1). Applying H¨ older’s inequality and Theorem 4.2.1, we have B

|du|s w1/λ dx

1/s

= ≤

B

(|du|w1/λs )s dx

B

|du|t dx

= dut,B

1/s

1/t

B

B

wdx

wt/λ(t−s) dx

(t−s)/st (4.2.19)

1/λs

≤ C1 diam(B)−1 u − ct,σB

B

wdx

1/λs

.

Next, set m = s/r, so that m < s < t. Note that if u is a solution of equation (4.2.1) and c is a closed form, then u − c is also a solution of equation (4.2.1). Using Lemma 3.1.1, we obtain u − ct,σB ≤ C2 |B|(m−t)/mt u − cm,ρB ,

(4.2.20)

where ρ = σ 2 . Substituting (4.2.20) into (4.2.19) we ﬁnd that B

|du|s w1/λ dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt u − cm,ρB

wdx B

1/λs

(4.2.21) .

126

4 Caccioppoli inequalities

Now using H¨ older’s inequality with 1/m = 1/s + (s − m)/ms, we have u − cm,ρB

m 1/m = ρB |u − c|w1/s w−1/s dx ≤ ≤

ρB

|u − c|s wdx

|u − c|s wdx ρB

1/s

1 m/(s−m) w

ρB

1/s

1 1/(r−1) w

ρB

(s−m)/ms

(4.2.22)

dx (r−1)/s

dx

for all balls B with ρB ⊂ Ω and all closed forms c. Combining (4.2.21) and (4.2.22), we obtain B

|du|s w1/λ dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt 1/s × ρB |u − c|s wdx .

wdx

B

1/λs

1 1/(r−1)

ρB

w

(r−1)/s dx

(4.2.23) Next, using the condition w ∈ Aλr (Ω), we ﬁnd

wdx B ≤

1/λs

(1/w) ρB

ρB

≤ |ρB|

wdx

r−1 1 s + λs

ρB

(1/w)

1 |ρB|

(r−1)/s

1/(r−1)

B

dx

1/(r−1)

wdx

λ(r−1) 1/λs dx

1 |ρB|

(1/w)

ρB

1/(r−1)

λ(r−1) 1/λs dx

≤ C4 |ρB|(r−1)/s+1/λs = C5 |B|(r−1)/s+1/λs . (4.2.24) Finally, substituting (4.2.24) into (4.2.23), we obtain 1/s

|du| w s

1/λ

dx

−1

≤ Cdiam(B)

B

1/s |u − c| wdx s

.

ρB

4.2.4 Parametric version The following more general Ar (Ω)-weighted result is proved in [180] recently. Theorem 4.2.5. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be an A-harmonic tensor in a domain Ω ⊂ Rn and ρ > 1. Assume that 1 < s < ∞ is the

4.3 Local and global two-weight cases

127

exponent associated with the A-harmonic equation and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s |du| w dx s

α

B

C ≤ diam(B)

1/s |u − c| w dx s

α

(4.2.25)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Here α is any constant with 0 < α ≤ 1. Proof. Theorem 4.2.5 follows from Theorem 4.2.2 and equation (1.4.2).

4.2.5 Inequalities with two parameters The following parametric version of the Ar (λ, Ω)-weighted Caccioppoli-type inequality was obtained recently. This is a very general result since it contains two parameters, α and λ. In fact, for some particular values of α and λ our result reduces to several known results. Theorem 4.2.6. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be an A-harmonic tensor in a domain Ω ⊂ Rn , ρ > 1, and 0 < α < 1. Assume that 1 < s < ∞ is a ﬁxed exponent associated with the A-harmonic equation and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s |du| w s

B

αλ

dx

C ≤ diam(B)

1/s |u − c| w dx s

α

(4.2.26)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c.

4.3 Local and global two-weight cases We have presented some Caccioppoli inequalities with one weight in the previous section The purpose of this section is to study two-weighted local Caccioppoli inequalities for solutions of the nonhomogeneous A-harmonic equation. For this, we recall that throughout this chapter M is a compact Riemannian manifold on Rn . As usual, we begin our discussion with unweighted inequalities.

4.3.1 An unweighted inequality We ﬁrst prove the following elementary version of the local Caccioppoli inequality.

128

4 Caccioppoli inequalities

Theorem 4.3.1. Let u ∈ D (Ω, ∧l ) be a solution to the nonhomogeneous Aharmonic equation (1.2.10) in a bounded domain Ω, and σ > 1 be a constant. Then, there exists a constant C, independent of u, such that dup,B ≤ Cdiam(B)−1 u − cp,σB

(4.3.1)

for all balls or cubes B with σB ⊂ Ω and all closed forms c. Here 1 1 be a real number and η ∈ C0∞ (σB), η ≡ 1 in B, and |∇η| ≤

C1 |B|−1/n , diam(B)

where C1 is a constant. Choosing the test form ϕ = −uη p for (1.2.10), we have dϕ = −η p du − puη p−1 dη. From the deﬁnition of solutions to equation (1.2.10) in Section 1.2, we obtain A(x, du) · (−η p du) + A(x, du) · (−puη p−1 dη) + B(x, du) · (−uη p ) = 0. B

B

B

(4.3.2) Thus, it follows that A(x, du) · (η p du) σB = − σB A(x, du) · (puη p−1 dη) − σB B(x, du) · (uη p ). Applying (1.2.8), we have |A(x, du) · (η p du)| σB = σB |η p A(x, du) · du| = σB |A(x, du) · du| ≥ σB |du|p .

(4.3.3)

(4.3.4)

Note the fact that |∇η| = |dη| and diam(B) ≤ diam(Ω) < ∞. Using (4.3.4), (4.3.3), (1.2.8), and the H¨ older inequality, we obtain |η|p |du|p σB ≤ σB |A(x, du) · (η p du)| ≤ σB |A(x, du)| · (p|u||η|p−1 |dη|) + σB |B(x, du)| · (|u||η|p ) ≤ σB a|du|p−1 · (p|u||η|p−1 |∇η|) + σB b|du|p−1 · (|u||η|p ) ≤

C2 diam(B)

σB

|du|p−1 |η|p−1 |u| + C3

σB

|du|p−1 |u||η|p

4.3 Local and global two-weight cases

≤

C2 diam(B) up,σB

+C3 ηup,σB ≤ ≤

129

(|du|η)p−1

σB

σB

(|du|η)p−1

p−1 p

p−1 p

C2 diam(B) up,σB

+ C3 ηup,σB

C2 diam(B) up,σB

+ C3 up,σB

≤

C2 +C3 diam(B) up,σB diam(B)

≤

C4 diam(B) up,σB

σB

p−1 p

p−1 p

(ηdu)p

|η|p |du|p σB

(η|du|)p σB

(η|du|)p σB

p−1 p

p−1 p

(4.3.5)

p−1 p

p−1 p

which is equivalent to ηdup,σB ≤ Thus, we have

C4 up,σB . diam(B)

dup,B = ηdup,B ≤ ηdup,σB ≤

C4 diam(B) up,σB .

It is clear that if u is a solution of (1.2.10) and c is any closed form, then u − c is also a solution of (1.2.10). Thus, it follows that dup,B ≤ Cdiam(B)−1 u − cp,σB .

4.3.2 Two-weight inequalities Using the elementary version of the local Caccioppoli inequality obtained above, we prove the following two-weight Caccioppoli inequality. Theorem 4.3.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be a solution of the nonhomogeneous A-harmonic equation in a domain Ω ⊂ Rn and ρ > 1. Assume that 1 < s < ∞ is the exponent associated with the A-harmonic equation and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

130

4 Caccioppoli inequalities

1/s |du|s w1α dx

≤

1/s |u − c|s w2α dx

(4.3.6)

dus,B,w1α ≤ Cdiam(B)−1 u − cs,ρB,w2α

(4.3.6)

B

or

C diam(B)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Here α is any constant with 0 < α < λ. Proof. Choose t = λs/(λ − α). Since 1/s = 1/t + (t − s)/st, from the H¨older inequality and Theorem 4.3.1, we obtain

1/s |du|s w1α dx s 1/s α/s = B |du|w1 dx

B

≤

|du| dx t

B

≤ dut,B ·

1/t B

B

w1λ dx

α/s w1

(t−s)/st

st/(t−s)

(4.3.7)

dx

α/λs

≤ C1 diam(B)−1 u − ct,σB

B

w1λ dx

α/λs

for all balls B with σB ⊂ Ω and all closed forms c. Let m = λs/(λ+α(r −1)), so that m < s < t. Using Lemma 3.1.1 and the fact that u−c is also a solution of (1.2.10), we have u − ct,σB ≤ C2 |B|(m−t)/mt u − cm,ρB ,

(4.3.8)

where ρ > σ > 1. Substituting (4.3.8) into (4.3.7), we obtain B

|du|s w1α dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt u − cm,ρB

B

w1λ dx

α/λs

.

(4.3.9)

The H¨older inequality with 1/m = 1/s + (s − m)/sm again yields u − cm,ρB 1/m = ρB |u − c|m dx m 1/m α/s −α/s = ρB |u − c|w2 w2 dx 1/s λ/(r−1) α(r−1)/λs 1 s α ≤ ρB |u − c| w2 dx dx ρB w2

(4.3.10)

4.3 Local and global two-weight cases

131

for all balls B with ρB ⊂ Ω and all closed forms c. Substituting (4.3.10) into (4.3.9), we ﬁnd B

|du|s w1α dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt w1 λ,B 1/w2 λ/(r−1),ρB 1/s × ρB |u − c|s w2α dx . α/s

α/s

(4.3.11)

Using the condition (w1 , w2 ) ∈ Ar,λ (M ), we obtain α/s

α/s

w1 λ,B 1/w2 λ/(r−1),ρB r−1 α/λs λ λ/(r−1) ≤ w dx (1/w2 ) dx ρB 1 ρB

= |ρB|

r

1 |ρB|

wλ dx ρB 1

1 |ρB|

ρB

1 w2λ

1 r−1

r−1 α/λs

(4.3.12)

dx

≤ C4 |B|αr/λs . Combining (4.3.11) and (4.3.12), we have

1/s |du|

s

w1α dx

B

C5 ≤ diam(B)

1/s |u − c|

s

w2α dx

(4.3.13)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Note that Theorem 4.3.2 contains two weights, w1 (x) and w2 (x), and two parameters, λ and α. These features make this result more ﬂexible and useful. In fact, the existing versions of the Caccioppoli-type inequality in [186, 267, 71] are special cases of Theorem 4.3.2 with suitable choices of weights and parameters. Theorem 4.3.3. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n − 1, be a solution of the nonhomogeneous A-harmonic equation in a domain Ω ⊂ Rn , ρ > 1, and 0 < α < 1. Assume that 1 < s < ∞ is a ﬁxed exponent associated with the A-harmonic equation and (w1 , w2 ) ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s |du|

s

B

w1αλ dx

C ≤ diam(B)

1/s |u − c|

s

w2α dx

(4.3.14)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Proof. Choose t = s/(1 − α), so that 1 < s < t. Since 1/s = 1/t + (t − s)/st, by the H¨ older inequality and Theorem 4.2.1, it follows that

132

4 Caccioppoli inequalities

1/s |du|s w1αλ dx s 1/s αλ/s = B |du|w1 dx st/(t−s) (t−s)/st

1/t αλ/s t w1 ≤ B |du| dx dx B

B

≤ dut,B ·

B

w1λ dx

(4.3.15)

α/s

≤ C1 diam(B)−1 u − ct,σB

B

w1λ dx

α/s

for all balls B with σB ⊂ Ω and all closed forms c. Now since c is a closed form and u is an A-harmonic tensor, u − c is also an A-harmonic tensor. Taking m = s/(1 + α(r − 1)), so that m < s < t. Thus, an application of Lemma 3.1.1 yields u − ct,σB ≤ C2 |B|(m−t)/mt u − cm,σ2 B (4.3.16)

= C2 |B|(m−t)/mt u − cm,ρB , where ρ = σ 2 . Substituting (4.3.16) into (4.3.15), we have B

|du|s w1αλ dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt u − cm,ρB

B

w1λ dx

α/s

(4.3.17) .

Now since 1/m = 1/s+(s−m)/sm, by the H¨ older inequality again, we obtain u − cm,ρB 1/m = ρB |u − c|m dx = ≤

ρB

α/s

|u − c|

s

ρB

−α/s

|u − c|w2 w2 w2α dx

m

1/m

(4.3.18)

dx

1/s

ρB

1 w2

α(r−1)/s

1/(r−1) dx

for all balls B with ρB ⊂ Ω and all closed forms c. Combining (4.3.17) and (4.3.18), it follows that B

|du|s w1αλ dx

1/s

≤ C3 diam(B)−1 |B|(m−t)/mt w1 λ,B 1/w2 1/(r−1),ρB 1/s × ρB |u − c|s w2α dx . αλ/s

α/s

(4.3.19)

4.4 Inequalities with Orlicz norms

133

Since (w1 , w2 ) ∈ Ar (λ, Ω), we have αλ/s

α/s

w1 λ,B · 1/w2 1/(r−1),ρB r−1 α/s λ 1/(r−1) ≤ w dx (1/w ) dx 2 ρB 1 ρB |ρB|

=

r

1 |ρB|

wλ dx ρB 1

1 |ρB|

ρB

1 w2

r−1 α/s (4.3.20)

1/(r−1) dx

≤ C4 |B|αr/s . Substituting (4.3.20) into (4.3.19), we ﬁnd 1/s

|du|

s

B

w1αλ dx

C ≤ diam(B)

1/s |u − c|

s

w2α dx

(4.3.21)

ρB

for all balls B with ρB ⊂ Ω and all closed forms c. Note that if λ = 1, then Ar (λ, Ω) = Ar (1, Ω) becomes the usual Ar (Ω) weight.

4.4 Inequalities with Orlicz norms The purpose of this section is to develop various versions of the Caccioppoli inequalities with Orlicz norms · Lp (log L)α (E) . We begin with unweighted · Lp (log L)α (E) norm inequalities for du. Then, we discuss the Ar (M )-weighted cases. All the results of this section can be extended rather easily to the two-weight cases.

4.4.1 Basic · Lp (log L)α (E) estimates The purpose of this section is to prove the following Caccioppoli-type estimate with Orlicz norms for solutions to the nonhomogeneous A-harmonic equation on a Riemannian manifold. Theorem 4.4.1. Let u ∈ Lp (log L)α (Ω, ∧l ), l = 0, 1, . . . , n − 1, be a solution to the nonhomogeneous A-harmonic equation in a domain Ω ⊂ Rn . Then, there exists a constant C, independent of u, such that duLp (log L)α (B) ≤ C|B|−1/n u − cLp (log L)α (σB)

(4.4.1)

134

4 Caccioppoli inequalities

for some constant σ > 1 and all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0. Here d0 , 1 0 are constants. Proof. Let B ⊂ M be a ball with diam(B) ≥ d0 > 0. Let ε > 0 be small enough and the constant C1 large enough, so that |B|−ε/p ≤ C1 . 2

(4.4.2)

Applying Lemma 2.9 in [201], we have dup+ε,B ≤ C2 |B|(p−(p+ε))/p(p+ε) dup,σB

(4.4.3)

for some σ > 1. We may assume that |du|/||du||p,B ≥ 1 on B. Otherwise, setting B1 = {x ∈ B : |du|/||du||p,B ≥ 1}, B2 = {x ∈ B : |du|/||du||p,B < 1} and using the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ), where s > 0 is an arbitrary constant, we obtain duLp (log L)α (B) 1/p |du| dx = B |du|p logα e + ||du|| p,B |du| dx = B1 |du|p logα e + ||du|| p,B 1/p |du| α p e + |du| log ||du||p,B dx B2 p1 |du| dx ≤ 21/p |du|p logα e + ||du|| B1 p,B +

+

α p e+ |du| log B2

|du| ||du||p,B

dx

p1

(4.4.4)

.

Now, we shall estimate the ﬁrst term on the right-hand side of (4.4.4). Since |du|/||du||p,B ≥ 1 on B1 , for ε > 0 in (4.4.2), there exists C3 > 0 such that ε |du| |du| α ≤ C3 . (4.4.5) log e + dup,B ||du||p,σ1 B Combining (4.4.2), (4.4.3), and (4.4.5), we obtain

α p e+ |du| log B1

≤ C4

1

du εp,σ

1B

|du| ||du||p,B

1/p dx 1/p

|du|p+ε dx B1

4.4 Inequalities with Orlicz norms

≤ C4 =

1

du εp,σ

1B

≤

1/p

1B

C4 ε/p

du p,σ

135

|du|p+ε dx B

|du|p+ε dx B

1 (p+ε)/p

p+ε

|B|(p−(p+ε))/p(p+ε) dup,σ1 B

C5 ε/p

du p,σ

(p+ε)/p

(4.4.6)

1B

≤ C6 dup,σ1 B , where σ1 > 1 is a constant. Next, since |du| logα e + ≤ M1 logα (e + 1) ≤ M2 , ||du||p,B

x ∈ B2 ,

we can estimate the second term, to ﬁnd

|du| log p

B2

α

|du| e+ ||du||p,B

1/p dx

≤ C7 dup,σ2 B ,

(4.4.6)

where σ2 > 1 is a constant. From (4.4.4), (4.4.6), and (4.4.6) , we have duLp (log L)α (B) ≤ C8 dup,σ3 B ,

(4.4.7)

where σ3 = max{σ1 , σ2 }. Note that diam(B) = C9 |B|1/n , and hence from Theorem 4.3.2, we obtain dup,σ3 B ≤ C10 |B|−1/n u − cp,σ4 B for some σ4 > σ3 and all closed forms c. It is easy to see that |u − c| α ≥ 1, α > 0. log e + ||u − c||p,σ2 B

(4.4.8)

Now combining last three inequalities, we ﬁnd duLp (log L)α (B) ≤ C11 |B|−1/n u − cp,σ4 B ≤ C11 |B|−1/n u − cLp (log L)α (σ4 B) .

(4.4.9)

If we revise (4.4.5) and (4.4.8) in the proof of Theorem 4.4.1, we obtain the following version of the Caccioppoli-type estimate. Corollary 4.4.2. Let u ∈ Lp (log L)α (Ω, ∧l ), l = 0, 1, . . . , n − 1, be a solution to the nonhomogeneous A-harmonic equation in a domain Ω ⊂ Rn . Then, there exists a constant C, independent of u, such that

136

4 Caccioppoli inequalities

B

≤

|du|p logα e + C diam(B)

|du| ||du||p,M

1/p dx

α p e+ |u − c| log σB

|u−c| ||u−c||p,M

1/p dx

for some constant σ > 1 and all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0. Here d0 , 1 0 are constants.

4.4.2 Weak reverse H¨ older inequalities We now prove the weak reverse H¨ older inequalities with · Ls (log L)α (E) norms. These results will be used later to establish some weighted inequalities. Lemma 4.4.3. Let u be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1 and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that uLs (log L)α (B) ≤ C|B|(t−s)/st uLt (log L)β (σB) for all constants α > 0 and β > 0, and all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0, where d0 is a ﬁxed constant. Proof. For any ball B ⊂ M with diam(B) ≥ d0 > 0, we may choose ε > 0 small enough and the constant C1 large enough so that |B|−ε/st ≤ C1 .

(4.4.10)

us+ε,B ≤ C2 |B|(t−(s+ε))/t(s+ε) ut,σB

(4.4.11)

From Lemma 3.1.1, we have

for some σ > 1. Similar to (4.4.5) in the proof of Theorem 4.4.1, we can |u| assume that u

≥ 1 on B. For above ε > 0, there exists C3 > 0 such that t,B log

α

|u| e+ us,B

≤ C3

|u| ||u||t,σB

ε .

From (4.4.11) and (4.4.12), it follows that uLs (log L)α (B) =

α s e+ |u| log B

≤ C4

1

u εt,σB

|u| ||u||s,B

|u|s+ε dx B

1/s dx

1/s

(4.4.12)

4.4 Inequalities with Orlicz norms

≤

C4 ε/s

u t,σB

≤

C5 ε/s

u t,σB

≤ C6 |B|

B

|u|s+ε dx

137 1 (s+ε)/s

s+ε

|B|(t−(s+ε))/t(s+ε) ut,σB

t−s−ε st

(s+ε)/s

(4.4.13)

ut,σB .

Now from (4.4.10) and (4.4.13) and using |u| β ≥ 1, β > 0, log e + ||u||t,σB we obtain uLs (log L)α (B) ≤ C6 |B|

t−s−ε st

≤ C7 |B|

t−s−ε st

≤ C7 |B|

t−s−ε st

≤ C8 |B|

t−s st

ut,σB

β t e+ |u| log σB

|u| ||u||t,σB

1/t dx

(4.4.14)

uLt (log L)β (σB)

uLt (log L)β (σB) .

Using a similar method developed in the proof of Lemma 4.4.3 and from Lemma 2.9 in [268], we can prove the following version of the weak reverse H¨ older inequality with Orlicz norms. Note that this result cannot be obtained by replacing u by du in Lemma 4.4.3 since du may not be a solution of (1.2.10). Lemma 4.4.4. Let u be an A-harmonic function in a domain Ω ⊂ Rn , σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that duLs (log L)α (B) ≤ C|B|(t−s)/st duLt (log L)β (σB) for all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0. Here d0 is a ﬁxed constant, α > 0 and β > 0 are arbitrary constants.

4.4.3 Ar (M )-weighted cases Theorem 4.4.5. Let u ∈ Lp (log L)α (Ω, ∧l ), l = 0, 1, . . . , n − 1, be a solution to the nonhomogeneous A-harmonic equation in a domain Ω ⊂ Rn and w ∈ Ar (Ω) for some r > 1. Also assume that diamΩ ≤ ∞. Then, there exists a constant C, independent of u, such that

138

4 Caccioppoli inequalities

duLp (log L)α (B,w) ≤ C|B|−1/n u − cLp (log L)α (σB,w)

(4.4.15)

for any closed form c, some constant σ > 1, and all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0. Here d0 , 1 0 are constants. Proof. Let B be a ball with σB ⊂ Ω and diam(B) ≥ d0 > 0. Also d0 ≤ diam(B) ≤ diam(Ω) < ∞. Thus, 0 < v1 ≤ |B| ≤ v2 < ∞ for some constants v1 and v2 . By Lemma 3.1.1, we have m1 us,ρ1 B ≤ ut,B ≤ m2 us,ρ2 B

(4.4.16)

for any solution u of (1.2.10) and any constants s, t > 0, where 0 < ρ1 < 1, ρ2 > 1, 0 < m1 < 1, and m2 > 1 are some constants. By Lemma 1.4.7, there exist constants k > 1 and C0 > 0, such that w k,B ≤ C0 |B|(1−k)/k w 1,B .

(4.4.17)

Choose s = pk/(k − 1), so that 1 < p < s and k = s/(s − p). We know that u ∈ Lp (log L)α (Ω, ∧l ) implies u ∈ Lp (Ω, ∧l ). Thus, for any closed form c, it follows that u − c ∈ Lp (Ω, ∧l ) since M is compact. By Caccioppoli inequality with Lp -norms, we know that du ∈ Lp (M, ∧l ) which implies dup,Ω = N < ∞. If dup,B = 0, then du = 0 a.e. on B and Theorem 4.4.3 follows. Thus, we assume that 0 < m1 ≤ dus,B < M1 and 0 < m2 ≤ dup,B < M2 by (4.4.16). Since 1/p = 1/s + (s − p)/ps, by the H¨older inequality, (4.4.17), and (3.4.14), we ﬁnd duLp (log L)α (B,w) = B |du|p logα e + = ≤

α |du| log p e +

B

|du| log s

B

≤ C1

αs p

≤ C2 |B|

1−k kp

e+ αs p

1

p w 1,B

|du| ||du||p,B

e+

1/p wdx

|du| ||du||p,B

|du|s log B

|du| ||du||p,B

w1/p

dx

p

1s B

|du| ||du||s,B

dx

|du|s log B

1/p dx

αs p

w

s s−p

dx

s−p sp

1s

wk dx B

e+

|du| ||du||s,B

(4.4.18)

1/k p1

dx

1s

.

Now an application of Theorem 4.4.1 yields

|du|s log B

αs p

e+

|du| ||du||s,B

1/s dx

≤ C3 |B|−1/n u − cLs (log L)αs/p (σ1 B) .

(4.4.19)

4.4 Inequalities with Orlicz norms

139 p r.

Here c is any closed form. Next, choose t = 4.4.3 with β = αr , we obtain B

|du|s log

αs p

e+

|du| ||du||p,B

Using (4.4.19) and Lemma

1/s dx (4.4.20)

≤ C4 |B|

−1/n

|B|

(t−s)/st

u − cLt (log L)β (σ2 B)

older inequality again with 1/t = 1/p + for some σ2 > σ1 . Using the H¨ (p − t)/pt, we have u − cLt (log L)β (σ2 B) =

σ2 B

|u − c|t logβ e +

≤

=

σ2 B

×

β

|u − c| log t

|u−c| ||u−c||t,σ2 B

e+

|u−c| ||u−c||t,σ2 B

βp/t p e+ |u − c| log σ2 B

t 1 p−t

σ2 B

w

dx

1/t dx

|u−c| ||u−c||t,σ2 B

w1/p w−1/p wdx

1/t

t dx

p1

(4.4.21)

p−t pt

≤ u − cLp (log L)α (σ2 B,w)

1 1 r−1

σ2 B

w

(r−1)/p dx

.

Combining (4.4.18), (4.4.20), and (4.4.21), we ﬁnd duLp (log L)α (B,w) ≤ C5 |B|−r/p−1/n u − cLp (log L)α (σ2 B,w)

×

wdx B

1 1 r−1

σ2 B w

(r−1) 1/p dx

≤ C5 |B|−r/p−1/n u − cLp (log L)α (σ2 B,w)

1/p × w1,σ2 B · 1/w1/(r−1),σ2 B . Now since w ∈ Ar (Ω), it follows that

1/p w1,σ2 B · 1/w1/(r−1),σ2 B r−1 1/p 1/(r−1) = wdx (1/w) dx σ2 B σ2 B

(4.4.22)

140

4 Caccioppoli inequalities

=

r−1 p1 1 |σ2 B|r |σ21B| σ2 B wdx |σ21B| σ2 B w1 r−1 dx

(4.4.23)

≤ C6 |B|r/p . Finally, substituting (4.4.23) into (4.4.22), we obtain duLp (log L)α (B,w) ≤ C7 |B|−1/n u − cLp (log L)α (σ2 B,w) .

4.5 Inequalities with the codiﬀerential operator In this section, we study the properties of the codiﬀerential operator d that is applied to a diﬀerential (l + 1)-form v ∈ D (Ω, ∧l+1 ) satisfying the conjugate A-harmonic equation A(x, du) = d v, (4.5.1) where A : Ω × ∧l (Rn ) → ∧l (Rn ) satisﬁes the following conditions: |A(x, ξ)| ≤ a|ξ|p−1

and

< A(x, ξ), ξ > ≥ |ξ|p

(4.5.2)

for almost every x ∈ Ω and all ξ ∈ ∧l (Rn ). Here a > 0 is a constant and 1 < p < ∞ is a ﬁxed exponent associated with (4.5.1).

4.5.1 Lq -estimate for dv We have explored some properties of a pair of solutions (u, v) to the above equation in Chapter 1. In this section, we focus on the integral properties of the codiﬀerential operator d . Theorem 4.5.1. Let u ∈ D (Ω, ∧l−1 ) and v ∈ D (Ω, ∧l+1 ) be a pair of Aharmonic tensors in a domain Ω ⊂ Rn and σ > 1 be a constant. Then, there exists a constant C, independent of v, such that d vq,B ≤ C|B|−1/n v − c1 q,σB

(4.5.3)

for all balls or cubes B with σB ⊂ Ω and all co-closed forms c1 . Here 1 < q < ∞. Proof. Choosing s = p, t = q, and Q = B in the local Hardy–Littlewood inequality (Theorem 1.5.2), we have ||u − uB ||pp,B ≤ C1 |B|

p−q n

||v − c1 ||qq,σ1 B .

(4.5.4)

4.5 Inequalities with the codiﬀerential operator

From (4.5.1) and (4.5.2), we obtain |d v|q dx = B |A(x, du)|q dx B ≤ C2 B |du|q(p−1) dx = C2 B |du|p dx,

141

(4.5.5)

and hence d vqq,B ≤ C2 dupp,B . From Theorem 4.3.1, it follows that d vqq,B ≤ C3 diam(B)−p u − cpp,σ2 B for any closed form c. Since uB is a closed form for any ball, we may choose c = uB . Now since diam(B) = C4 |B|1/n , we have d vqq,B ≤ C5 |B|−p/n u − uB pp,σ2 B .

(4.5.6)

Finally, a combination of (4.5.4) and (4.5.6) yields d vqq,B ≤ C6 |B|−q/n v − c1 qq,σ3 B which is the same as d vq,B ≤ C|B|−1/n v − c1 q,σ3 B .

4.5.2 Two-weight estimate for dv Theorem 4.5.2. Let u ∈ D (Ω, ∧l−1 ) and v ∈ D (Ω, ∧l+1 ) be a pair of Aharmonic tensors in a domain Ω ⊂ Rn and σ > 1 be a constant. Assume that (w1 , w2 ) ∈ Ar,λ (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of v, such that d vq,B,w1α ≤ C|B|−1/n v − c1 q,σB,w2α

(4.5.7)

for all balls or cubes B with σB ⊂ Ω and all co-closed forms c1 . Here 1 < q < ∞ and 0 < α < λ are constants. Proof. Choose s = λq/(λ − α). Since 1/q = 1/s + (s − q)/sq, using the H¨older inequality, we obtain

1/q |d v|q w1α dx q 1/q α/q = B |d v|w1 dx

B

142

4 Caccioppoli inequalities

≤

|d v|s dx B

= d vs,B ·

1/s

α/q

B

(s−q)/sq

sq/(s−q)

w1

dx (4.5.8)

α/λq wλ dx B 1

for all balls B with B ⊂ M . Now let t = λq/(λ + α(r − 1)), so that t < q < s. Using the weak reverse H¨older inequality for d v, it follows that d vs,B ≤ C1 |B|(t−s)/st d vt,σ1 B .

(4.5.9)

Substituting (4.5.9) into (4.5.8) and using Theorem 4.5.1, we have

|d v|q w1α dx

B

1/q

≤ C2 |B|(t−s)/st d vt,σ1 B

w1λ dx

B

α/λq

≤ C3 |B|(t−s)/st |B|−1/n v − c1 t,σ2 B

B

w1λ dx

α/λq

(4.5.10) .

Using the H¨ older inequality with 1/t = 1/q + (q − t)/qt again, we obtain v − c1 t,σ2 B 1/t = σ2 B |v − c1 |t dx = ≤

σ2 B

|v −

α/q −α/q c1 |w2 w2

|v − c1 |q w2α dx σ2 B

1/t

t

(4.5.11)

dx

1/q

σ2 B

1 w2

α(r−1)/λq

λ/(r−1) dx

for all balls B with σ2 B ⊂ Ω and all co-closed forms c1 . Substituting (4.5.11) into (4.5.10), we ﬁnd B

|d v|q w1α dx

1/q

≤ C4 |B|−1/n |B|(t−s)/st w1 λ,B 1/w2 λ/(r−1),σ2 B 1/q × σ2 B |v − c1 |q w2α dx . α/q

α/q

Now using the condition (w1 , w2 ) ∈ Ar,λ (Ω), we have α/q

α/q

w1 λ,B 1/w2 λ/(r−1),σ2 B r−1 α/λq λ λ/(r−1) ≤ w dx (1/w2 ) dx σ2 B 1 σ2 B

(4.5.12)

4.5 Inequalities with the codiﬀerential operator

|σ2 B|

r

=

1 |σ2 B|

wλ dx σ2 B 1

143

1 |σ2 B|

σ2 B

1 w2λ

r−1 α/λq

1/(r−1) dx

≤ C5 |B|αr/λq . (4.5.13) Finally, combining (4.5.12) and (4.5.13) and noting that t − s αr + = 0, st λq we obtain 1/q q α −1/n |d v| w1 dx ≤ C6 |B| B

1/q |v − c1 |

q

w2α dx

(4.5.14)

σ2 B

for all balls B with σ2 B ⊂ Ω and all co-closed forms c1 . Notes to Chapter 4. Many mathematicians have made signiﬁcant contriˇ butions to this topic. In particular, I. Peri´c and D. Zubrini´ c established a Caccioppoli-type estimate for quasilinear elliptic operator in [161]; M. Giaquinta and J. Souˇcek obtained a new Caccioppoli inequality in [269]; G.A. Ser¨egin provided a local estimate of the Caccioppoli-type inequality for extremal variational problems of Hencky plasticity in [173], and R.F. Gariepy gave a new proof of the partial regularity minimizers based on the Caccioppoli inequality [270]. For more versions of Caccioppoli-type inequalities, see [205, 271–273, 136, 170].

Chapter 5

Imbedding theorems

5.1 Introduction In recent years various versions of imbedding theorems for diﬀerential forms have been established. The imbedding theorems for functions can be found in almost every book on partial diﬀerential equations, see Sections 7.7 and 7.8 in [274], for example. For diﬀerent versions of imbedding theorems, see [20, 275–280, 32, 268, 81]. Many results for Sobolev functions have been extended to diﬀerential forms in Rn . The imbedding theorems play crucial role in generalizing the theory of Sobolev functions into the theory of diﬀerential forms. The objective of this chapter is to discuss several other versions of imbedding theorems for diﬀerential forms. We also explore some imbedding theorems related to operators, such as the homotopy operator T and Green’s operator G. We ﬁrst study the imbedding theorems for quasiconformal mappings in Section 5.2. Then, we establish an imbedding theorem for diﬀerential forms satisfying the nonhomogeneous A-harmonic equation in Section 5.3. In Section 5.4, we present the Ar (Ω)-weighted imbedding theorems related to the gradient operator and the homotopy operator. In Section 5.5, we explore some Ar (λ, Ω)-weighted imbedding theorems and Aλr (Ω)-weighted imbedding theorems. In Section 5.6, we develop some Ls -estimates and imbedding theorems for the compositions of operators. Finally, in Section 5.7, we study the two-weight imbedding inequalities.

5.2 Quasiconformal mappings The following three theorems can be found in [59]. In Theorem 5.2.1, the measure μ is deﬁned by Ar -weights and in Theorem 5.2.3, the measure μ is deﬁned by the Jacobian J(x, f ) of a K-quasiconformal mapping f : Rn → Rn . Theorem 5.2.2 implies that the Jacobian J(x, f ) of a K-quasiconformal mapping f belongs to Ar for some r > 1. R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 5,

145

146

5 Imbedding theorems

Theorem 5.2.1. Let w ∈ Ar , r0 < q < r < nq, and κ = n/(n − r/q) > 1. Then, there exists a constant C such that

1 μ(B)

1/κr

|ϕ| dμ

≤ Cdiam(B)

κr

B

1 μ(B)

1/r

|∇ϕ| dμ r

,

B

where B is a ball, dμ = w(x)dx, and ϕ ∈ C0∞ (B). Theorem 5.2.2. The Jacobian J(x, f ) of a K-quasiconformal mapping f : Rn → Rn is an A∞ -weight with constants depending only on n and K. Theorem 5.2.3. Let 1 < r < n, f : Rn → Rn be a K-quasiconformal mapping and w(x) = J(x, f )1−r/n . Then, there are constants κ = κ(n, K) > 1 and C = C(n, K, r) > 0 such that

1 μ(B)

1/κr

|ϕ| dμ κr

B

≤ Cdiam(B)

1 μ(B)

1/r

|∇ϕ| dμ r

,

B

where B is a ball, dμ = w(x)dx, and ϕ ∈ C0∞ (B).

5.3 Solutions to the nonhomogeneous equation We have presented the imbedding results for quasiconformal mappings in Section 5.2. In the remaining part of this chapter, we focus our attention on the weighted imbedding theorems for diﬀerential forms satisfying the nonhomogeneous A-harmonic equation d A(x, dω) = B(x, dω)

(5.3.1)

which we have introduced in Section 1.2. We will also develop the imbedding inequalities for some operators. Theorem 5.3.1. Let u ∈ D (Ω, ∧l ) be a solution of the nonhomogeneous A-harmonic equation (5.3.1) in a bounded domain Ω ⊂ Rn and du ∈ Ls (Ω, ∧l+1 ), l = 0, 1, . . . , n−1. Assume that 1 < s < ∞ and w ∈ Ar (Ω) for some r > 1. Then, u − uB W 1,s (B),w ≤ Cdus,σB,w for all balls B with σB ⊂ Ω, σ > 1. Here C is some constant independent of u and du. Proof. Replacing u by du in (1.5.5) and (1.5.6), we obtain ∇T (du)s,B ≤ C1 |B|dus,B

(5.3.2)

5.4 Imbedding inequalities for operators

147

and T (du)s,B ≤ C2 |B|diam(B)dus,B .

(5.3.3)

Using the similar skill developed in the proofs of inequalities (5.4.5) and (5.4.20) and the weak reverse H¨ older inequality for du (Theorem 6.2.12), we can extend inequalities (5.3.2) and (5.3.3) into the weighted versions ∇T (du)s,B,w ≤ C3 |B|dus,σ1 B,w

(5.3.4)

T (du)s,B,w ≤ C4 |B|diam(B)dus,σ2 B,w ,

(5.3.5)

and where σ1 , σ2 > 1. From (3.3.15), (5.3.4), and (5.3.5), we obtain u − uB W 1,s (B),w = T (du)s,B,w = (diam(B))−1 T (du)s,B,w + ∇(T (du))s,B,w ≤ (diam(B))−1 C4 |B|diam(B)dus,σ1 B,w + C3 |B|dus,σ2 B,w ≤ C5 |B|dus,σB,w , ≤ C6 dus,σB,w , that is u − uB W 1,s (B),w ≤ C6 dus,σB,w , where σ = max{σ1 , σ2 }. It is well known that if w ∈ Ar (E) and α is any constant with 0 < α ≤ 1, then wα ∈ Ar (E). Thus, inequality (5.3.2) still holds if we replace w(x) by wα (x) in (5.3.2). Hence, for any solution u of the nonhomogeneous Aharmonic equation (5.3.1), we have u − uB W 1,s (B),wα ≤ Cdus,σB,wα

(5.3.6)

for any constant α with 0 < α ≤ 1.

5.4 Imbedding inequalities for operators We have established the Ar (Ω)-weighted imbedding inequality for solutions of the nonhomogeneous A-harmonic equation in the previous section. The main purpose of this section is to develop the Ar (Ω)-weighted imbedding inequality for the homotopy operator. For this, ﬁrst some estimates related

148

5 Imbedding theorems

to the gradient operator and the composition of the gradient operator and the homotopy operator will be obtained.

5.4.1 The gradient and homotopy operators Let u be a diﬀerential form in the Sobolev space W 1,p (E, ∧l ) of l-forms deﬁned over a bounded and convex E, 0 < p < ∞ and w(x) be a weight. The weighted norm of T u is denoted by T uW 1,p (E,∧l ) and deﬁned as T uW 1,p (E,∧l ),

wα

= diam(E)−1 T up,E,wα + ∇(T u)p,E,wα ,

(5.4.1)

where α is a real number. Some elementary estimates for T u and ∇(T u) are known, in particular from [99], we have the following lemma. Lemma 5.4.1. Let u ∈ Lsloc (B, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form in a ball B ⊂ Rn . Then, ∇(T u)s,B ≤ C|B|us,B ,

(5.4.2)

T us,B ≤ C|B|diam(B)us,B .

(5.4.3)

Now we prove the following local weighted imbedding theorem for diﬀerential forms under the homotopy operator T . Theorem 5.4.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

wα

≤ C|B|us,ρB,wα

(5.4.4)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. In order to prove Theorem 5.4.2, we need the following Ar (Ω)-weighted imbedding theorem [277]. Theorem 5.4.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/s s α s α |∇(T u)| w dx ≤ C|B| |u| w dx (5.4.5) B

for any real number α with 0 < α ≤ 1.

ρB

5.4 Imbedding inequalities for operators

149

Note that (5.4.5) can be written as (5.4.5)

∇(T u)s,B,wα ≤ C|B|us,ρB,wα .

Proof. First, we shall show that inequality (5.4.5) holds for 0 < α < 1. Let t = s/(1 − α). From Theorem 1.1.4, we have

|∇(T u)|s wα dx

B

=

1/s

B

|∇(T u)|wα/s

≤ ∇(T u)t,B = ∇(T u)t,B

s

1/s dx

wtα/(t−s) dx B

B

wdx

α/s

(5.4.6)

(t−s)/st

.

Thus, from Lemma 5.4.1, we obtain ∇(T u)t,B ≤ C1 |B|ut,B .

(5.4.7)

Choose m = s/(1 + α(r − 1)), so that m < s. Substituting (5.4.7) into (5.4.6) and using Lemma 3.1.1, we ﬁnd B

|∇(T u)|s wα dx

≤ C1 |B|ut,B

1/s

B

wdx

α/s (5.4.8)

≤ C2 |B||B|(m−t)/mt um,ρB

B

wdx

α/s

.

From Theorem 1.1.4 with 1/m = 1/s + (s − m)/sm, we have um,ρB 1/m = ρB |u|m dx = ≤

|u|wα/s w−α/s

ρB

ρB

|u|s wα dx

m

1/m

(5.4.9)

dx

1/s

1 1/(r−1) ρB

w

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Substituting (5.4.9) into (5.4.8), we obtain B

|∇(T u)|s wα dx

1/s

≤ C2 |B||B|(m−t)/mt

|u|s wα dx ρB

1/s

150

5 Imbedding theorems

×

B

wdx

α/s

1 1/(r−1)

ρB

w

α(r−1)/s dx

.

(5.4.10)

Since w ∈ Ar (Ω), we ﬁnd α/s

α/s

w1,B · 1/w1/(r−1),ρB r−1 α/s 1/(r−1) ≤ wdx (1/w) dx ρB ρB =

|ρB|r

1 |ρB|

ρB

wdx

1 1/(r−1)

1 |ρB|

ρB

w

r−1 α/s

(5.4.11)

dx

≤ C3 |B|αr/s . Combining (5.4.11) and (5.4.10), it follows that

1/s |∇(T u)| w dx s

1/s

≤ C4 |B|

α

B

|u| w dx s

α

(5.4.12)

ρB

for all balls B with ρB ⊂ Ω. Thus, (5.4.5) holds for 0 < α < 1. Next, we shall prove (5.4.5) for α = 1, that is, we need to show that ∇(T u)s,B,w ≤ C|B|us,ρB,w .

(5.4.13)

By Lemma 1.4.7, there exist constants β > 1 and C5 > 0, such that w β,B ≤ C5 |B|(1−β)/β w 1,B

(5.4.14)

for any cube or ball B ⊂ Rn . Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by Theorem 1.1.4, Lemma 5.4.1, and (5.4.14), we have

1/s |∇(T u)|s wdx

s 1/s = B |∇(T u)|w1/s dx

B

≤

|∇(T u)|t dx B

1/t B

w1/s

st/(t−s)

1/s

≤ C6 ∇(T u)t,B · wβ,B 1/s

≤ C6 |B|ut,B · wβ,B 1/s

≤ C7 |B||B|(1−β)/βs w1,B · ut,B ≤ C7 |B||B|−1/t w1,B · ut,B . 1/s

(t−s)/st dx (5.4.15)

5.4 Imbedding inequalities for operators

151

In Lemma 3.1.1, let m = s/r, to have ut,B ≤ C8 |B|(m−t)/mt um,ρB .

(5.4.16)

Now Theorem 1.1.4 yields um,ρB

m 1/m = ρB |u|w1/s w−1/s dx ≤

ρB

|u|s wdx

1/s ρB

1 1/(r−1) w

(r−1)/s

(5.4.17)

dx

for all balls B with ρB ⊂ Ω. Next, since w ∈ Ar (Ω), it follows that 1/s

1/s

w1,B · 1/w1/(r−1),ρB r−1 1/s 1/(r−1) ≤ wdx (1/w) dx ρB ρB =

|ρB|r

1 |ρB|

ρB

wdx

1 |ρB|

ρB

1 1/(r−1) w

r−1 1/s

(5.4.18)

dx

≤ C9 |B|r/s . Combining (5.4.15), (5.14.16), (5.4.17), and (5.4.18), we have ∇(T u)s,B,w ≤ C10 |B||B|−1/t w1,B |B|(m−t)/mt um,ρB 1/s

≤ C10 |B||B|−1/m w1,B · 1/w1/(r−1),ρB us,ρB,w 1/s

1/s

(5.4.19)

≤ C11 |B|us,ρB,w for all balls B with ρB ⊂ Ω. Hence, (5.4.13) holds. At this point, we should notice that Theorem 5.4.3 can also be proved in a diﬀerent way. Indeed, we ﬁrst prove that inequality (5.4.5) holds for any Ar (Ω)-weight w(x), and then using the property of Ar (Ω)-weight, we can replace w(x) by wα (x) with 0 < α ≤ 1 as we did in the proof of inequality (5.3.6). However, the method that we have used to prove Theorem 5.4.3 requires some insight into the treatment of weights. Using the method similar to the proof of Theorem 5.4.3, we obtain T us,B,wα ≤ C|B|diam(B)us,ρB,wα , where α is any real number with 0 < α ≤ 1 and ρ > 1.

(5.4.20)

152

5 Imbedding theorems

Proof of Theorem 5.4.2. From (5.4.1), (5.4.5) , and (5.4.20), we have T uW 1,s (B),

wα −1

= diam(B)

T us,B,wα + ∇(T u)s,B,wα

≤ diam(B)−1 (C1 |B|diam(B)us,ρB,wα ) + C2 |B|us,ρB,wα ≤ C1 |B|us,ρB,wα + C2 |B|us,ρB,wα ≤ C3 |B|us,ρB,wα , which is equivalent to (5.4.4).

5.4.2 Some special cases Note that the parameter α in both Theorems 5.4.2 and 5.4.3 is any real number with 0 < α ≤ 1. Therefore, we can deduce diﬀerent versions of the weighted imbedding inequality by choosing diﬀerent values of α. For example, when t = 1 − α in Theorem 5.4.3 and dμ = w(x)dx, then inequality (5.4.5) becomes 1/s 1/s s −t s −t |∇(T u)| w dμ ≤ C|B| |u| w dμ . (5.4.21) B

ρB

If we choose α = 1/r in Theorem 5.4.3, then (5.4.5) reduces to 1/s

|∇(T u)|s w1/r dx

1/s

≤ C|B|

|u|s w1/r dx

B

.

(5.4.22)

ρB

If we choose α = 1/s in Theorem 5.4.3, then 0 < α < 1 since 1 < s < ∞, and (5.4.5) reduces to the following symmetric version: 1/s

|∇(T u)| w s

1/s

dx

1/s

≤ C|B|

B

|u| w s

1/s

dx

.

(5.4.23)

ρB

Finally, if we choose α = 1 in Theorem 5.4.3, we have the following weighted imbedding inequality: ∇(T u)s,B,w ≤ C|B|us,ρB,w .

(5.4.24)

Similarly, when α = 1 in Theorem 5.4.2, we have T uW 1,s (B),

w

≤ C|B|us,B,w .

(5.4.25)

5.4 Imbedding inequalities for operators

153

5.4.3 Global imbedding theorems We shall prove the following global Ar (Ω)-weighted imbedding theorem for the operators applied to the diﬀerential forms satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a domain Ω. Theorem 5.4.4. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that ∇(T u)s,Ω,wα ≤ Cus,Ω,wα ,

(5.4.26)

T uW 1,s (Ω),wα ≤ Cus,Ω,wα

(5.4.27)

for any real number α with 0 < α ≤ 1. Proof. From the facts on page 16 in [204], we know that any open subset Ω of Rn is the union of a sequence of mutually disjoint Whitney cubes. Also, cubes are convex. Thus, the deﬁnition of the homotopy operator T can be extended into any open subset Ω of Rn . Using (5.4.5) and the covering lemma (Theorem 1.5.3), we ﬁnd that ∇(T u)s,Ω,wα

1/s = Ω |∇(T u)|s wα dx 1/s ≤ Q∈V C1 |Q| ρQ |u|s wα dx ≤ C1 |Ω| ≤ C2 ≤ C3

Q∈V

Q∈V

Ω

Ω

ρQ

|u|s wα dx

|u|s wα dx

|u|s wα dx

1/s

(5.4.28)

1/s

1/s

= C3 us,Ω,wα since Ω is bounded. Thus, (5.4.26) holds. Similarly, using Theorem 1.5.3 and (5.4.20), we have T us,Ω,wα ≤ C4 diam(Ω)us,Ω,wα . Now combining (5.4.1), (5.4.28), and (5.4.29), we obtain

(5.4.29)

154

5 Imbedding theorems

T uW 1,s (Ω),

wα

= diam(Ω)−1 T us,Ω,wα + ∇(T u)s,Ω,wα ≤ C4 us,Ω,wα + C3 us,Ω,wα

(5.4.30)

≤ C5 us,Ω,wα and hence (5.4.27) holds. Remark. Similar to the local case, when α assumes some particular values in (5.4.26) and (5.4.27), we obtain diﬀerent versions of the global result. For example, if we choose α = 1, we ﬁnd that (5.4.26) and (5.4.27) reduce to ∇(T u)s,Ω,w ≤ Cus,Ω,w ,

(5.4.31)

≤ Cus,Ω,w ,

(5.4.32)

T uW 1,s (Ω),

w

respectively.

5.5 Other weighted cases In the previous section, we have discussed both local and global Ar (Ω)weighted imbedding theorems for diﬀerential forms satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a domain Ω. In this section, we study the Ar (λ, Ω)-weighted imbedding theorems for operators applied to the solutions of the nonhomogeneous A-harmonic equation (5.3.1).

5.5.1 Ls-estimates for T First, we recall the deﬁnition of Ar (λ, Ω)-weights which was introduced in [121]. A weight w(x) is called an Ar (λ, Ω)-weight for some r > 1 and λ > 0 in a domain Ω, write w ∈ Ar (λ, Ω), if w(x) > 0 a.e., and sup B

1 |B|

λ

w dx B

1 |B|

1/(r−1) (r−1) 1 dx <∞ w B

(5.5.1)

for any ball B ⊂ Ω. It is worth noticing that there exists some diﬀerence between Ar (λ, Ω)weights and Ar (λ)-weights. We say that a weight w(x) is an Ar (λ)-weight if inequality (5.5.1) holds for any ball B ⊂ Rn . The condition w(x) ∈ Ar (λ, Ω), however, requires that (5.5.1) holds for all balls B ⊂ Ω. We should also note

5.5 Other weighted cases

155

that Ar (1, Ω) = Ar (Ω) is the usual Ar (Ω)-weights which have been widely studied. The following results are obtained in [268] recently. Theorem 5.5.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that 1/s

|T u| w s

αλ

1/s

≤ C|B|diam(B)

dx

|u| w dx s

B

α

(5.5.2)

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Note that (5.5.2) can be written as (5.5.2)

T us,B,wαλ ≤ C|B|diam(B)us,ρB,wα .

Proof. Let t = s/(1 − α). Using the H¨ older inequality and Lemma 5.4.1, we have

1/s |T u|s wαλ dx B

s 1/s = B |T u|wαλ/s dx (5.5.3)

(t−s)/st ≤ T ut,B B wtαλ/(t−s) dx

α/s ≤ C1 |B|diam(B)ut,B B wλ dx . Choose m = s/(1 + α(r − 1)), so that m < s. Using Lemma 3.1.1, we obtain ut,B ≤ C2 |B|(m−t)/mt um,ρB ,

(5.5.4)

where ρ > 1. Substituting (5.5.4) into (5.5.3), we ﬁnd B

|T u|s wαλ dx

1/s

≤ C3 |B|diam(B)|B|

(m−t)/mt

um,ρB

λ

B

w dx

α/s

(5.5.5) .

Now since 1/m = 1/s+(s−m)/sm, by the H¨ older inequality again, we obtain um,ρB = =

|u|m dx ρB

ρB

1/m

|u|wα/s w−α/s

m

1/m dx

156

5 Imbedding theorems

≤

ρB

|u|s wα dx

1/s

1 1/(r−1) ρB

w

α(r−1)/s dx

(5.5.6)

for all balls B with ρB ⊂ Ω. Combining (5.5.5) and (5.5.6), it follows that B

|T u|s wαλ dx

1/s αλ/s

α/s

≤ C3 |B|diam(B)|B|(m−t)/mt wλ,B 1/w1/(r−1),ρB 1/s × ρB |u|s wα dx .

(5.5.7)

Since w ∈ Ar (λ, Ω), we have αλ/s

α/s

wλ,B · 1/w1/(r−1),ρB r−1 α/s λ 1/(r−1) ≤ w dx (1/w) dx ρB ρB |ρB|

=

r

1 |ρB|

λ

ρB

w dx

1 |ρB|

1 1/(r−1)

ρB

w

r−1 α/s

(5.5.8)

dx

≤ C4 |B|αr/s . Substituting (5.5.8) into (5.5.7), we ﬁnd that 1/s

|T u|s wαλ dx

1/s

≤ C|B|diam(B)

|u|s wα dx

B

(5.5.9)

ρB

for all balls B with ρB ⊂ Ω.

5.5.2 Ls-estimates for ∇ ◦ T Theorem 5.5.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that 1/s

|∇(T u)| w s

B

αλ

dx

1/s

≤ C|B|

|u| w dx s

α

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1.

(5.5.10)

5.5 Other weighted cases

157

Note that (5.5.10) can be written as (5.5.10)

∇(T u)s,B,wαλ ≤ C|B|us,ρB,wα . Proof. Let t = s/(1 − α). Using H¨ older’s inequality, we have

1/s |∇(T u)|s wαλ dx

s 1/s = B |∇(T u)|wαλ/s dx

B

≤ ∇(T u)t,B = ∇(T u)t,B

tαλ/(t−s)

B

w

B

wλ dx

α/s

dx

(t−s)/st

(5.5.11)

.

Thus, from Lemma 5.4.1, it follows that ∇(T u)t,B ≤ C1 |B|ut,B .

(5.5.12)

Set m = s/(1 + α(r − 1)), so that m < s. Substituting (5.5.12) into (5.5.11) and using Lemma 3.1.1, we have

1/s |∇(T u)|s wαλ dx

α/s ≤ C1 |B|ut,B B wλ dx

B

≤ C2 |B||B|(m−t)/mt um,ρB

(5.5.13) B

wλ dx

α/s

.

Using the H¨ older inequality again with 1/m = 1/s + (s − m)/sm, we ﬁnd um,ρB 1/m = ρB |u|m dx = ≤

ρB

|u|wα/s w−α/s

|u|s wα dx ρB

m

1/m

(5.5.14)

dx

1/s

1 1/(r−1) w

ρB

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Substituting (5.5.14) into (5.5.13), we obtain B

|∇(T u)|s wαλ dx

1/s

≤ C2 |B||B|(m−t)/mt ×

B

wλ dx

α/s

ρB

|u|s wα dx

1 1/(r−1) ρB

w

1/s (5.5.15) α(r−1)/s dx

.

158

5 Imbedding theorems

Now using the condition w ∈ Ar (λ, Ω), we obtain αλ/s

α/s

wλ,B · 1/w1/(r−1),ρB r−1 α/s λ 1/(r−1) ≤ w dx (1/w) dx ρB ρB |ρB|r

=

1 |ρB|

ρB

wλ dx

1 |ρB|

1 1/(r−1)

w

ρB

r−1 α/s

(5.5.16)

dx

≤ C3 |B|αr/s . Combining (5.5.16) and (5.5.15), we ﬁnd that 1/s

|∇(T u)| w s

αλ

dx

1/s

≤ C4 |B|

|u| w dx s

B

α

(5.5.17)

ρB

for all balls B with ρB ⊂ Ω. In Theorems 5.5.1 and 5.5.2, the parameters α and λ are any real numbers with 0 < α < 1 and λ > 0. Therefore, we can deduce diﬀerent versions of the weighted imbedding inequalities by choosing particular values of α and λ. In fact, some existing results are the special cases of our theorems. For example, when α = 1/r in Theorem 5.5.2, then (5.5.10) reduces to 1/s

|∇(T u)| w s

λ/r

1/s

≤ C|B|

dx

|u| w s

B

1/r

dx

.

(5.5.18)

ρB

If we choose α = 1/s in Theorem 5.5.2, then 0 < α < 1 since 1 < s < ∞, and (5.5.10) reduces to 1/s

|∇(T u)| w s

λ/s

dx

1/s

≤ C|B|

|u| w s

B

1/s

dx

,

(5.5.19)

ρB

where w ∈ Ar (λ, Ω). If w ∈ Ar (1, Ω), from (5.5.19) we obtain the following symmetric version: 1/s

|∇(T u)|s w1/s dx B

1/s

≤ C|B|

|u|s w1/s dx

.

(5.5.20)

ρB

5.5.3 Ar (λ, Ω)-weighted imbedding theorems Using Theorems 5.5.1 and 5.5.2, it is easy to prove the following local Ar (λ, Ω)-weighted imbedding inequality for diﬀerential forms under the homotopy operator T .

5.5 Other weighted cases

159

Theorem 5.5.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(5.5.21)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Proof. Applying (5.5.2) and (5.5.10) , we obtain T uW 1,s (B,∧l ),

wαλ

= diam(B)−1 T us,B,wαλ + ∇(T u)s,B,wαλ ≤ diam(B)−1 [C1 |B|diam(B)us,ρB,wα ] + C2 |B|us,ρB,wα ≤ C1 |B|us,ρB,wα + C2 |B|us,ρB,wα ≤ C3 |B|us,ρB,wα , which is equivalent to inequality (5.5.21). Using Theorem 5.5.3, we can prove the following Sobolev–Poincar´e imbedding inequality for diﬀerential forms. Theorem 5.5.4. Let du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn . Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that u − uB W 1,s (B),

wαλ

≤ C|B|dus,ρB,wα

(5.5.22)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Proof. Since ω = du ∈ Lsloc (Ω, ∧l+1 ) satisﬁes (5.3.1), using (5.5.21) and uB = u − T (du), we have u − uB W 1,s (B),

wαλ

= T (du)W 1,s (B),

wαλ

≤ C|B|dus,ρB,wα .

5.5.4 Some corollaries Choosing λ = 1, α = 1/s, and α = 1/λ with λ > 1 in Theorem 5.5.1, we have the following corollaries, respectively.

160

5 Imbedding theorems

Corollary 5.5.5. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s |T u| w dx s

1/s

≤ C|B|diam(B)

α

|u| w dx s

B

α

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Corollary 5.5.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s |T u|s wλ/s dx

1/s

≤ C|B|diam(B)

|u|s w1/s dx

B

ρB

for all balls B with ρB ⊂ Ω. Corollary 5.5.7. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 1. Then, there exists a constant C, independent of u, such that 1/s

|T u| wdx

1/s

≤ C|B|diam(B)

s

B

|u| w s

1/λ

dx

ρB

for all balls B with ρB ⊂ Ω. Next, choosing λ = 1 in Corollary 5.5.6, we have the following symmetric imbedding inequality. Corollary 5.5.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s |T u|s w1/s dx B

for all balls B with ρB ⊂ Ω.

1/s

≤ C|B|diam(B)

|u|s w1/s dx ρB

5.5 Other weighted cases

161

Now assume that w ∈ Ar (Ω) for some r > 1. Then, from Theorem 5.5.3, we have (5.5.23) T uW 1,s (B), wα ≤ C|B|us,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Putting α = 1/s, s > 1, in (5.5.23), we ﬁnd T uW 1,s (B),

w1/s

≤ C|B|us,ρB,w1/s .

(5.5.24)

Let λ > 1 and α = 1/λ in Theorem 5.5.3. Then, inequality (5.5.21) becomes (5.5.25) T uW 1,s (B), w ≤ C|B|us,ρB,w1/λ , where B is any ball with ρB ⊂ Ω and w ∈ Ar (λ, Ω) for some r > 1 and λ > 1. Setting α = 1/s in Theorem 5.5.3, we have the following corollary. Corollary 5.5.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

wλ/s

≤ C|B|us,ρB,w1/s

for all balls B with ρB ⊂ Ω. Similarly, from Theorem 5.5.4, we have the following Sobolev–Poincar´e imbedding inequalities for diﬀerential forms. Corollary 5.5.10. Let du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn . Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that u − uB W 1,s (B),

wα

≤ C|B|dus,ρB,wα

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Corollary 5.5.11. Let du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn . Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 1. Then, there exists a constant C, independent of u, such that u − uB W 1,s (B), for all balls B with ρB ⊂ Ω.

w

≤ C|B|dus,ρB,w1/λ

162

5 Imbedding theorems

Corollary 5.5.12. Let du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn . Assume that ρ > 1 and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that u − uB W 1,s (B),

wλ/s

≤ C|B|dus,ρB,w1/s

for all balls B with ρB ⊂ Ω. Choosing λ = 1 in Corollary 5.5.12, we have the following symmetric inequality. Corollary 5.5.13. Let du ∈ Lsloc (Ω, ∧l+1 ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn . Assume that ρ > 1 and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that u − uB W 1,s (B),

w1/s

≤ C|B|dus,ρB,w1/s

for all balls B with ρB ⊂ Ω.

5.5.5 Global imbedding theorems From Theorem 1.5.3 (covering lemma) and the local imbedding inequalities, we have the following global Ar (λ, Ω)-weighted imbedding inequalities in a bounded domain Ω. Theorem 5.5.14. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T us,Ω,wαλ ≤ Cus,Ω,wα

(5.5.26)

∇(T u)s,Ω,wαλ ≤ Cus,Ω,wα

(5.5.27)

and

for any real number α with 0 < α < 1.

5.5 Other weighted cases

163

Proof. Applying (5.5.2) and Theorem 1.5.3, we ﬁnd that Ω

≤

|T u|s wαλ dx

1/s

Q∈V

C1 |Q|diam(Q)

≤ C1 |Ω|diam(Ω) ≤ C2

|u|s wα dx

1/s ≤ C3 Ω |u|s wα dx , Q∈V

ρQ

Q∈V

|u| w dx s

ρQ

α

|u|s wα dx

1/s

1/s

1/s

Ω

which shows that (5.5.26) holds. From (5.5.10) and Theorem 1.5.3, we can prove (5.5.27) similarly. Theorem 5.5.15. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T uW 1,s (Ω),

wαλ

≤ Cus,Ω,wα

(5.5.28)

for any real number α with 0 < α < 1. Proof. Using (5.5.21) and Theorem 1.5.3, we ﬁnd that T uW 1,s (Ω), wαλ

≤ Q∈V T uW 1,s (Q), wαλ ≤ Q∈V (C1 |Q|us,ρQ,wα ) 1/s ≤ C1 |Ω| Q∈V ρQ |u|s wα dx ≤ C2

|u|s wα dx

1/s ≤ C3 Ω |u|s wα dx Q∈V

1/s

Ω

which shows that inequality (5.5.28) holds. Next, we shall prove the following global Sobolev–Poincar´e imbedding inequality for diﬀerential functions.

164

5 Imbedding theorems

Theorem 5.5.16. Let du ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a diﬀerential function satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ), l = 1, 2 . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that u − uQ0 W 1,s (Ω),

wαλ

≤ Cdus,Ω,wα

(5.5.29)

for any real number α with 0 < α < 1. Here Q0 is some cube in Ω. Proof. From the local Sobolev–Poincar´e imbedding inequality (5.5.22) and Theorem 1.5.3, we ﬁnd that u − uQ0 W 1,s (Ω), wαλ

≤ Q∈V u − uB W 1,s (Q), wαλ 1/s ≤ Q∈V C1 |Q| ρQ |du|s wα dx ≤ C1 |Ω| ≤ C2

Q∈V

|du|s wα dx ρQ

|du|s wα dx

1/s ≤ C3 Ω |du|s wα dx . Q∈V

1/s

1/s

Ω

Thus, the imbedding inequality (5.5.29) holds.

5.5.6 Aλ r (Ω)-weighted estimates B. Liu studied the Aλr (Ω)-weighted estimates in [182] and obtained some versions of Aλr (Ω)-weighted imbedding inequalities for diﬀerential forms satisfying the homogeneous A-harmonic equation. These inequalities are similar to the results presented in Theorems 5.5.1, 5.5.2, and 5.5.3 for the nonhomogeneous A-harmonic equation. Using the known results for the solutions of the nonhomogeneous A-harmonic equation, we can prove Theorems 5.5.17 and 5.5.18 with Aλr (Ω)-weights. Theorem 5.5.17. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

5.6 Compositions of operators

165

1/s |T u|s wα dx

|u|s wαλ dx

B

and

1/s

≤ C|B|diam(B) ρB

1/s |∇(T u)| w dx s

α

1/s

≤ C|B|

B

|u| w s

αλ

dx

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Theorem 5.5.18. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) be the operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

wα

≤ C|B|us,ρB,wαλ

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. As we have discussed in the previous sections, the parameters α and λ make Theorems 5.5.17 and 5.5.18 more ﬂexible and useful.

5.6 Compositions of operators In this section, we study diﬀerent versions of the Ls -estimates for the compositions of T , d, G and ∇, T , G acted on diﬀerential forms which are solutions of the nonhomogeneous A-harmonic equation in a domain Ω.

5.6.1 Ar (Ω)-weighted estimates for T ◦ d ◦ G Theorem 5.6.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, T (d(G(u))) ∈ Lsloc (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that T (d(G(u)))s,B,wα ≤ C|B|diam(B)us,ρB,wα

(5.6.1)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. The Ls -integrability of T (d(G(u))) follows from inequality (5.6.1), and hence it suﬃces to show that (5.6.1) holds. From Lemma 5.4.1 and (3.3.3),

166

5 Imbedding theorems

we obtain T (d(G(u)))s,B ≤ C1 |B|diam(B)d(G(u))s,B

(5.6.2)

≤ C2 |B|diam(B)us,B . We shall ﬁrst prove (5.6.1) for 0 < α < 1. Let t = s/(1 − α). Using Theorem 1.1.4, we have T (d(G(u)))s,B,wα

s 1/s = B |T (d(G(u)))|wα/s dx ≤ T (d(G(u)))t,B = T (d(G(u)))t,B

B

B

wtα/(t−s) dx wdx

α/s

(t−s)/st

(5.6.3)

.

Thus, from (5.6.2), it follows that T (d(G(u)))t,B ≤ C1 |B|diam(B)ut,B .

(5.6.4)

Choose m = s/(1 + α(r − 1)), so that m < s. Substituting (5.6.4) into (5.6.3) and using Lemma 3.1.1, we obtain T (d(G(u)))s,B,wα ≤ C1 |B|diam(B)ut,B

B

wdx

α/s

≤ C2 |B|diam(B)|B|(m−t)/mt um,ρB

(5.6.5) B

wdx

α/s

.

Now using the H¨ older inequality with 1/m = 1/s + (s − m)/sm, we ﬁnd that um,ρB 1/m = ρB |u|m dx = ≤

ρB

|u|wα/s w−α/s

|u|s wα dx ρB

m

1/m

(5.6.6)

dx

1/s

1 1/(r−1) ρB

w

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Substituting (5.6.6) into (5.6.5), we obtain

5.6 Compositions of operators

167

T (d(G(u)))s,B,wα ≤ C2 |B|diam(B)|B|(m−t)/mt ×

B

wdx

α/s

|u|s wα dx ρB

1 1/(r−1) ρB

w

1/s (5.6.7)

α(r−1)/s dx

.

Next, since w ∈ Ar (Ω), it is easy to conclude that α/s

α/s

w1,B · 1/w1/(r−1),ρB r−1 α/s 1/(r−1) ≤ wdx (1/w) dx ρB ρB =

|ρB|r

1 |ρB|

ρB

wdx

1 |ρB|

1 1/(r−1)

ρB

w

r−1 α/s

(5.6.8)

dx

≤ C3 |B|αr/s . Combining (5.6.8) and (5.6.7), we have T (d(G(u)))s,B,wα ≤ C4 |B|diam(B)us,ρB,wα

(5.6.9)

for all balls B with ρB ⊂ Ω. Thus, (5.6.1) holds for 0 < α < 1. Next, we shall prove (5.6.1) for α = 1, that is, we need to show that T (d(G(u)))s,B,w ≤ C|B|diam(B)us,ρB,w .

(5.6.10)

By Lemma 1.4.7, there exist constants β > 1 and C5 > 0, such that w β,B ≤ C5 |B|(1−β)/β w 1,B .

(5.6.11)

Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by the H¨older inequality, (5.6.2), and (5.6.11), it follows that

1/s |T (d(G(u)))|s wdx

s 1/s = B |T (d(G(u)))|w1/s dx

B

≤

B

|T (d(G(u)))|t dx

1/t B

w1/s

st/(t−s)

(5.6.12)

1/s

≤ C6 T (d(G(u)))t,B · wβ,B 1/s

≤ C6 |B|diam(B)ut,B · wβ,B 1/s

≤ C7 |B|diam(B)|B|(1−β)/βs w1,B · ut,B ≤ C7 |B|diam(B)|B|−1/t w1,B · ut,B . 1/s

(t−s)/st dx

168

5 Imbedding theorems

Set m = s/r in Lemma 3.1.1, to have ut,B ≤ C8 |B|(m−t)/mt um,ρB .

(5.6.13)

Using the H¨ older inequality again, we ﬁnd um,ρB

m 1/m = ρB |u|w1/s w−1/s dx ≤

ρB

|u|s wdx

1/s

1 1/(r−1)

ρB

w

(5.6.14)

(r−1)/s dx

.

Now since w ∈ Ar (Ω), we have 1/s

1/s

w1,B · 1/w1/(r−1),ρB r−1 1/s 1/(r−1) ≤ wdx (1/w) dx ρB ρB =

|ρB|r

1 |ρB|

ρB

wdx

1 |ρB|

1 1/(r−1)

ρB

w

r−1 1/s

(5.6.15)

dx

≤ C9 |B|r/s . Combining (5.6.12), (5.6.13), (5.6.14), and (5.6.15), we obtain T (d(G(u)))s,B,w ≤ C10 |B|diam(B)|B|−1/t w1,B |B|(m−t)/mt um,ρB 1/s

≤ C10 |B|diam(B)|B|−1/m w1,B · 1/w1/(r−1),ρB us,ρB,w 1/s

1/s

≤ C11 |B|diam(B)us,ρB,w for all balls B with ρB ⊂ Ω. Hence, (5.6.10) holds. Theorem 5.6.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (d(G(u)))W 1,s (B) ≤ C|B|us,B for all balls B with B ⊂ Ω. Proof. From (3.3.14), Lemma 5.4.1, and (3.3.3), we obtain

(5.6.16)

5.6 Compositions of operators

169

T (d(G(u)))W 1,s (B) = diam(B)−1 T (d(G(u)))s,B + ∇(T (d(G(u))))s,B ≤ diam(B)−1 · C1 |B|diam(B)d(G(u))s,B + C2 |B|d(G(u))s,B ≤ C3 |B|d(G(u))s,B ≤ C4 |B|us,B .

5.6.2 Ar (Ω)-weighted estimates for ∇ ◦ T ◦ G Similar to Theorem 5.6.1, we have the following result for ∇ ◦ T ◦ G. Theorem 5.6.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, ∇(T (G(u))) ∈ Lsloc (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that ∇(T (G(u)))s,B,wα ≤ C|B|us,ρB,wα

(5.6.17)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. From (5.6.17), it is clear that ∇(T (G(u))) ∈ Lsloc (Ω, ∧l ). Hence, we only need to prove (5.6.17). From Lemma 5.4.1 and (3.3.3), we ﬁnd that ∇(T (G(u)))s,B ≤ C|B|G(u)s,B ≤ C|B|us,B .

(5.6.18)

The rest of the proof is similar to that of Theorem 5.6.1, and hence omitted. Theorem 5.6.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (d(G(u)))W 1,s (B),wα ≤ C|B|us,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. Using (3.3.15), and Theorems 5.6.1 and 5.6.3, we obtain

(5.6.19)

170

5 Imbedding theorems

T (d(G(u)))W 1,s (B),wα = diam(B)−1 T (d(G(u)))s,B,wα + ∇(T (d(G(u))))s,B,wα ≤ diam(B)−1 C1 diam(B)|B|us,ρ1 B,wα + C2 |B|us,ρ2 B,wα

(5.6.20)

≤ C3 |B|us,ρB,wα , where ρ = max{ρ1 , ρ2 }. Next, as applications of the local results, we prove the following global Ar (Ω)-weighted estimates for the compositions of operators. Theorem 5.6.5. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that T (d(G(u)))s,Ω,wα ≤ C|Ω|diam(Ω)us,Ω,wα

(5.6.21)

for any real number α with 0 < α ≤ 1. Proof. Let V = {Qi } be a Whitney cover of Ω. Applying the covering lemma and Theorem 5.6.1, we obtain T (d(G(u)))s,Ω,wα ≤ Q∈V T (d(G(u)))s,Q,wα ≤ Q∈V (C1 |Q|diam(Q)us,ρQ,wα ) ≤ Q∈V (C2 |Ω|diam(Ω)us,Ω,wα )

(5.6.22)

≤ (C2 |Ω|diam(Ω)us,Ω,wα ) · N ≤ C3 |Ω|diam(Ω)us,Ω,wα . Theorem 5.6.6. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that ∇(T (G(u)))s,Ω,wα ≤ C|Ω|us,Ω,wα for any real number α with 0 < α ≤ 1.

(5.6.23)

5.6 Compositions of operators

171

Proof. The proof requires Theorems 1.5.3 and 5.6.3 and is similar to that of Theorem 5.6.5.

5.6.3 Imbedding for T ◦ d ◦ G At this stage, we are ready to prove the following global Ar (Ω)-weighted Sobolev–Poincar´e imbedding theorem for the composition of operators applied to the solutions of the nonhomogeneous A-harmonic equation. Theorem 5.6.7. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω. Assume that T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1) and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, T (d(G(u))) ∈ W 1,s (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that (5.6.24) T (d(G(u)))W 1,s (Ω),wα ≤ C|Ω|us,Ω,wα for any real number α with 0 < α ≤ 1. Proof. Applying (3.3.15) and (3.3.3), and Theorems 5.6.5 and 5.6.6, we ﬁnd that T (d(G(u)))W 1,s (Ω),wα = diam(Ω)−1 T (d(G(u)))s,Ω,wα + ∇(T (d(G(u))))s,Ω,wα ≤ C1 diam(Ω)−1 · |Ω|diam(Ω)us,Ω,wα + C2 |Ω|us,Ω,wα ≤ C1 |Ω|us,Ω,wα + C2 |Ω|us,Ω,wα ≤ C3 |Ω|us,Ω,wα .

For α = 1, Theorems 5.6.5, 5.6.6, and 5.6.7 reduce to the following interesting corollaries. Corollary 5.6.8. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that T (d(G(u)))s,Ω,w ≤ C|Ω|diam(Ω)us,Ω,w .

(5.6.25)

Corollary 5.6.9. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω and

172

5 Imbedding theorems

T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that ∇(T (G(u)))s,Ω,w ≤ C|Ω|us,Ω,w .

(5.6.26)

Corollary 5.6.10. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω. Assume that T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is a homotopy operator deﬁned in (1.5.1) and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, T (d(G(u))) ∈ W 1,s (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that (5.6.27) T (d(G(u)))W 1,s (Ω),w ≤ C|Ω|us,Ω,w . Next, since 1 < s < ∞, when α = 1/s in (5.6.21), (5.6.23), and (5.6.24), we obtain, respectively T (d(G(u)))s,Ω,w1/s ≤ C|Ω|diam(Ω)us,Ω,w1/s , ∇(T (G(u)))s,Ω,w1/s ≤ C|Ω|us,Ω,w1/s , T (d(G(u)))W 1,s (Ω),w1/s ≤ C|Ω|us,Ω,w1/s . Similarly, when α = s/(r + s) in (5.6.21), (5.6.23), and (5.6.24), we ﬁnd that T (d(G(u)))s,Ω,ws/(r+s) ≤ C|Ω|diam(Ω)us,Ω,ws/(r+s) , ∇(T (G(u)))s,Ω,ws/(r+s) ≤ C|Ω|us,Ω,ws/(r+s) , T (d(G(u)))W 1,s (Ω),ws/(r+s) ≤ C|Ω|us,Ω,ws/(r+s) .

Finally, from Theorems 5.6.5 and 5.6.6, the following corollary for the compositions of operators immediately follows. Corollary 5.6.11. The compositions T ◦ d ◦ G and ∇ ◦ T ◦ G are bounded operators if Ω is a bounded, convex domain.

5.7 Two-weight cases We have already discussed various versions of the Ls -estimates and imbedding inequalities for the solutions of the A-harmonic equation and the related

5.7 Two-weight cases

173

operators applied to these solutions. In this section, we study the twoweighted imbedding inequalities.

5.7.1 Two-weight imbedding for the operator T We begin with the following local weighted imbedding theorem for diﬀerential forms under the homotopy operator T . Theorem 5.7.1. Assume that u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, is a diﬀerential form satisfying the nonhomogeneous A-harmonic equation (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1). If ρ > 1 and (w1 , w2 ) ∈ Ar (Ω) for some r > 1, then there exists a constant C, independent of u, such that T uW 1,s (B),

w1α

≤ C|B|us,ρB,w2α

(5.7.1)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Moreover, if (w1 , w2 ) ∈ Ar (Ω) and w1 ∈ RH(Ω), we have T uW 1,s (B),

w1

≤ C|B|us,ρB,w2

(5.7.2)

for all balls B with ρB ⊂ Ω. In order to prove Theorem 5.7.1, ﬁrst we need to prove the following twoweighted imbedding inequality. Theorem 5.7.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/s |∇(T u)|s w1α dx ≤ C|B| |u|s w2α dx (5.7.3) B

ρB

for any real number α with 0 < α < 1. Moreover, if (w1 , w2 ) ∈ Ar (Ω) and w1 ∈ RH(Ω), we have 1/s 1/s |∇(T u)|s w1 dx ≤ C|B| |u|s w2 dx (5.7.4) B

ρB

for all balls B with ρB ⊂ Ω. Note that (5.7.3) can be written as ∇(T u)s,B,w1α ≤ C|B|us,ρB,w2α .

(5.7.4)

174

5 Imbedding theorems

Proof. First, we shall prove that inequality (5.7.3) holds for 0 < α < 1. Let t = s/(1 − α). From Theorem 1.1.4, we ﬁnd that

1/s |∇(T u)|s w1α dx s 1/s α/s = B |∇(T u)|w1 dx

B

≤ ∇(T u)t,B

tα/(t−s)

B

w1

B

w1 dx

= ∇(T u)t,B

α/s

(t−s)/st

(5.7.5)

dx .

Thus, from Lemma 5.4.1, it follows that ∇(T u)t,B ≤ C1 |B|ut,B .

(5.7.6)

Choose m = s/(1 + α(r − 1)), so that m < s. Substituting (5.7.6) into (5.7.5) and using Lemma 3.1.1, we obtain

|∇(T u)|s w1α dx

B

≤ C1 |B|ut,B

1/s

w1 dx

B

α/s

≤ C2 |B||B|(m−t)/mt um,ρB

(5.7.7)

w1 dx

B

α/s

.

Now from Theorem 1.1.4 with 1/m = 1/s + (s − m)/sm, we have um,ρB 1/m = ρB |u|m dx = ≤

−α/s

α/s

|u|w2 w2

ρB

|u|

s

ρB

w2α dx

m

1/m

(5.7.8)

dx

1/s

ρB

1 w2

α(r−1)/s

1/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (5.7.8) into (5.7.7), it follows that B

|∇(T u)|s w1α dx

1/s

≤ C2 |B||B|(m−t)/mt ×

B

w1 dx

|u|s w2α dx ρB

α/s

ρB

1 w2

1/s (5.7.9)

α(r−1)/s

1/(r−1) dx

.

5.7 Two-weight cases

175

Next, since (w1 , w2 ) ∈ Ar (Ω), we ﬁnd that α/s

α/s

w1 1,B · 1/w2 1/(r−1),ρB r−1 α/s 1/(r−1) ≤ w dx (1/w2 ) dx ρB 1 ρB =

|ρB|

r

1 |ρB|

ρB

w1 dx

1 |ρB|

ρB

1 w2

r−1 α/s (5.7.10)

1/(r−1) dx

≤ C3 |B|αr/s . Combination of (5.7.10) and (5.7.9) yields 1/s s α |∇(T u)| w1 dx ≤ C4 |B| B

1/s |u|

s

w2α dx

(5.7.11)

ρB

for all balls B with ρB ⊂ Ω. Thus, (5.7.3) holds if 0 < α < 1. Next, we show that (5.7.4) holds if w1 ∈ RH(Ω). Since w1 ∈ RH(Ω), there exist constants β > 1 and C5 > 0, such that w1 β,B ≤ C5 |B|(1−β)/β w1 1,B

(5.7.12)

for any cube or ball B ⊂ Rn . Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by Theorem 1.1.4, Lemma 5.4.1, and (5.7.12), we obtain

1/s |∇(T u)|s w1 dx s 1/s 1/s = B |∇(T u)|w1 dx

B

≤

|∇(T u)|t dx B

1/t B

1/s

(t−s)/st

st/(t−s)

w1

dx

1/s

≤ C6 ∇(T u)t,B · w1 β,B

(5.7.13)

1/s

≤ C6 |B|ut,B · w1 β,B 1/s

≤ C7 |B||B|(1−β)/βs w1 1,B · ut,B ≤ C7 |B||B|−1/t w1 1,B · ut,B . 1/s

Now, from Lemma 3.1.1 with m = s/r, we have ut,B ≤ C8 |B|(m−t)/mt um,ρB ,

(5.7.14)

176

5 Imbedding theorems

and hence from Theorem 1.1.4, we ﬁnd that um,ρB = ≤

ρB

−1/s

1/s

|u|w2 w2

ρB

|u|s w2 dx

m

1/m dx

1/s

ρB

1 w2

(r−1)/s

1/(r−1)

(5.7.15)

dx

for all balls B with ρB ⊂ Ω. Using the condition (w1 , w2 ) ∈ Ar (Ω), we have 1/s

1/s

w1 1,B · 1/w2 1/(r−1),ρB ≤

w dx ρB 1

=

|ρB|r

1 |ρB|

(1/w2 )1/(r−1) dx ρB

ρB

w1 dx

1 |ρB|

r−1 1/s

ρB

1 w2

r−1 1/s (5.7.16)

1/(r−1) dx

≤ C9 |B|r/s . Combining (5.7.13), (5.7.14), (5.7.15), and(5.7.16), if follows that ∇(T u)s,B,w1 ≤ C10 |B||B|−1/t w1 1,B |B|(m−t)/mt um,ρB 1/s

≤ C10 |B||B|−1/m w1 1,B · 1/w2 1/(r−1),ρB us,ρB,w2 1/s

1/s

(5.7.17)

≤ C11 |B|us,ρB,w2 for all balls B with ρB ⊂ Ω. Hence, (5.7.4) holds. Using the method similar to the proof of Theorem 5.7.2, we obtain T us,B,w1α ≤ C|B|diam(B)us,ρB,w2α ,

(5.7.18)

where (w1 , w2 ) ∈ Ar (Ω) for some r > 1, w1 ∈ RH(Ω), and α is any real number with 0 < α ≤ 1 and ρ > 1.

5.7 Two-weight cases

177

Proof of Theorem 5.7.1. From (5.4.1), (5.7.4) , and (5.7.18), we have T uW 1,s (B),

w1α

= diam(B)−1 T us,B,w1α + ∇(T u)s,B,w1α

≤ diam(B)−1 C1 |B|diam(B)us,ρB,w2α + C2 |B|us,ρB,w2α ≤ C1 |B|us,ρB,w2α + C2 |B|us,ρB,w2α ≤ C3 |B|us,ρB,w2α which is equivalent to (5.7.1). Note that the parameter α in both Theorems 5.7.1 and 5.7.2 is any real number with 0 < α < 1. Therefore, we can have diﬀerent versions of the weighted imbedding inequality by choosing some particular values of α. For example, when t = 1 − α, dμ = w1 (x)dx, and dν = w2 (x)dx, then inequality (5.7.3) becomes

|∇(T u)|s w1−t dμ

1/s

|u|s w2−t dν

≤ C|B|

B

1/s .

(5.7.19)

ρB

If we choose α = 1/r, then (5.7.3) reduces to 1/s

|∇(T u)|

s

1/r w1 dx

1/s

≤ C|B|

|u|

s

B

1/r w2 dx

.

(5.7.20)

ρB

Similarly, if we choose α = 1/s, then 0 < α < 1 since 1 < s < ∞, and (5.7.3) reduces to the following symmetric version: 1/s

|∇(T u)|

s

1/s w1 dx

1/s

≤ C|B|

B

|u|

s

1/s w2 dx

.

(5.7.21)

ρB

Finally, if we choose α = r/(r + s) in Theorem 5.7.2, we have the following weighted imbedding inequality: ∇(T u)s,B,wr/(r+s) ≤ C|B|us,ρB,wr/(r+s) . 1

(5.7.22)

2

Similarly, selecting α = s/(r + s) in Theorem 5.7.1, we have T uW 1,s (B),

s/(r+s)

w1

≤ C|B|us,B,ws/(r+s) . 2

(5.7.23)

178

5 Imbedding theorems

5.7.2 Ar,λ(E)-weighted imbedding Theorem 5.7.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |∇(T u)|s w1α dx

1/s

≤ C|B|

|u|s w2α dx

B

(5.7.24)

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that (5.7.24) can be written as ∇(T u)s,B,w1α ≤ C|B|us,ρB,w2α .

(5.7.25)

Proof. Choose t = λs/(λ − α). Since 1/s = 1/t + (t − s)/st, from the H¨older inequality, we obtain B

= ≤

|∇(T u)|s w1α dx

1/s

B

α/s

|∇(T u)|w1

|∇(T u)|t dx B

≤ ∇(T u)t,B ·

s

1/s dx

1/t B

B

w1λ dx

α/s

(t−s)/st

st/(t−s)

w1

(5.7.26)

dx

α/λs

for all balls B ⊂ Ω. Thus, from Lemma 5.4.1, it follows that ∇(T u)t,B ≤ C1 |B|ut,B .

(5.7.27)

Choose m = λs/(λ + α(r − 1)), so that m < s < t. Substituting (5.7.27) into (5.7.26) and using Lemma 3.1.1, we ﬁnd that B

|∇(T u)|s w1α dx

≤ C1 |B|ut,B

1/s

B

w1λ dx

α/λs

≤ C2 |B||B|(m−t)/mt um,ρB

(5.7.28) B

w1λ dx

α/λs

.

5.7 Two-weight cases

179

Now, using Theorem 1.1.4 with 1/m = 1/s + (s − m)/sm, we have um,ρB

=

ρB

=

|u|m dx

m

−α/s

α/s

|u|w2 w2

ρB

≤

1/m

|u|s w2α dx ρB

1/m

(5.7.29)

dx

1/s

ρB

1 w2

α(r−1)/λs

λ/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (5.7.29) into (5.7.28), it follows that

|∇(T u)|s w1α dx

B

1/s

≤ C3 |B||B|(m−t)/mt ×

|u|s w2α dx ρB

B

w1λ dx

α/λs ρB

1 w2

1/s (5.7.30)

α(r−1)/λs

λ/(r−1) dx

.

Next, using the condition (w1 , w2 ) ∈ Ar,λ (Ω), we obtain

wλ dx B 1 ≤

α/λs ρB

wλ dx ρB 1

= |ρB|r

1 |ρB|

ρB

1 w2

α(r−1)/λs

λ/(r−1) dx

(1/w2 )λ/(r−1) dx ρB w1λ dx

1 |ρB|

ρB

r−1 α/λs (5.7.31) 1 w2λ

1 r−1

r−1 α/λs dx

≤ C4 |B|αr/λs . Combining (5.7.30) and (5.7.31), we have 1/s

|∇(T u)|s w1α dx B

1/s

≤ C|B|

|u|s w2α dx

(5.7.32)

ρB

for all balls B with ρB ⊂ Ω. We are now ready to prove the following imbedding inequality with two-weights.

180

5 Imbedding theorems

Theorem 5.7.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

w1α

≤ C|B|us,ρB,w2α

(5.7.33)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Proof. Using the same method developed in the proof of Theorem 5.7.3, we have T us,B,w1α ≤ C1 |B|diam(B)us,ρB,w2α ,

(5.7.34)

where α is any real number with 0 < α < λ and ρ > 1. Combining (1.1.1), (5.7.25), and (5.7.34), we obtain T uW 1,s (B),

w1α

= diam(B)−1 T us,B,w1α + ∇(T u)s,B,w1α

≤ diam(B)−1 C2 |B|diam(B)us,ρB,w2α + C3 |B|us,ρB,w2α ≤ C2 |B|us,ρB,w2α + C3 |B|us,ρB,w2α ≤ C4 |B|us,ρB,w2α which is equivalent to (5.7.33). Note that Theorem 5.7.3 contains two weights, w1 (x) and w2 (x), and two parameters, λ and α. This makes the imbedding inequalities more ﬂexible and useful. In fact, many existing versions of the imbedding inequality are special cases of Theorem 5.7.3 with suitable choices of weights and parameters. For example, if we consider the case (w1 , w2 ) ∈ Ar,λ (Ω) for some λ > 1 in Theorem 5.7.3, then we can choose α = 1 and obtain the following corollary. Corollary 5.7.5. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ > 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that 1/s

|∇(T u)| w1 dx s

B

for all balls B with ρB ⊂ Ω.

1/s

≤ C|B|

|u| w2 dx s

ρB

(5.7.35)

5.7 Two-weight cases

181

If we let w1 (x) = w2 (x) = w(x) in (5.7.35), we obtain 1/s

|∇(T u)| wdx

1/s

≤ C|B|

s

|u| wdx s

B

,

(5.7.36)

ρB

where the weight w(x) satisﬁes sup B⊂E

1 |B|

1/λr

(w)λ dx B

1 |B|

λr /r 1/λr 1 dx < ∞, w B

which is a generalization of the usual Ar -weights. We know that the Ar,λ (Ω)-weight reduces to the usual Ar (Ω)-weight if w1 (x) = w2 (x) and λ = 1. Hence, when w1 (x) = w2 (x) = w(x) and λ = 1 in Theorem 5.7.3, it reduces to the following local Ar (Ω)-weighted imbedding inequality. Theorem 5.7.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some r with 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |∇(T u)|s wα dx

1/s

≤ C|B|

|u|s wα dx

B

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Similarly, if we choose α = 1/s in Theorem 5.7.3, then inequality (5.7.25) reduces to 1/s 1/s 1/s 1/s |∇(T u)|s w1 dx ≤ C|B| |u|s w2 dx . (5.7.37) B

ρB

Finally, for α = 1/r in Theorem 5.7.3, inequality (5.7.25) reduces to 1/s

1/r

|∇(T u)|s w1 dx B

1/s 1/r

≤ C|B|

|u|s w2 dx

.

(5.7.38)

ρB

5.7.3 Ar (λ, E)-weighted imbedding Theorem 5.7.7. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (5.3.1) in a bounded,

182

5 Imbedding theorems

convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that 1/s

|∇(T u)|s w1αλ dx

1/s

≤ C|B|

|u|s w2α dx

B

(5.7.39)

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Note that (5.7.39) can be written as (5.7.39)

∇(T u)s,B,w1αλ ≤ C|B|us,ρB,w2α .

Proof. Select t = s/(1−α), so that 1 < s < t. Noticing 1/s = 1/t+(t−s)/st and using H¨ older’s inequality, we obtain B

= ≤

|∇(T u)|s w1αλ dx

1/s

B

αλ/s

|∇(T u)|w1

|∇(T u)|t dx B

≤ ∇(T u)t,B ·

s

1/s dx

1/t B

B

w1λ dx

αλ/s

(t−s)/st

st/(t−s)

w1

(5.7.40)

dx

α/s

for all balls B ⊂ Ω. Applying Lemma 5.4.1, we have ∇(T u)t,B ≤ C1 |B|ut,B .

(5.7.41)

Choose m = s/(1 + α(r − 1)), so that m < s < t. Substituting (5.7.41) into (5.7.40) and using Lemma 3.1.1, we obtain B

|∇(T u)|s w1αλ dx

≤ C1 |B|ut,B

1/s

B

w1λ dx

α/s

≤ C2 |B||B|(m−t)/mt um,ρB

(5.7.42) B

w1λ dx

α/s

.

5.7 Two-weight cases

183

Now using the H¨ older inequality with 1/m = 1/s + (s − m)/sm, we ﬁnd that um,ρB 1/m = ρB |u|m dx

=

ρB

≤

−α/s

α/s

|u|w2 w2

|u|s w2α dx ρB

m

1/m

(5.7.43)

dx

1/s

ρB

1 w2

α(r−1)/s

1/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (5.7.43) into (5.7.42), it follows that B

|∇(T u)|s w1αλ dx

1/s

≤ C3 |B||B|(m−t)/mt ×

|u|s w2α dx ρB

w1λ dx

B

α/s ρB

1 w2

1/s (5.7.44)

α(r−1)/s

1/(r−1) dx

.

Next, using the condition (w1 (x), w2 (x)) ∈ Ar (λ, Ω), we obtain

α/s wλ dx B 1 ≤

ρB

wλ dx ρB 1

= |ρB|r

1 w2

ρB

dx

1/(r−1)

ρB

1 |ρB|

α(r−1)/s

1/(r−1)

r−1 α/s

(1/w2 )

dx

w1λ dx

1 |ρB|

(5.7.45) ρB

1 w2

1/(r−1)

r−1 α/s dx

≤ C4 |B|αr/s . Combining (5.7.44) and (5.7.45), we ﬁnd that 1/s

|∇(T u)|

s

w1αλ dx

B

1/s

≤ C|B|

|u|

s

w2α dx

(5.7.46)

ρB

for all balls B with ρB ⊂ Ω. Applying Theorem 5.7.7 and the same method used in the proof of Theorem 5.7.4, we can prove the following imbedding inequality with two-weights in the weight class Ar (λ, Ω).

184

5 Imbedding theorems

Theorem 5.7.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (5.3.1) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that (5.7.47) T uW 1,s (B), w1αλ ≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1.

5.7.4 Global imbedding theorem It should be noticed that, using the covering lemma, many of the above local imbedding inequalities can be extended to the global cases on the bounded, convex domains. For example, Theorems 5.7.7 and 5.7.8 can be stated as follows. Theorem 5.7.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (5.3.1) on the bounded, convex domain Ω in Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |∇(T u)|

s

w1αλ dx

≤ C|Ω|

Ω

1/s |u|

s

w2α dx

(5.7.48)

Ω

for any real number α with 0 < α < 1. Theorem 5.7.10. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (5.3.1) on the bounded, convex domain Ω in Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that (5.7.49) T uW 1,s (Ω), w1αλ ≤ C|Ω|us,Ω,w2α for any real number α with 0 < α < 1. Using Theorem 3.3.8 and imitating the proof of Theorem 3.2.10, we obtain the following global Sobolev imbedding inequality for Green’s operator that is applied to the solutions of the nonhomogeneous A-harmonic equation in the John domain, see [103].

5.7 Two-weight cases

185

Theorem 5.7.11. Let u ∈ D (Ω, ∧0 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10), G be Green’s operator, and w ∈ Ar (Ω) for some 1 < r < ∞. Assume that s is a ﬁxed exponent associated with the Aharmonic equation (1.2.10), r < s < ∞. Then, there exists a constant C, independent of u, such that G(u) − (G(u))Q0 W 1,s (Ω),w ≤ Cdus,Ω,w for any δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube.

(5.7.50)

Chapter 6

Reverse H¨ older inequalities

In this chapter, we will present various versions of the reverse H¨ older inequality which serve as powerful tools in mathematical analysis. The original study of the reverse H¨older inequality can be traced back in Muckenhoupt’s work in [145]. During recent years, diﬀerent versions of the reverse H¨older inequality have been established for diﬀerent classes of functions, such as eigenfunctions of linear second-order elliptic operators [281], functions with discrete-time variable [282], and continuous exponential martingales [119].

6.1 Preliminaries We begin this chapter with some preliminaries which form the bases for the study of the reverse H¨ older inequalities. These results will be extended to the solutions of the A-harmonic equation in this chapter.

6.1.1 Gehring’s lemma We ﬁrst state Gehring’s lemma, which plays a crucial role in the study of the reverse H¨older inequalities for Ar (Ω)-weights. This lemma has also been used to improve the degree of integrability of solutions of certain partial diﬀerential equations. Theorem 6.1.1. Let 1 < p < ∞, and f be a nonnegative measurable function on a domain Ω ⊂ Rn satisfying 1 |B|

f p dx ≤ Cp B

1 |B|

p

f dx

(6.1.1)

B

R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 6,

187

188

6 Reverse H¨ older inequalities

for all balls B with σB ⊂ Ω, where Cp is a constant independent of the balls B and σ > 1. Then, there exist another exponent s > p and a constant Cs depending only on n, p, and Cp such that s 1 1 f s dx ≤ Cs f dx (6.1.2) |B| B |B| B for every ball B ⊂ Ω. Remark. Let s, t, σ, and C1 be some positive numbers with 0 < t < s < ∞ and σ > 1. If 1/s 1/t 1 1 s t f dx ≤ C1 f dx |B| B |B| σB holds for any ball B with σB ⊂ Ω, then there exists ε > 0 such that

1 |B|

1/(s+ε)

f

s+ε

B

dx

≤ C1

1 |B|

1/r

r

f dx σB

for any number r > 0 and any ball B with σB ⊂ Ω. In 1985, C. Nolder proved the following theorem which can be used to establish the global integrability of measurable functions in a John domain Ω. Theorem 6.1.2. Suppose that f and g are measurable functions in a John domain Ω, q > 0, σ > 1, and w ∈ Ar (Ω). If there exists a constant A such that q |f − fB,μ | dμ ≤ A |g|q dμ (6.1.3) B

σB

for all balls B with σB ⊂ Ω, then there is a constant B, independent of f and g, such that |f − fB0 ,μ |q dμ ≤ B |g|q dμ (6.1.4) Ω

Ω

for some ball B0 ⊂ Ω.

6.1.2 Inequalities for supersolutions In this chapter, we shall focus our attention to diﬀerent versions of the weak H¨ older inequality for the solutions of the A-harmonic equation. For this, ﬁrst we shall state the weak H¨older inequality for the positive supersolutions. 1,p (Ω, μ) is called Recall that a function u in the weighted Sobolev space Wloc a (weak) solution of the A-harmonic equation

6.2 The ﬁrst weighted case

189

−div A(x, ∇u) = 0

(6.1.5)

in Ω if

A(x, ∇u) · ∇ϕ(x)dx = 0

(6.1.6)

Ω 1,p whenever ϕ ∈ C0∞ (Ω). A function u in the weighted Sobolev space Wloc (Ω, μ) is called a supersolution of the A-harmonic equation (6.1.5) in Ω if

−div A(x, ∇u) ≥ 0 weakly in Ω, i.e.,

A(x, ∇u) · ∇ϕ(x)dx ≥ 0

(6.1.7)

Ω

whenever ϕ ∈ C0∞ (Ω) is nonnegative. A function v is a subsolution of (6.1.5) if −v is a supersolution of (6.1.5). We shall need the following result which has been established in [59]. Theorem 6.1.3. Suppose that u is a positive supersolution in 2B, where B is a ball. If 0 < q ≤ s < κ(p − 1), then

1 μ(B)

1s

s

u dμ

≤C

B

1 μ(2B)

q

u dμ

q1 ,

2B

where C is a constant independent of u and

κ=

⎧ n ⎨ n−p

if

p
⎩

if

p ≥ n.

2

Now we state the following reverse H¨older inequality for Ar (Ω)-weights. Theorem 6.1.4. If w ∈ Ar (Ω), then there exist constants β > 1 and C, such that β1 1 1 β w dx ≤C wdx (6.1.8) |B| B |B| B for all balls B ⊂ Ω.

6.2 The ﬁrst weighted case In [71], C. Nolder proved the weak reverse H¨ older inequality which has played an important role in the establishment of the integral estimates for

190

6 Reverse H¨ older inequalities

A-harmonic tensors. Nolder’s result has been introduced earlier in this monograph; however, for the convenience of readers, we restate the weak reverse H¨ older inequality as follows. Theorem 6.2.1. Let u be an A-harmonic tensor in Ω, σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that us,B ≤ C|B|(t−s)/st ut,σB for all balls or cubes B with σB ⊂ Ω.

6.2.1 Ar (Ω)-weighted inequalities In 1999, S. Ding [267] extended the above weak reverse H¨ older inequality to the following Ar (Ω)-weighted version. Theorem 6.2.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s

|u| wdx

≤ C|B|

s

B

1/t |u| w

(t−s)/st

t

t/s

dx

(6.2.1)

σB

for all balls B with σB ⊂ Ω. Proof. By Theorem 6.1.4, there exist constants β > 1 and C1 > 0, such that w β,B ≤ C1 |B|(1−β)/β w 1,B

(6.2.2)

for any cube or ball B ⊂ Rn . Select k = sβ/(β − 1), so that s < k and β = k/(k − s). From (6.2.2) and the H¨ older inequality, we obtain B

= ≤

|u|s wdx B

1/s

(|u|w1/s )s dx

|u|k dx B

1/s

1/k B

w1/s

ks/(k−s)

1/s

≤ uk,B · wβ,B 1/s

≤ C2 |B|(1−β)/βs w1,B uk,B = C2 |B|−1/k w1,B uk,B 1/s

(k−s)/ks dx (6.2.3)

6.2 The ﬁrst weighted case

191

for all balls B with B ⊂ Ω. Choosing m = st/(s + t(r − 1)), then from Theorem 6.2.1, we have uk,B ≤ C3 |B|(m−k)/km um,σB .

(6.2.4)

Combination of (6.2.3) and (6.2.4) gives us,B,w ≤ C4 |B|−1/m w1,B um,σB . 1/s

(6.2.5)

Now since m < t, using the H¨ older inequality, it follows that um,σB

m 1/m = σB |u|w1/s w−1/s dx ≤

|u|t wt/s dx σB

1/t

1 mt/(s(t−m)) w

σB

1/s

≤ 1/wmt/(s(t−m)),σB

σB

|u|t wt/s dx

1/t

(t−m)/mt

(6.2.6)

dx .

From the choice of m, we have r − 1 = s(t − m)/mt. Since w ∈ Ar , we ﬁnd that 1/s

1/s

w1,B · 1/wmt/(s(t−m)),σB

1 mt/(s(t−m)) s(t−m)/mt 1/s = wdx dx B σB w ≤

|σB|1+s(t−m)/mt

1 |σB|

wdx B

1 |σB|

1 1/(r−1)

σB

w

r−1 1/s dx

≤ C5 |B|1/s+1/m−1/t . (6.2.7) Now the combination of (6.2.5), (6.2.6), and (6.2.7) yields us,B,w ≤ C4 |B|−1/m w1,B 1/wmt/(s(t−m)),σB 1/s

≤ C6 |B|

1/s−1/t σB

1/s

|u|t wt/s dx

1/t

σB

|u|t wt/s dx

1/t (6.2.8)

.

It is easy to see that (6.2.8) is equivalent to inequality (6.2.1). In 1999, B. Liu and S. Ding established the following local Ar (Ω)-weighted weak reverse H¨older inequality for A-harmonic tensors on any subdomain of Ω ⊂ Rn .

192

6 Reverse H¨ older inequalities

Theorem 6.2.3. Let u be an A-harmonic tensor in a domain Ω ⊂Rn . Assume that 0 < s, t < ∞, σ > 1, and w ∈ Ar for some r > 1, then there exists a constant C, independent of u, such that

1 μ(B)

1/s

s

|u| wdx

≤C

B

1 μ(σB)

1/t

t

|u| wdx

(6.2.9)

σB

for all balls B with σB ⊂ Ω. Proof. Since w ∈ Ar (Ω) for some r > 1, from Theorem 6.1.4, there exists a constant β > 1, such that (6.1.8) holds. For any s > 0, we can write β−1 1 1 = + . s βs βs Thus, by the H¨ older inequality, there exists a constant C1 such that

1 s |u| wdx s 1s 1 = B (|u| w s )s dx

B

≤

B

|u|

βs β−1

dx

β−1 βs

(1−β)/βs

≤ C1 |B|

B

β

w dx

1

βs

(6.2.10)

1/s

w1,B uβs/(β−1),B .

Now, we choose k = s(r − 1)/t and m = st/(s + tk), so that m < t. Thus, from Theorem 6.2.1, we have u

βs β−1 ,B

≤ C2 |B|

(m(β−1)−βs)/βsm

um,σB

which yields

s

B

|u| wdx

≤ C3 |B|

1s

m(β−1)−βs 1−β βs + βsm

−1/m

= C3 |B|

1

s w1,B um,σB

(6.2.11)

(μ(B))1/s um,σB .

Therefore,

1 μ(B)

s

|u| wdx B

1s

≤ C3 |B|

−1/m

um,σB .

(6.2.12)

On the other hand, noting that (t − m)/m = r − 1 and using the H¨ older inequality again, we obtain

6.2 The ﬁrst weighted case

193

um,σB = ≤ =

(|u| w t w− t )m dx σB 1

1

(|u| w1/t )t dx σB

t

|u| wdx σB

1/m

1/t

mt

(w−1/t ) t−m dx σB

1t " " "w−1 "1/t

1/(r−1),σB

(t−m)/mt

(6.2.13)

.

Now using the property that w ∈ Ar (Ω), it follows that 1

1

t 1/w t 1 w1,σB

r−1 ,σB

=

σB

wdx

r

≤ |σB|

1 |σB|

r/t

≤ C4 |σB|

1 ( 1 ) r−1 dx σB w

σB

wdx

r−1 1t

1 |σB|

(1) σB w

1 r−1

(6.2.14)

r−1 1t dx

,

which gives 1

1/w t 1

r

r−1 ,σB

−1

−1

r

t t ≤ C4 |σB| t w1,σB ≤ C5 |B| t w1,σB .

(6.2.15)

Combining (6.2.12), (6.2.13), (6.2.14), and (6.2.15), and noticing that r/t − 1/m = 0, we ﬁnally obtain

1 μ(B)

s

|u| wdx B

1/s

≤ C6

1 μ(σB)

1/t

t

|u| wdx

.

σB

6.2.2 Inequalities in Ls(μ)-averaging domains As an application of the local result, Theorem 6.2.3, we prove the following global reverse H¨older inequality in Ls (μ)-averaging domains. However, we remark that this result can also be proved by the generalized H¨ older inequality. Theorem 6.2.4. Let u be an A-harmonic tensor in an Ls (μ)-averaging domain Ω ⊂ Rn and 1 < s ≤ t < ∞. Assume that w ∈ Ar for some r > 1 and the measure μ is deﬁned by dμ = w(x)dx. Then, there exists a constant C, independent of u, such that

194

6 Reverse H¨ older inequalities

1 μ(Ω)

1/s

s

|u| wdx

≤C

Ω

1/t

1 μ(Ω)

t

|u| wdx

.

(6.2.16)

Ω

Proof. We may assume that sup 2B⊂Ω

1 μ(B)

1/t

t

|u| wdx

< ∞.

B

Choose a ball B0 ⊂ Ω and a constant C1 large enough, such that

1 μ(B)

sup2B⊂Ω

t

|u| wdx B

1/t

1/t t 1 ≤ sup2B⊂Ω μ(B) |u| wdx σB 1/t t 1 ≤ C1 μ(B |u| wdx . σB0 0)

Moreover, since

(6.2.17)

w(x)dx ≤

μ(B) =

w(x)dx = μ(σB),

B

σB

from Theorem 6.2.3, we obtain

1 μ(Ω)

= ≤ ≤ ≤

s

|u| wdx Ω

1 μ(Ω) 1 μ(Ω) 1 μ(Ω) 1 μ(Ω)

≤ C4 ≤ C5 ≤ C6

#

= C6

1s

1s #

1 μ(B0 ) μ(B 0)

1s 1s 1s

1 μ(Ω) 1 μ(Ω) 1 μ(Ω)

1 μ(Ω)

s

Ω

|u| wdx #

1

(μ(B0 )) s sup2B⊂Ω 1

#

1

#

(μ(B0 )) s sup2B⊂Ω

1 μ(B)

C3 μ(B)

1s

1s 1s

1

#

1 μ(B0 )

(μ(B0 )) s − t 1

1

(μ(Ω)) s − t 1

t

|u| wdx Ω

1

#

#

s

B

1 μ(σB)

(μ(B0 )) s sup2B⊂Ω (μ(B0 )) s

$ 1s

|u| wdx

t

|u| wdx σB t

|u| wdx σB t

|u| wdx σB0 t

|u| wdx σB0 t

|u| wdx Ω

$ 1t

which implies that inequality (6.2.16) holds.

$ 1s

$ 1t

$ 1t

$ 1t

$ 1t

$ 1t

6.2 The ﬁrst weighted case

195

6.2.3 Inequalities in John domains We know that a δ-John domain is an Ls (μ)-averaging domain when w satisﬁes the Ar (Ω)-condition. Thus, Theorem 6.2.4 holds if Ω ⊂Rn is a δ-John domain. Therefore, the following result is a special case of Theorem 6.2.4. However, as a further application of Theorem 6.2.3, we provide its proof. Once again we notice that it can also be proved by the generalized H¨ older inequality. Theorem 6.2.5. Let Ω ⊂Rn be a δ-John domain, u be any A-harmonic tensor, 0 < s < t < ∞, and w ∈ Ar for some r > 1. Then, there exists a constant C, independent of u, such that

1 μ(Ω)

1/s

s

|u| wdx

≤C

Ω

1/t

1 μ(Ω)

t

|u| wdx

.

(6.2.18)

Ω

Proof. By Theorem 6.2.3 and the covering lemma, there exists a covering V of Ω such that s s us,Ω,w = Ω |u| wdx s ≤ Q∈V Q |u| wdx ≤ C1 ≤ C1

− st Q∈V μ(Q)μ(σQ) s

1− t Q∈V μ(σQ)

≤ C1 μ(Ω)(t−s)/t ≤ C2 μ(Ω)(t−s)/t Thus,

Q∈V

|u| wdx

t

t

t

st

st

.

1/t t

≤ C3 μ(Ω)

Ω

st

st

|u| wdx σQ

|u| wdx Ω

1/s s

t

|u| wdx σQ

|u| wdx σQ

|u| wdx

(t−s)/st Ω

which is equivalent to (6.2.18). Remark. (1) Our local result, Theorem 6.2.3, holds in any type of domains. (2) We only applied Theorem 6.2.3 to Ls (μ)-averaging domains and δ-John domains, and this has resulted in Theorems 6.2.4 and 6.2.5, respectively. Clearly, Theorem 6.2.3 can also be applied to other kind of domains, such as the domains with Whitney covers.

196

6 Reverse H¨ older inequalities

6.2.4 Parametric inequalities In 2001, S. Ding proved the following Ar (Ω)-weighted weak reverse H¨older inequality for A-harmonic tensors with a parameter α. Theorem 6.2.6. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1 |B|

1/s

|u| w dx s

≤C

α

B

1 |B|

1/t

|u| w t

αt/s

dx

(6.2.19)

σB

for all balls B with σB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. First, we suppose that 0 < α < 1. Let k = s/(1 − α). From H¨ older’s inequality, we ﬁnd that B

|u|s wα dx

= ≤

1/s

B

|u|wα/s

|u| dx k

B

= uk,B

s

1/s dx

1/k

B

B

wdx

w

α/s ks/(k−s)

(k−s)/ks

(6.2.20)

dx

α/s

for all balls B with B ⊂ Ω. Let m = st/(s + αt(r − 1)). By Theorem 6.2.1, we obtain (6.2.21) uk,B ≤ C1 |B|(m−k)/km um,σB . Now the H¨ older inequality with 1/m = 1/t + (t − m)/mt yields um,σB = ≤ =

σB

|u|wα/s w−α/s

σB

|u|t wαt/s dx

σB

|u|t wαt/s dx

1/m

m

dx

1/t σB

1/t σB

(1/w)

αmt/s(t−m)

(1/w)

1/(r−1)

(t−m)/mt dx

α(r−1)/s dx

Combining (6.2.20), (6.2.21), and (6.2.22), we ﬁnd that

.

(6.2.22)

6.2 The ﬁrst weighted case

B

|u|s wα dx

197

1/s

1/t ≤ C1 |B|(m−k)/km σB |u|t wαt/s dx α(r−1)/s

α/s 1/(r−1) × B wdx (1/w) dx . σB

(6.2.23)

Next, since w ∈ Ar (Ω), it follows that

wdx B

α/s

(1/w) σB

=

wdx B

≤ |σB|αr/s

1/(r−1)

(1/w) σB

1 |σB|

α(r−1)/s dx

1/(r−1)

wdx B

r−1 α/s dx

1 |σB|

(1/w) σB

1/(r−1)

r−1 α/s dx

≤ C2 |σB|αr/s = C3 |B|αr/s . (6.2.24) Substituting (6.2.24) into (6.2.23), we obtain

1/s

|u|s wα dx

1/t

≤ C4 |B|(t−s)/st

B

|u|t wαt/s dx

.

σB

Therefore, inequality (6.2.19) holds for 0 < α < 1. For the case α = 1, by Theorem 6.1.4, there exist constants β > 1 and C5 > 0, such that w β,B ≤ C5 |B|(1−β)/β w 1,B

(6.2.25)

for any cube or ball B ⊂ Rn . Choose k = sβ/(β − 1), so that s < k and β = k/(k − s). By (6.2.25) and H¨ older’s inequality, we ﬁnd that B

≤

|u|s wdx

1/s

|u|k dx B

1/k B

w1/s

sk/(k−s)

1/s

= uk,B · wβ,B

(6.2.26) 1/s

≤ C6 |B|(1−β)/βs w1,B · uk,B = C6 |B|−1/k w1,B · uk,B . 1/s

(k−s)/sk dx

198

6 Reverse H¨ older inequalities

Choosing m = st/(s + t(r − 1)) and repeating the same procedure as for the case 0 < α < 1, we ﬁnd that (6.2.19) is also true for α = 1. As a particular case, we choose α = 1 in Theorem 6.2.6 to obtain the following version of the reverse H¨ older inequality. Corollary 6.2.7. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/t 1 1 |u|s wdx ≤C |u|t wt/s dx |B| B |B| σB for all balls B with σB ⊂ Ω. The choice α = s, 0 < s ≤ 1, in Theorem 6.2.6 leads to the following symmetric version of the reverse H¨older inequality. Corollary 6.2.8. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < t < ∞, 0 < s ≤ 1, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s

1 |B|

|u|s ws dx

≤C

B

1 |B|

1/t

|u|t wt dx σB

for all balls B with σB ⊂ Ω. Finally, Theorem 6.2.6 for α = 1/t, t ≥ 1 reduces to the following corollary. Corollary 6.2.9. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that t ≥ 1, 0 < s < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1 |B|

1/s

|u| w s

1/t

dx

B

≤C

1 |B|

1/t

|u| w t

1/s

dx

σB

for all balls B with σB ⊂ Ω. Now we shall prove the following global reverse H¨older inequality with Ar (Ω)-weights for A-harmonic tensors. Theorem 6.2.10. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn with |Ω| < ∞. Assume that 0 < s ≤ t < ∞ and w ∈ Ar (Ω) for some r > 1. Then,

6.2 The ﬁrst weighted case

1 |Ω|

199

1/s

|u|s wα dx

≤

Ω

1 |Ω|

1/t

|u|t wαt/s dx

(6.2.27)

Ω

for any real number α with 0 < α ≤ 1. Proof. It is clear that (6.2.27) holds if s = t. Now we assume that s < t. Using the H¨ older inequality with 1/s = 1/t + (t − s)/st, we have

1/s |u|s wα dx

s 1/s = Ω |u|wα/s dx

Ω

≤

1dx Ω

(t−s)/st Ω

|u|wα/s

t

1/t

(6.2.28)

dx

1/t = |Ω|(t−s)/st Ω |u|t wαt/s dx , which is equivalent to (6.2.27). Remark. As in Theorem 6.2.4, Theorem 6.2.10 can be proved as an application of Theorem 6.2.6. Although, Theorem 6.2.10 requires a stronger condition 0 < s ≤ t < ∞, the resulting inequality (6.2.27) is sharper than (6.2.19). We also remark that in Theorem 6.2.10 particular values of α lead to global results which are analogous to those we have listed for the local case. We have discussed the weak reverse H¨older inequality for diﬀerential forms u satisfying the A-harmonic equation, which is useful when we estimate the Ls -norm us,B . But, when we study the solutions of the A-harmonic equation, we often need to estimate the Ls -norm dus,B . In Theorem 6.2.12, we discuss the weak reverse H¨older inequality for du, which was established in [201]. In order to prove this result, we need the following lemma which is due to C. Nolder and his co-worker. Lemma 6.2.11. Suppose that |v| ∈ Lsloc (Ω), σ > 1, and 0 < t < s. If there exists a constant A such that vs,Q ≤ A|Q|(t−s)/st vt,2Q

(6.2.29)

for all cubes Q with 2Q ⊂ Ω, then for all r > 0, there exists a constant B, depending only on σ, n, s, t, r, and A, such that vs,Q ≤ B|Q|(r−s)/sr vr,σQ for all cubes Q with σQ ⊂ Ω.

200

6 Reverse H¨ older inequalities

6.2.5 Estimates for du Theorem 6.2.12. Suppose that u is a solution of (1.2.10), σ > 1, and p, q > 0. Then, there exists a constant C, depending only on σ, n, p, a, b, and q, such that (6.2.30) dup,Q ≤ C|Q|(q−p)/pq duq,σQ for all Q with σQ ⊂ Ω. Proof. By Theorems 4.2.1, 6.2.1, and 3.2.3 with p = (p + 1)/2, we have dup,Q ≤ C1 |Q|1/n u − uσQ p,√σQ

≤ C2 |Q|(p −p)/pp u − uσQ p ,σQ ≤ C3 |Q|(p −p)/pp dup ,σQ . Thus, du satisﬁes the reverse H¨older inequality (6.2.29), and hence (6.2.30) follows from Lemma 6.2.11. At this point, we should notice that, using Theorem 6.2.12 and the similar methods developed in the proofs of the Ar (Ω)-weighted inequalities, all versions of the Ar (Ω)-weighted weak reverse H¨older inequality for diﬀerential form u can be extended to du. For example, Theorem 6.2.12 implies that the proof holds if we replace u by du in Theorem 6.2.6. Therefore, we have the following Ar (Ω)-weighted weak reverse H¨older inequality for du when u is an A-harmonic tensor in a domain Ω ⊂ Rn . Theorem 6.2.13. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/t 1 1 |du|s wα dx ≤C |du|t wαt/s dx (6.2.31) |B| B |B| σB for all balls B with σB ⊂ Ω and any real number α with 0 < α ≤ 1. Now, if we choose α = 1, α = s for 0 < s ≤ 1 and α = 1/t for t ≥ 1 in Theorem 6.2.13, respectively, we obtain the following special cases for du. Corollary 6.2.14. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/t 1 1 s t t/s |du| wdx ≤C |du| w dx |B| B |B| σB for all balls B with σB ⊂ Ω.

6.3 The second weighted case

201

Corollary 6.2.15. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < t < ∞, 0 < s ≤ 1, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/t 1 1 s s t t |du| w dx ≤C |du| w dx |B| B |B| σB for all balls B with σB ⊂ Ω. Corollary 6.2.16. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that t ≥ 1, 0 < s < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1 |B|

1/s

|du|s w1/t dx

≤C

B

1 |B|

1/t

|du|t w1/s dx σB

for all balls B with σB ⊂ Ω.

6.3 The second weighted case Here we shall prove the following Aλr (Ω)-weighted reverse H¨older inequalities which were established by S. Ding in 2001.

6.3.1 Inequalities with Aα r (Ω)-weights Theorem 6.3.1. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Aα r (Ω) for some r > 1 and α > 1. Then, there exists a constant C, independent of u, such that 1/s 1/t s 1/α (t−s)/st t t/s |u| w dx ≤ C|B| |u| w dx (6.3.1) B

σB

for all balls B with σB ⊂ Ω. Note that (6.3.1) has the following symmetric version: 1/s 1/t 1 1 s 1/α t t/s |u| w dx ≤C |u| w dx . |B| B |B| σB

(6.3.1)

Proof. Choose k = αs/(α − 1). It is easy to see that s < k and 1/s = 1/k + (k − s)/ks. From the H¨ older inequality, we obtain

202

6 Reverse H¨ older inequalities

1/s |u|s w1/α dx

1/s = B (|u|w1/αs )s dx

1/k 1/αs ks/(k−s) (k−s)/ks w ≤ B |u|k dx dx B

B

≤ uk,B ·

B

wdx

(6.3.2)

1/αs

for all balls B with B ⊂ Ω. Choosing m = st/(s + t(r − 1)), by Theorem 6.2.1, we have (6.3.3) uk,B ≤ C1 |B|(m−k)/km um,σB . Since 1/m = 1/t + (t − m)/mt, using the H¨ older inequality again, we obtain um,σB

m 1/m = σB |u|w1/s w−1/s dx ≤ ≤

|u|t wt/s dx σB

1/t

1 mt/s(t−m) w

σB

1/(r−1)

(1/w) σB

dx

(r−1)/s σB

(t−m)/mt dx

|u|t wt/s dx

1/t

(6.3.4)

.

Combining (6.3.2), (6.3.3), and (6.3.4), we arrive at the estimate

1/s |u|s w1/α dx B

1/αs ≤ C1 |B|(m−k)/km B wdx (r−1)/s

1/t 1/(r−1) × σB (1/w) dx |u|t wt/s dx . σB

(6.3.5)

Now, since w ∈ Aα r (Ω), it follows that B

wdx

1/αs σB

≤

B

wdx

(1/w)

1/(r−1)

(r−1)/s dx

1/(r−1)

σB

(1/w)

α(r−1) 1/αs dx

1 ≤ |σB|α(r−1)+1 |σB| wdx B ×

1 |σB|

1 1/(r−1)

σB

w

≤ C2 |σB|(r−1)/s+1/αs ≤ C3 |B|(r−1)/s+1/αs .

α(r−1) 1/αs dx

(6.3.6)

6.3 The second weighted case

203

Finally, substituting (6.3.6) into (6.3.5) and noticing 1 1 1 1 1 r−1 m−k = − = − − − , km k m s αs t s we obtain

1/s |u| w s

1/α

≤ C|B|

dx

1/t |u| w

(t−s)/st

B

t

t/s

dx

.

σB

The constant α in Theorem 6.3.1 is arbitrary, and hence diﬀerent values of α lead to diﬀerent versions of the weak reverse H¨older inequality. For example, when α = t/s, so that t > s, we have the following symmetric form of the weighted weak reverse H¨older inequality. Corollary 6.3.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s < t < ∞ and t/s w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

1/s |u| w s

s/t

≤ C|B|

dx

B

1/t |u| w

(t−s)/st

t

t/s

dx

(6.3.7)

σB

for all balls B with σB ⊂ Ω. Now we shall prove the following interesting result. Theorem 6.3.3. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor s/t in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ Ar (Ω) for some r > 1 and 0 < α < 1. Then, there exists a constant C, independent of u, such that

1/s |u|s wα dx

≤ C|B|(t−s)/st

B

1/t |u|t wα dx

(6.3.8)

σB

for all balls B with σB ⊂ Ω. Proof. Choose k = s/(1 − α). Using the H¨older inequality, we obtain

1/s |u|s wα dx

1/s = B (|u|wα/s )s dx

1/k

(k−s)/ks ≤ B |u|k dx wαk/(k−s) dx B

α/s ≤ uk,B · B wdx .

B

(6.3.9)

204

6 Reverse H¨ older inequalities

Now, let m = t/(1 + α(r − 1)). From Theorem 6.2.1, we have uk,B ≤ C1 |B|(m−k)/km um,σB .

(6.3.10)

Using the H¨ older inequality again, we ﬁnd that um,σB = ≤ ≤

|u|wα/t w−α/t

σB

σB

|u|t wα dx

σB

|u|t wα dx

m

1/m dx

1/t

(1/w)

αm/(t−m)

σB

(1/w)

1/(r−1)

σB

1/t

(t−m)/mt dx

(6.3.11)

α(r−1)/t dx

.

s/t

On the other hand, w ∈ Ar (Ω) yields

wdx B ≤

α/s

B

wdx

α(r−1)/s

1/(r−1)

(1/w) σB

dx

1/(r−1)

σB

(1/w)

s(r−1)/t α/s dx

1 ≤ |σB|s(r−1)/t+1 |σB| wdx B

×

1 |σB|

1 1/(r−1)

σB

w

(6.3.12)

s(r−1)/t α/s dx

≤ C2 |σB|α(r−1)/t+α/s ≤ C3 |B|α(r−1)/t+α/s . Finally, combining (6.3.9), (6.3.10), (6.3.11), and (6.3.12), we obtain B

|u|s wα dx

1/s

≤ C1 |B|(m−k)/km ×

B

|u|t wα dx (t−s)/st

wdx

α/s

1/t

σB

≤ C4 |B|

σB

σB

|u|t wα dx

1/t

which is equivalent to inequality (6.3.8).

(1/w)

1/(r−1)

α(r−1)/t dx

6.3 The second weighted case

205

6.3.2 Inequalities with two parameters In 2003, Y. Xing proved the following version of the Aλr (Ω)-weighted weak reverse H¨older inequality for diﬀerential forms. Theorem 6.3.4. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Aλr (Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1/s |u| w dx s

≤ C|B|

β

1/t |u| w

(t−s)/st

t

B

βλt/s

dx

(6.3.13)

σB

for all balls B with σB ⊂ Ω and any real number β with 0 < β < 1. Note that (6.3.13) has the following symmetric version:

1 |B|

1/s

|u|s wβ dx

1 |B|

1/t

|u|t wβλt/s dx

(6.3.14)

us,B,wβ ≤ C|B|(t−s)/st ut,σB,wβλt/s .

(6.3.14)

B

≤C

σB

or

Proof. Choose k = s/(1 − β), so that s < k and 1/s = 1/k + (k − s)/ks. Using Theorem 1.1.4, we obtain

1/s |u|s wβ dx

s 1/s = B |u|wβ/s dx

B

1/k

(k−s)/ks |u|k dx wkβ/(k−s) dx B B

β/s = uk,B B wdx ≤

(6.3.15)

for all balls B with B ⊂ Ω. Next, choose m = st/(s + βλt(r − 1)), so that m < t. Using Lemma 3.1.1, we obtain uk,B ≤ C1 |B|(m−k)/km um,σB . Since 1/m = 1/t + (t − m)/mt, by Theorem 1.1.4 again, we have um,σB

m 1/m = σB |u|wβλ/s w−βλ/s dx

(6.3.16)

206

6 Reverse H¨ older inequalities

≤

|u|t wβλt/s dx

σB

=

|u|t wβλt/s dx σB

1/t σB

(1/w)

1/t

(1/w) σB

βλmt/s(t−m)

1/(r−1)

(t−m)/mt dx

βλ(r−1)/s dx

(6.3.17) .

Combining (6.3.15), (6.3.16), and (6.3.17), we arrive at the following estimate: B

|u|s wβ dx

1/s

1/t ≤ C1 |B|(m−k)/km σB |u|t wβλt/s dx βλ(r−1)/s

β/s 1/(r−1) × B wdx (1/w) dx . σB

(6.3.18)

Now, since w ∈ Aλr (Ω), we ﬁnd that B

wdx

β/s σB

≤

σB

wdx

(1/w)

σB

1/(r−1)

(1/w)

βλ(r−1)/s dx

1/(r−1)

λ(r−1) β/s dx

1 ≤ |σB|λ(r−1)+1 |σB| wdx B ×

1 |σB|

σB

(1/w)

1/(r−1)

(6.3.19)

λ(r−1) β/s dx

≤ C2 |σB|β(λ(r−1)+1)/s = C3 |B|βλ(r−1)/s+β/s . Finally, substituting (6.3.19) into (6.3.18) and using 1 1 1 − α 1 βλ(r − 1) m−k = − = − − , km k m s t s we obtain

1/s |u|s wβ dx

1/t

≤ C|B|(t−s)/st

B

|u|t wβλt/s dx

.

σB

In Theorem 6.3.4, if we let β = 1/p, where p > 1 is a real number, then inequality (6.3.13) reduces to

|u| w s

B

1/s

1/p

dx

≤ C|B|

1/t |u| w

(t−s)/st

t

σB

λt/ps

dx

.

(6.3.20)

6.3 The second weighted case

207

Similarly, when β = s/(s + t) in Theorem 6.3.4, we have the following symmetric inequality:

1 |B|

1/s

|u|s ws/(s+t) dx

≤C

B

1 |B|

1/t

|u|t wλt/(s+t) dx

.

(6.3.21)

σB

Choosing λ = 1 in (6.3.21), we ﬁnd the following version of the weak reverse H¨older inequality. Corollary 6.3.5. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the Aharmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Ar (Ω) for some r > 1, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u|s ws/(s+t) dx

≤C

B

1/t

1 |B|

|u|t wt/(s+t) dx σB

for all balls B with σB ⊂ Ω. Selecting β = 1/λ with λ > 1 in Theorem 6.3.4, we obtain the following result. Corollary 6.3.6. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Aλr (Ω) for some r > 1 and λ > 1, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u|s w1/λ dx

≤C

B

1 |B|

1/t

|u|t wt/s dx σB

for all balls B with σB ⊂ Ω. Assuming β = s with 0 < s < 1 in Theorem 6.3.4, we have the following inequality. Corollary 6.3.7. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < t < ∞, σ > 1, and 0 < s < 1. If w ∈ Aλr (Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u| w dx s

s

B

≤C

1 |B|

1/t

|u| w dx t

λt

σB

for all balls B with σB ⊂ Ω. Letting λ = 1 in Corollary 6.3.7, we ﬁnd the following symmetric inequality.

208

6 Reverse H¨ older inequalities

Corollary 6.3.8. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < t < ∞, σ > 1, and 0 < s < 1. If w ∈ Ar (Ω) for some r > 1, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u| w dx s

s

≤C

B

1 |B|

1/t

|u| w dx t

t

(6.3.22)

σB

for all balls B with σB ⊂ Ω. Assuming t > 1 in Theorem 6.3.4 and letting β = 1/t in (6.3.13), we have the following corollary. Corollary 6.3.9. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s < ∞, σ > 1, and t > 1. If w ∈ Aλr (Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u| w s

1/t

≤C

dx

B

1 |B|

1/t

|u| w t

λ/s

dx

(6.3.23)

σB

for all balls B with σB ⊂ Ω. Choosing λ = 1 in Corollary 6.3.9, inequality (6.3.23) reduces to

1 |B|

1/s

|u|s w1/t dx

≤C

1 |B|

1/t

|u|t w1/s dx

(6.3.24)

us,B,w1/t ≤ C|B|(t−s)/st ut,σB,w1/s

(6.3.24)

B

σB

or

which has a nice symmetric property. Applying Theorem 6.3.4 and the covering lemma, we can also prove the following global result. Theorem 6.3.10. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn with |Ω| < ∞. Assume that 0 < s ≤ t < ∞ and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then,

1 |Ω|

1/s

|u|s wβ dx Ω

≤

C |Ω|

for any real number β with 0 < β < 1.

1/t

|u|t wβλt/s dx Ω

(6.3.25)

6.4 The third weighted case

209

6.3.3 Aλ r (Ω)-weighted inequalities for du By the virtue of Theorem 6.2.12 and using methods similar to the proofs of Theorems 6.3.3 and 6.3.4, we can prove the following versions of the weighted weak reverse H¨older inequality for du. Theorem 6.3.11. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1. Assume that 0 < s, t < ∞, and w ∈ s/t Ar (Ω) for some r > 1 and 0 < α < 1. Then, there exists a constant C, independent of u, such that

1/s |du|s wα dx

1/t

≤ C|B|(t−s)/st

|du|t wα dx

B

(6.3.26)

σB

for all balls B with σB ⊂ Ω. Theorem 6.3.12. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Aλr (Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1/s

|du| w dx s

β

≤ C|B|

1/t |du| w

(t−s)/st

t

B

βλt/s

dx

(6.3.27)

σB

for all balls B with σB ⊂ Ω and any real number β with 0 < β < 1. Note that (6.3.27) has the following symmetric version:

1 |B|

1/s

|du|s wβ dx B

≤C

1 |B|

1/t

|du|t wβλt/s dx

(6.3.28)

dus,B,wβ ≤ C|B|(t−s)/st dut,σB,wβλt/s .

(6.3.28)

σB

or

From the above results it is clear that all versions of the weak reverse H¨ older inequality for diﬀerential forms u obtained in this section can be extended to diﬀerential forms du.

6.4 The third weighted case In the last two sections, we have discussed Ar (Ω)-weighted and Aλr (Ω)weighted weak reverse H¨older inequalities. In this section, we study Ar (λ, Ω)weighted weak reverse H¨older inequalities.

210

6 Reverse H¨ older inequalities

6.4.1 Local inequalities In 2002, S. Ding and D. Sylvester [268] established the following Ar (λ, Ω)weighted weak reverse H¨older inequality. Theorem 6.4.1. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Ar (λ, Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1/s |u| w s

αλ

≤ C|B|

dx

1/t |u| w

(t−s)/st

t

B

αt/s

dx

(6.4.1)

σB

for all balls B with σB ⊂ Ω and any real number α with 0 < α < 1. Note that (6.4.1) has the following symmetric version:

1 |B|

1/s

|u|s wαλ dx

≤C

B

1 |B|

1/t

|u|t wαt/s dx

.

(6.4.2)

σB

Proof. Choose k = s/(1 − α), so that s < k and 1/s = 1/k + (k − s)/ks. Applying H¨ older’s inequality, we ﬁnd that B

= ≤

|u|s wαλ dx

1/s

B

|u|wαλ/s

|u| dx k

B

= uk,B

s

1/s dx

1/k

B

wλ dx

kαλ/(k−s)

w B

α/s

dx

(k−s)/ks

(6.4.3)

for all balls B with B ⊂ Ω. Next, choose m = st/(s + αt(r − 1)). Then, from Theorem 6.2.1, we obtain uk,B ≤ C1 |B|(m−k)/km um,σB .

(6.4.4)

Since 1/m = 1/t + (t − m)/mt, by the H¨ older inequality again, we have um,σB

m 1/m = σB |u|wα/s w−α/s dx

6.4 The third weighted case

≤ =

σB

211

|u|t wαt/s dx

|u|t wαt/s dx σB

1/t σB

1/t

(1/w)

(1/w) σB

αmt/s(t−m)

1/(r−1)

(t−m)/mt dx

α(r−1)/s dx

(6.4.5) .

Combining (6.4.3), (6.4.4), and (6.4.5), we arrive at the following estimate: B

|u|s wαλ dx

1/s

≤ C1 |B|(m−k)/km ×

σB

B

|u|t wαt/s dx

wλ dx

1/t

α/s

1 1/(r−1) w

σB

α(r−1)/s dx

(6.4.6)

.

Now since w ∈ Ar (λ, Ω), it follows that

wλ dx

B

≤

α/s

σB λ

σB

w dx

(1/w)

σB

1/(r−1)

(1/w)

α(r−1)/s dx

1/(r−1)

(r−1) α/s dx

1 λ ≤ |σB|(r−1)+1 |σB| w dx B ×

1 |σB|

σB

(1/w)

1/(r−1)

(6.4.7)

(r−1) α/s dx

≤ C2 |σB|αr/s = C3 |B|αr/s . Finally, substituting (6.4.7) into (6.4.6) and using (m−k)/km = 1/k−1/m = 1/s − 1/t − αr/s, we obtain

1/s

|u| w s

B

αλ

dx

≤ C|B|

1/t |u| w

(t−s)/st

t

αt/s

dx

.

σB

If we choose α = 1/p where p > 1 in Theorem 6.4.1, then inequality (6.4.1) reduces to 1/s 1/t s λ/p (t−s)/st t t/ps |u| w dx ≤ C|B| |u| w dx . (6.4.8) B

σB

Setting α = s/(s + t) in Theorem 6.4.1, we have the following symmetric inequality:

212

6 Reverse H¨ older inequalities

1 |B|

1/s

|u|s wλs/(s+t) dx

≤C

B

1 |B|

1/t

|u|t wt/(s+t) dx

.

(6.4.9)

σB

Choosing λ = 1 in (6.4.9), we obtain the following symmetric weak reverse H¨ older inequality. Corollary 6.4.2. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the Aharmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Ar (Ω) for some r > 1, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u|s ws/(s+t) dx

≤C

B

1 |B|

1/t

|u|t wt/(s+t) dx σB

for all balls B with σB ⊂ Ω. Selecting α = 1/λ with λ > 1 in Theorem 6.4.1, we have the following corollary. Corollary 6.4.3. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Ar (λ, Ω) for some r > 1 and λ > 1, then there exists a constant C, independent of u, such that

1 |B|

1/s

|u| wdx s

≤C

B

1 |B|

1/t

|u| w t

t/λs

dx

σB

for all balls B with σB ⊂ Ω.

6.4.2 Global inequality As an application of the local result, Theorem 6.4.1, we shall prove the following global weak reverse H¨older inequality which can also be proved by using Theorem 1.1.4 directly. Theorem 6.4.4. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a bounded domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 1 < s ≤ t < ∞ and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1 |Ω|

1/s

|u|s wαλ dx Ω

≤C

1 |Ω|

1/t

|u|t wαt/s dx Ω

(6.4.10)

6.4 The third weighted case

213

for any real number α with 0 < α < 1. Proof. By (6.4.1) and Theorem 1.5.3, we obtain Ω

≤ ≤

|u|s wαλ dx

1/s

|u|s wαλ dx Q

Q∈V

Q∈V

C1 |Q|(t−s)/st

≤ C1 |Ω|(t−s)/st ≤ C1 |Ω|(t−s)/st

1/s

|u|t wαt/s dx σQ

Q∈V

σQ

|u|t wαt/s dx

|u|t wαt/s dx

1/t ≤ C2 |Ω|(t−s)/st Ω |u|t wαt/s dx , Q∈V

1/t 1/t

1/t

Ω

which is equivalent to (6.4.10).

6.4.3 Analogies for du Using Theorem 6.2.12 and the method we developed in the proof of Theorem 6.4.1, we obtain the following weak reverse H¨ older inequality for du, where u satisﬁes the A-harmonic equation (1.2.4). Theorem 6.4.5. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 0 < s, t < ∞, σ > 1. If w ∈ Ar (λ, Ω) for some r > 1 and λ > 0, then there exists a constant C, independent of u, such that

1/s |du| w s

αλ

dx

≤ C|B|

1/t |du| w

(t−s)/st

t

B

αt/s

dx

(6.4.11)

σB

for all balls B with σB ⊂ Ω and any real number α with 0 < α < 1. Note that (6.4.11) has the following symmetric version:

1 |B|

1/s

|du|s wαλ dx B

≤C

1 |B|

1/t

|du|t wαt/s dx

.

(6.4.11)

σB

From Theorem 6.4.5 and the method to the proof of Theorem 6.4.4, we have the following global weak reverse H¨ older inequality for du in a bounded domain Ω ⊂ Rn .

214

6 Reverse H¨ older inequalities

Theorem 6.4.6. Let u ∈ D (Ω, ∧l ) be a diﬀerential form satisfying the A-harmonic equation (1.2.4) in a bounded domain Ω ⊂ Rn , l = 0, 1, . . . , n. Suppose that 1 < s ≤ t < ∞ and w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s

1 |Ω|

|du| w s

αλ

dx

≤C

Ω

1 |Ω|

1/t

|du| w t

αt/s

dx

(6.4.12)

Ω

for any real number α with 0 < α < 1.

6.5 Two-weight inequalities Many two-weight versions of the weak reverse H¨older inequality have been established recently. In this section, we discuss these results for Ar,λ (Ω)-weights and Ar (λ, Ω)-weights.

6.5.1 Ar,λ(Ω)-weighted cases In this section, we prove the Ar,λ (Ω)-weighted weak reverse H¨older inequalities. The following two-weighted inequality is due to Y. Xing and G. Bao [84]. Theorem 6.5.1. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , ρ > 1. Assume that 0 < s, t < ∞, and (w1 , w2 ) ∈ Ar,λ (Ω) for some 1 < r < ∞ and λ ≥ 1. Then, there exists a constant C, independent of u, such that

1 |B|

1/s

|u|s w1 dx

≤C

B

1 |B|

1/t

t/s

|u|t w2

dx

(6.5.1)

ρB

for all balls B with ρB ⊂ Ω and 0 < < λ. stλr tr+sλr ,

Proof. Let k = sλ/(λ − ) and m = Theorem 1.1.4, we have

so that s < k and m < t. Using

1/s |u|s w1 dx s 1/s /s = B |u|w1 dx

B

≤

|u| dx B

= uk,B

k

1/k

B

B

w1λ dx

/s w1

/sλ

.

(k−s)/ks

ks/(k−s) dx

(6.5.2)

6.5 Two-weight inequalities

215

From Theorem 6.2.1, it follows that uk,B ≤ C1 |B|(m−k)/km um,ρB .

(6.5.3)

Using the generalized H¨ older inequality with 1/m = 1/t + (t − m)/mt again, we obtain um,ρB ≤ =

|u|

t

ρB

t/s w2 dx

t/s

ρB

|u|t w2

dx

1/t

ρB

1/t

ρB

1 w2

1 w2

(t−m)/mt

mt/(st−sm) dx

(6.5.4)

r/sλr

λr /r dx

.

Combining (6.5.2), (6.5.3), and (6.5.4), we ﬁnd that

|u|s w1 dx

B

1/s

≤ C1 |B|(m−k)/km ×

ρB

1 w2

B

w1λ dx

/sλ

t/s

ρB

|u|t w2

1/t dx

(6.5.5)

r/sλr

λr /r dx

.

Now since (w1 , w2 ) ∈ Ar,λ (Ω), we obtain

/λs wλ dx B 1 ≤ |ρB|

r/λs

ρB

1 |ρB|

1 w2

r/sλr

λr /r dx

wλ dx ρB 1

1/λr

1 |ρB|

ρB

1 w2

1/λr r/s

λr /r dx

≤ C2 |B|r/λs . (6.5.6) Finally, substituting (6.5.6) into (6.5.5), we have B

|u|s w1 dx

1/s

≤ C3 |B|(m−k)/mk+r/sλ = C3 |B|(1/s−1/t)

t/s

ρB

|u|t w2

t/s

ρB

|u|t w2

1/t dx

1/t dx

.

(6.5.7)

216

6 Reverse H¨ older inequalities

In Theorem 6.5.1, particular values of the parameter lead to diﬀerent versions of the weak reverse H¨ older inequality. For example, choosing = s with s < λ, we obtain the following symmetric weak reverse H¨ older inequality

1/s

1 |B|

|u|

s

≤C

w1s dx

B

1 |B|

1/t

|u|

t

w2t dx

(6.5.8)

ρB

for all balls B with ρB ⊂ Ω. Similarly, if we select = 1/t with λt > 1, inequality (6.5.1) reduces to

1 |B|

1/s

1/t

|u|s w1 dx

≤C

B

1 |B|

1/t

1/s

|u|t w2 dx

(6.5.9)

ρB

for all balls B with ρB ⊂ Ω. Now, we prove the following global weak reverse H¨ older inequality. Theorem 6.5.2. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn with |Ω| < ∞. Assume that 1 < s ≤ t < ∞, 0 < < λ, and (w1 , w2 ) ∈ Ar,λ (Ω) for some 1 < r < ∞ and λ ≥ 1, then 1/s 1/t 1 1 s t t/s |u| w1 dx ≤C |u| w2 dx . (6.5.10) |Ω| Ω |Ω| Ω Proof. Using Theorems 6.5.1 and 1.5.3, we obtain

1/s |u|s w1 dx

1/s ≤ D∈V D |u|s w1 dx 1/t t/s ≤ D∈V Ck |D|(t−s)/st ρD |u|t w2 dx

Ω

≤

Ck |Ω|

(t−s)/st

D∈V

≤ C|Ω|(t−s)/st

Ω

|u|

t/s

|u|t w2 Ω

t

t/s w2 dx

1/t

(6.5.11)

1/t dx

.

6.5.2 Ar (λ, Ω)-weighted cases Theorem 6.5.3. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , ρ > 1. Assume that 0 < s, t < ∞, and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

6.5 Two-weight inequalities

217

1/s |u|s w1αλ dx

1/t αt/s

≤ C|B|(t−s)/st

|u|t w2

B

dx

(6.5.12)

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Proof. Choose k = s/(1 − α), so that s < k and 1/s = 1/k + (k − s)/ks. Applying H¨ older’s inequality, we have

1/s |u|s w1αλ dx s 1/s αλ/s = B |u|w1 dx

B

≤

|u|k dx B

= uk,B

1/k

B

w1λ dx

kαλ/(k−s)

w B 1

(k−s)/ks

(6.5.13)

dx

α/s

for all balls B with B ⊂ Ω. Next, choose m = st/(s + αt(r − 1)). Using Theorem 6.2.1, we obtain uk,B ≤ C1 |B|(m−k)/km um,σB .

(6.5.14)

Since 1/m = 1/t + (t − m)/mt, by the H¨ older inequality again, we have um,σB m 1/m α/s −α/s = σB |u|w2 w2 dx ≤ =

αt/s

dx

αt/s

dx

|u|t w2 σB |u|t w2 σB

1/t 1/t

(1/w2 ) σB

αmt/s(t−m)

(1/w2 ) σB

1/(r−1)

(t−m)/mt dx

(6.5.15)

α(r−1)/s dx

.

Combining (6.5.13), (6.5.14), and (6.5.15), we arrive at the following estimate: B

|u|s w1αλ dx

≤ C1 |B| ×

1/s

(m−k)/km

σB

αt/s

|u|t w2

α/s wλ dx B 1

σB

1 w2

1/t dx

.

Now since (w1 , w2 ) ∈ Ar (λ, Ω), we ﬁnd that

α(r−1)/s

1/(r−1) dx

(6.5.16)

218

6 Reverse H¨ older inequalities

B

≤

w1λ dx

α/s

1/(r−1)

σB

wλ dx σB 1

(1/w2 )

(1/w2 ) σB

α(r−1)/s dx

1/(r−1)

(r−1) α/s dx

1 λ ≤ |σB|(r−1)+1 |σB| w dx B 1 (r−1) α/s 1/(r−1) 1 dx × |σB| σB (1/w2 )

(6.5.17)

≤ C2 |σB|αr/s = C3 |B|αr/s . Finally, substituting (6.5.17) into (6.5.16) and using 1 1 1 1 αr m−k = − = − − , km k m s t s we obtain

1/s |u|

s

≤ C|B|

w1αλ dx

1/t |u|

(t−s)/st

B

t

αt/s w2 dx

.

σB

6.5.3 Two-weight inequalities for du From Theorem 6.2.12 (weak reverse H¨older inequality for du), we can prove the following Ar,λ (Ω)-weighted weak reverse H¨older inequality for du. Theorem 6.5.4. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , ρ > 1. Assume that 0 < s, t < ∞, and (w1 , w2 ) ∈ Ar,λ (Ω) for some 1 < r < ∞ and λ ≥ 1. Then, there exists a constant C, independent of u, such that

1 |B|

1/s

|du|

s

B

w1β dx

≤C

1 |B|

1/t

|du|

t

βt/s w2 dx

ρB

for all balls B with ρB ⊂ Ω and 0 < β < λ. Similarly, following the proof of Theorem 6.5.3, we can prove Ar (λ, Ω)-weighted weak reverse H¨older inequality for du. Theorem 6.5.5. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , ρ > 1. Assume that 0 < s, t < ∞, and (w1 , w2 ) ∈

6.6 Inequalities with Orlicz norms

219

Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |du|

s

w1αλ dx

≤ C|B|

B

1/t |du|

(s−t)/st

t

αt/s w2 dx

ρB

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1.

6.6 Inequalities with Orlicz norms In this section, we shall present the weak reverse H¨older inequality with Lp (log L)α -norm. For this, we recall that the Lp (log L)α -norm of a measurable function on Ω is deﬁned as |f |

α p p α p dx ≤ k , (6.6.1) |f | log e + f L log L = inf k : k Ω where 0 0 are real numbers. Let E be any subset of Rn . The functional of a measurable function f over E is deﬁned by |f | p1 dx , |f |p logα e + (6.6.2) [f ]Lp (log L)α (E) = ||f ||p E

1/p . From Theorem 1.9.1, we know that the where ||f ||p = E |f (x)|p dx norm f Lp logα L is equivalent to [f ]Lp (log L)α for 0 < p < 1 and α ≥ 0. Hence, we do not distinguish the norm f Lp logα L from the functional [f ]Lp (log L)α throughout this section, and for convenience write |f | p1 α p dx . |f |p logα e + (6.6.3) f L log L = ||f ||p E

6.6.1 Elementary inequalities It is easy to see that for any constant k, there exist constants m > 0 and M > 0, such that t ≤ M log(e + t), t > 0. (6.6.4) m log(e + t) ≤ log e + k From the weak reverse H¨older inequality (Lemma 3.1.1), we know that the norms us,B and ut,B are comparable when 0 < d1 ≤ diam(B) ≤ d2 < ∞. Hence, we may assume that 0 < m1 ≤ us,B ≤ M1 < ∞ and 0 < m2 ≤

220

6 Reverse H¨ older inequalities

ut,B ≤ M2 < ∞ for some constants mi and Mi , i = 1, 2. Thus, we have |u| ≤ C2 log(e + |u|) (6.6.5) C1 log (e + |u|) ≤ log e + us,B and

C3 log(e + |u|) ≤ log e +

|u| ut,B

≤ C4 log(e + |u|)

(6.6.6)

for any s > 0 and t > 0, where Ci are constants, i = 1, 2, 3, 4. Using (6.6.5) and (6.6.6), we obtain C5

B

|u|s logα e +

|u| ||u||t,B

≤ uLs (log L)α (B) = ≤ C6 and

B

1/s dx

α s e+ |u| log B |u| ||u||t,B

|u|s logα e +

|u| ||u||s,B

1/s dx

(6.6.7)

1/s dx

C7 uLt (log L)α (B) ≤ B |u|t logα e +

|u| ||u||s,B

1/t dx

(6.6.8)

≤ C8 uLt (log L)α (B) for any ball B and any s > 0, t > 0, and α > 0. Consequently, uLs (log L)α (B) < ∞ if and only if |u| log s

B

α

|u| e+ dx ||u||t,B

1/s < ∞.

6.6.2 Lp(log L)α-norm inequalities Now, we shall prove the following weak reverse H¨ older inequality with Lp (log L)α -norm. Theorem 6.6.1. Let u ∈ D (Ω, ∧l ), l = 0, 1, . . . , n, be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that uLs (log L)α (B) ≤ C|B|(t−s)/st uLt (log L)β (σB)

(6.6.9)

6.6 Inequalities with Orlicz norms

221

for any constants α > 0 and β > 0, and all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0, where d0 is a ﬁxed constant. Proof. For any ball B ⊂ Ω with diam(B) ≥ d0 > 0, we choose ε > 0 small enough and a constant C1 such that |B|−ε/st ≤ C1 .

(6.6.10)

us+ε,B ≤ C2 |B|(t−(s+ε))/t(s+ε) ut,σ1 B

(6.6.11)

By Lemma 3.1.1, we have

for some σ1 > 1. Now from (6.6.7), we have

α s e+ |u| log B

C3

|u| ||u||t,B

≤ uLs (log L)α (B) ≤ C4 B |u|s logα e +

1/s dx

|u| ||u||t,B

(6.6.12)

1/s dx

.

Setting B1 = {x ∈ B : |u|/||u||t,B ≥ 1}, B2 = {x ∈ B : |u|/||u||t,B < 1} and using (6.6.12) and the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ), where s > 0 is any constant, we ﬁnd that uLs (log L)α (B) 1/s |u| dx = B |u|s logα e + ||u|| s,B |u| dx = B1 |u|s logα e + ||u|| s,B +

B2

≤ 21/s +

|u|s logα e +

≤2

+ C6

|u| log

C5

1/s dx

|u| ||u||s,B

1/s

|u|s logα e + B1 s

B2

|u| ||u||s,B

α

e+

|u| ||u||s,B

α s e+ |u| log B1

|u|s logα e + B2

dx

dx

(6.6.13)

1s

|u| ||u||t,B

|u| ||u||t,B

1s

dx

dx

1s

1s

.

Next, since |u|/||u||t,B > 1 on B1 , for ε > 0 appeared in (6.6.10), there exists C7 > 0 such that

222

6 Reverse H¨ older inequalities

log

|u| e+ ut,σ1 B

α

≤ C7

|u| ||u||t,σ1 B

ε .

(6.6.14)

Combining (6.6.10), (6.6.11), and (6.6.14), we obtain B1

|u|s logα e +

≤ C8

u εt,σ

1B

≤ C8

1

u εt,σ B 1

1/s dx 1/s

1

|u| ||u||t,B

|u|

s+ε

B1

dx 1/s

|u|s+ε dx B =

C8 ε/s

u t,σ

1B

≤

C9 ε/s

u t,σ

B

|u|

s+ε

dx

(6.6.15)

1 (s+ε)/s

s+ε

|B|(t−(s+ε))/t(s+ε) ut,σ1 B

(s+ε)/s

1B

≤ C10 |B|(t−s)/st ut,σ1 B . Finally, since logα e + it follows that

|u| ||u||t,B

≤ M1 logα (e + 1) ≤ M2 ,

α s e+ |u| log B2

≤

|u| ||u||t,B

M2 |u|s dx B2

x ∈ B2

1/s dx

1/s

(6.6.16)

≤ C11 us,B which in view of Lemma 3.1.1 gives us,B ≤ C12 |B|(t−s)/st ut,σ2 B .

(6.6.17)

Substituting (6.6.17) into (6.6.16) yields |u|s logα e +

B2

|u| ||u||t,B

1/s dx

≤ C13 |B|(t−s)/st ut,σ2 B .

(6.6.18)

Combining (6.6.13), (6.6.15), and (6.6.18), we obtain uLs (log L)α (B) ≤ C14 |B|(t−s)/st ut,σ3 B ,

(6.6.19)

6.6 Inequalities with Orlicz norms

223

where σ3 = max{σ1 , σ2 }. Now since logβ e + obtain

|u| ||u||t,σ3 B

≥ 1 for β > 0, we

uLs (log L)α (B) ≤ C14 |B|(t−s)/st ut,σ3 B 1/t = C14 |B|(t−s)/st σ3 B |u|t dx ≤ C14 |B|(t−s)/st

σ3 B

|u|t logβ e +

|u| ||u||t,σ3 B

1/t dx

= C14 |B|(t−s)/st uLt (log L)β (σ3 B) .

Using a similar method developed in the proof of Theorem 6.6.1 and Theorem 6.2.12, we can prove the following version of the weak reverse H¨ older inequality with Lp (log L)α -norm. Theorem 6.6.2. Let u be an A-harmonic tensor in a domain Ω ⊂ Rn , σ > 1, and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that duLs (log L)α (B) ≤ C|B|(t−s)/st duLt (log L)β (σB)

(6.6.20)

for all balls B with σB ⊂ Ω and diam(B) ≥ d0 > 0. Here d0 is a ﬁxed constant, α > 0 and β > 0 are any constants. Note that the above version of the weak reverse H¨older inequality cannot be obtained by replacing u by du in Theorem 6.6.1 since du may not be a solution of the A-harmonic equation. Notes to Chapter 6. In this chapter, we have developed the weak reverse H¨older inequalities for diﬀerential forms satisfying some version of the A-harmonic equation. However, several other diﬀerent versions of the weak reverse H¨older inequality have been established during the recent years. For example, see [283] for reverse H¨older inequalities with boundary integrals and Lp -estimates for solutions of nonlinear elliptic and parabolic boundary-value problems. We encourage readers to see [284–297, 126, 268, 282, 171, 79, 84] for further interesting versions of the weak reverse H¨ older inequality.

Chapter 7

Inequalities for operators

The purpose of this chapter is to present a series of the local and global estimates for some operators, including the homotopy operator T , the Laplace– Beltrami operator Δ = dd + d d, Green’s operator G, the gradient operator ∇, the Hardy–Littlewood maximal operator, and the diﬀerential operator, which act on the space of harmonic forms deﬁned in a domain in Rn , and the compositions of some of these operators. We introduce the Hardy–Littlewood maximal operator IMs and the sharp maximal operator IMs applied to diﬀerential forms in Section 7.1. We develop some basic estimates for Green’s operator ∇ ◦ T and d ◦ T in Section 7.2. We establish some Ls -estimates and imbedding inequalities for the compositions of homotopy operator T and Green’s operator G in Section 7.3. In Section 7.4, we prove some Poincar´etype inequalities for T ◦ G and G ◦ T . In Section 7.5, we obtain Poincar´e-type inequalities for the homotopy operator T . In Section 7.6, we study various estimates for the composition T ◦ H. In Section 7.7, we provide the estimates for the compositions of three operators. Finally, in Section 7.8, we oﬀer some norm comparison theorems for the maximal operators.

7.1 Introduction We begin this section by introducing the Hardy–Littlewood maximal operator. For a locally Ls -integrable form u(y), the Hardy–Littlewood maximal operator IMs is deﬁned by IMs (u) = IMs u = IMs u(x) = sup r>0

1 |B(x, r)|

1/s

|u(y)| dy s

,

(7.1.1)

B(x,r)

where B(x, r) is the ball of radius r, centered at x, 1 ≤ s < ∞. We write IM(u) = IM1 (u) if s = 1. Similarly, for a locally Ls -integrable form u, we deﬁne the sharp maximal operator IMs by R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 7,

225

226

7 Inequalities for operators

IMs (u) = IMs u = IMs u(x) 1/s 1 s = supr>0 |B(x,r)| |u(y) − u | dy . B(x,r) B(x,r)

(7.1.2)

These operators and those mentioned above play an important role in many diverse ﬁelds, including partial diﬀerential equations and analysis. For example, the Laplace–Beltrami operator and Green’s operator are widely used in physics, potential theory, and nonlinear elasticity, etc. The gradient operator and the homotopy operator are eﬀective tools in analysis which have found many applications in diﬀerent areas of mathematics, including topology and diﬀerential geometry. Some estimates for the Laplace– Beltrami operator Δ, Green’s operator G, and projection operator have been established in previous chapters. Also, see [298–309, 60, 261, 127, 128, 113, 48, 163–165, 174, 133, 83, 86] for recent results related to these operators. In this chapter, we develop several other local and global estimates for these operators and their compositions. We also study the diﬀerential operator d and codiﬀerential operator d and some other related operators.

7.2 Some basic estimates The Ls -estimates for the compositions of Green’s operator G, diﬀerential operator d, and codiﬀerential operator d have been developed and used in Chapter 3. Now, we recall the following estimate which was established in [21].

7.2.1 Estimates related to Green’s operator Let u ∈ C ∞ (∧l M ), l = 1, 2, . . . , n − 1. For 1 < s < ∞, there exists a constant C, independent of u, such that dd G(u)s,M + d dG(u)s,M + dG(u)s,M +d G(u)s,M + G(u)s,M

(7.2.1)

≤ Cus,M . Proof of inequality (7.2.1). First, we shall prove (7.2.1) for the case s ≥ 2 by using the closed graph theorem. From [144], we have the following L2 estimate:

7.2 Some basic estimates

227

dd G(u)2,M + d dG(u)2,M + dG(u)2,M +d G(u)2,M + G(u)2,M

(7.2.2)

≤ C1 u2,M . Let un s,M + G(un ) − vs,M → 0 as n → ∞. Since Ls is imbedded in L2 , it follows that un 2,M + G(un ) − v2,M → 0 as n → ∞. From (7.2.2), we have G(un )2,M = G(un − H(un ))2,M ≤ C2 un − H(un )2,M ≤ C2 un 2,M + H(un )2,M

(7.2.3)

≤ C3 un 2,M →0 as n → ∞, where H is the projection operator. Thus, v = 0, and so by the closed graph theorem G is bounded. Repeating this procedure for the compositions d dG, dd G, dG, and d G, the proof for the case s ≥ 2 follows. Next, we assume that 1 < s < 2. Notice that for smooth forms, Green’s operator commutes with anything the Laplacian does (see [23]) and is selfadjoint. In particular, when η, u ∈ C ∞ (∧l M ), we have ⎧ = (η, G(u)), ⎨ (G(η), u) (7.2.4) dG(η) = Gd(η), ⎩ = Gd (u). d G(u) Let ηn = G(u)(|G(u)|2 + 1/n)(s−2)/2 and observe that ηn ∈ C ∞ . By the Lebesgue dominated convergence theorem (LDCT), we obtain ηn tt,M → G(u)ss,M

(7.2.5)

as n → ∞, where t is a positive number with 1/t + 1/s = 1. Next notice that |(G(u), ηn )| increases to G(u)ss,M (again by the LDCT). Therefore, given ε > 0, we may choose a large positive integer N , so that for n > N we have G(u)ss,M ≤ |(G(u), η)| + ε. Now since 1 < s < 2, it follows that t > 2. Hence, using H¨ older inequality and the case for s ≥ 2, we have G(u)ss,M ≤ |(G(u), η)| + ε = |(u, G(ηn ))| + ε ≤ us,M G(ηn )t,M + ε ≤ C4 us,M ηn t,M + ε s/t

→ C5 us,M G(u)t,M + ε.

(7.2.6)

228

7 Inequalities for operators

Thus, letting ε → 0, we obtain s/t

G(u)ss,M ≤ C6 us,M G(u)s,M .

(7.2.7) s/t

Assuming G(u)t,M > 0, and dividing (7.2.7) by G(u)s,M , we obtain G(u)s,M ≤ C7 us,M .

(7.2.8)

Notice that (7.2.8) obviously holds for the case G(u)t,M = 0. Next, we set ηn = dG(u)(|dG(u)|2 + 1/n)(s−2)/2 . As above, ηn ∈ C ∞ and by the LDCT, ηn tt,M → dG(u)ss,M

(7.2.9)

as n → ∞. Again we see that |(dG(u), ηn )| increases to dG(u)ss,M by the LDCT. Therefore, for ε > 0, we may choose a large positive integer N , such that for n > N , we have dG(u)ss,M ≤ |(dG(u), ηn )| + ε. Now we ﬁnd that dG(u)ss,M ≤ |(dG(u), ηn )| + ε = |(u, G(d ηn ))| + ε ≤ us,M G(d ηn )t,M + ε

(7.2.10)

≤ C8 us,M ηn t,M + ε s/t

→ C8 us,M dG(u)t,M + ε. Thus, as above, we obtain the following analog of (7.2.8): dG(u)s,M ≤ C9 us,M . Finally, choosing ηn = d G(u)(|d G(u)|2 + 1/n)(s−2)/2 , ηn = dd G(u)(|dd G(u)|2 + 1/n)(s−2)/2 , ηn = d dG(u)(|d dG(u)|2 + 1/n)(s−2)/2 , and following the same method, it follows that d G(u)s,M ≤ C10 us,M , dd G(u)s,M ≤ C11 us,M , d dG(u)s,M ≤ C12 us,M .

(7.2.11)

7.2 Some basic estimates

229

Combining these inequalities with (7.2.8) and (7.2.11), we obtain the required inequality (7.2.1).

7.2.2 Estimates for ∇ ◦ T Now, we prove the following basic Ar (Ω)-weighted estimate for ∇ ◦ T . Theorem 7.2.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s |∇(T u)|

s

1/s

≤ C|B|

w1α dx

|u|

w2αλ dx

(7.2.12)

∇(T u)s,B,w1α ≤ C|B|us,ρB,w2αλ .

(7.2.12)

B

s

ρB

for any real number α with 0 < α < 1. Note that (7.2.12) can be written as

Proof. Let t = s/(1 − α), so that 1 < s < t. Using the H¨older inequality, we have

1/s |∇(T u)|s w1α dx B =

B

α/s

|∇(T u)|w1

≤ ∇(T u)t,B = ∇(T u)t,B

s

1/s dx

tα/(t−s) w dx B 1

B

w1 dx

α/s

(t−s)/st

(7.2.13)

.

Thus, from inequality (1.5.5), we obtain ∇(T u)t,B ≤ C1 |B|ut,B .

(7.2.14)

Choose m=

s , 1 + αλ(r − 1)

so that m < s. Substituting (7.2.14) into (7.2.13) and using Lemma 3.1.1, we ﬁnd that

230

7 Inequalities for operators

1/s |∇(T u)|s w1α dx

α/s ≤ C1 |B|ut,B B w1 dx B

≤ C2 |B||B|(m−t)/mt um,ρB

(7.2.15) B

w1 dx

α/s

.

Now, using the H¨ older inequality with 1/m = 1/s + (s − m)/sm, we have um,ρB 1/m = ρB |u|m dx = ≤

ρB

αλ/s

|u|w2

−αλ/s

m

1/m

w2

dx

1/s

|u|s w2αλ dx ρB

ρB

1 w2

(7.2.16) αλ(r−1)/s

1/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (7.2.16) into (7.2.15), we obtain B

|∇(T u)|s w1α dx

1/s

≤ C2 |B||B|(m−t)/mt ×

B

w1 dx

α/s

ρB

= C2 |B||B|(m−t)/mt

ρB

1 w2

ρB

|u|s w2α dx

1/s

αλ(r−1)/s

1/(r−1)

(7.2.17)

dx

|u|s w2α dx

1/s

α/s

αλ/s

w1 1,B · w12 1/(r−1),ρB .

Next, since (w1 , w2 ) ∈ Aλr (Ω), we ﬁnd α/s

αλ/s

w1 1,B · w12 1/(r−1),ρB αλ(r−1)/s

α/s 1 1/(r−1) = B w1 dx dx ρB w2 ≤

ρB

w1 dx

ρB

1 w2

λ(r−1) α/s

1/(r−1) dx

# 1 = |ρB|λ(r−1)+1 |ρB| w dx ρB 1 ×

1 |ρB|

ρB

1 w2

λ(r−1) α/s

1/(r−1)

≤ C3 |ρB|αλ(r−1)/s+α/s ≤ C4 |B|αλ(r−1)/s+α/s .

dx

(7.2.18)

7.2 Some basic estimates

231

Combining (7.2.18) and (7.2.17) and using α αλ(r − 1) m−t =− − , mt s s we ﬁnd

1/s |∇(T u)|s w1α dx

1/s

≤ C5 |B|

B

|u|s w2αλ dx

(7.2.19)

ρB

for all balls B with ρB ⊂ Ω. Thus, inequality (7.2.12) holds. The method similar to the proof of Theorem 7.2.1 also leads to the following inequality: T us,B,w1α ≤ C|B|diam(B)us,ρB,w2αλ ,

(7.2.20)

where α is any real number with 0 < α ≤ 1 and ρ > 1. Next, we prove the following local weighted imbedding theorem for diﬀerential forms under the homotopy operator T . Theorem 7.2.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : Ls (Ω, ∧l ) → W 1,s (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T uW 1,s (B),

w1α

≤ C|B|us,ρB,w2αλ

(7.2.21)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Proof. From the deﬁnition of the · W 1,s (B), (7.2.20), we have T uW 1,s (B),

w1α

norm, (7.2.12) , and

w1α

= diam(B)−1 T us,B,w1α + ∇(T u)s,B,w1α ≤ diam(B)−1 C1 |B|diam(B)us,ρB,w2αλ + C2 |B|us,ρB,w2αλ ≤ C1 |B|us,ρB,w2αλ + C2 |B|us,ρB,w2αλ ≤ C3 |B|us,ρB,w2αλ , which is equivalent to (7.2.21). In Theorems 7.2.1 and 7.2.2, the parameters α and λ are arbitrary real numbers with 0 < α < 1 and λ > 0. Therefore, for diﬀerent values of α

232

7 Inequalities for operators

and λ we obtain diﬀerent versions of the weighted imbedding inequality. For example, let λ = 1 and t = 1 − α in Theorem 7.2.1 and write dμ = w1 (x)dx and dν = w2 (x)dx. Then, inequality (7.2.12) reduces to |∇(T u)|

s

w1−t dμ

1/s

≤ C|B|

|u|

s

B

w2−t dν

1/s .

(7.2.22)

ρB

If we choose α = 1/r in Theorem 7.2.1, then (7.2.12) reduces to 1/s

|∇(T u)|

s

1/r w1 dx

1/s

≤ C|B|

|u|

s

B

λ/r w2 dx

,

(7.2.23)

ρB

where r > 1 and λ > 0. If we choose α = 1/s in Theorem 7.2.1, then 0 < α < 1 since 1 < s < ∞. Thus, (7.2.12) reduces to the following symmetric version: 1/s

|∇(T u)|

s

1/s w1 dx

1/s

≤ C|B|

B

|u|

s

λ/s w2 dx

,

(7.2.24)

ρB

where λ > 0. Next, set λ = s in (7.2.24), we have

1/s |∇(T u)|

s

1/s w1 dx

1/s

≤ C|B|

B

|u| w2 dx s

,

(7.2.25)

,

(7.2.26)

ρB

where s > 1. Finally, select λ = r in (7.2.23), to obtain

1/s 1/r

|∇(T u)|s w1 dx B

1/s

≤ C|B|

|u|s w2 dx ρB

where r > 1. In addition to the condition (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0, if w1 (x) ∈ Ar (Ω) in Theorems 7.2.1 and 7.2.2, then using Theorem 6.1.4, we can also prove that inequalities (7.2.12), (7.2.20), and (7.2.21) hold, that is, ∇(T u)s,B,w1 ≤ C|B|us,ρB,w2λ ,

(7.2.27)

T us,B,w1 ≤ C|B|diam(B)us,ρB,w2λ ,

(7.2.28)

T uW 1,s (B),w1 ≤ C|B|us,B,w2λ .

(7.2.29)

and

7.2 Some basic estimates

233

Using the local weighted inequalities developed above, we shall now prove the following global estimates. Theorem 7.2.3. Let u ∈ Ls (D, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain D ⊂ Rn and T : Ls (D, ∧l ) → W 1,s (D, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that ∇(T u)s,D,w1α ≤ Cus,D,w2αλ

(7.2.30)

≤ Cus,D,w2αλ

(7.2.31)

and T uW 1,s (D),

w1α

for any real number α with 0 < α < 1. Proof. Using (7.2.12) and Theorem 1.5.3, we ﬁnd that ∇(T u)s,D,w1α

1/s = D |∇(T u)|s w1α dx 1/s s αλ ≤ Q∈V C1 |Q| ρQ |u| w2 dx ≤ C1 |D|

Q∈V

|u|s w2αλ dx ρQ

1/s

(7.2.32)

1/s ≤ C1 |D| Q∈V D |u|s w2αλ dx

1/s ≤ C3 D |u|s w2αλ dx = C3 us,D,w2αλ since D is bounded. Thus, (7.2.30) holds. Similarly, using Theorem 1.5.3 and (7.2.20), we have T us,D,w1α ≤ C4 diam(D)us,D,w2αλ . From the deﬁnition of the · W 1,s (D), obtain T uW 1,s (D),

wα

(7.2.33)

norm, (7.2.32), and (7.2.33), we

w1α

= diam(D)−1 T us,D,w1α + ∇(T u)s,D,w1α ≤ C4 us,D,w2αλ + C3 us,D,w2αλ ≤ C5 us,D,w2αλ , which implies that (7.2.31) holds.

(7.2.34)

234

7 Inequalities for operators

Remark. Choosing some particular values of α in (7.2.30) and (7.2.31) leads to global results similar to the local case. Further, in addition to the condition (w1 , w2 ) ∈ Aλr (Ω), if w1 ∈ Ar (Ω) in Theorems 7.2.3, we ﬁnd that (7.2.30) and (7.2.31) become ∇(T u)s,D,w1 ≤ Cus,D,w2λ ,

(7.2.35)

T uW 1,s (D),w1 ≤ Cus,D,w2λ ,

(7.2.36)

respectively.

7.2.3 Estimates for d ◦ T We shall conclude this section with the local and global weighted estimates for d ◦ T , which will be used later. Theorem 7.2.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that σ > 1 and w ∈ Ar (Ω) for some r > 1. Then,

1/s |d(T u)| w dx s

α

≤ C|B|

1/s |u| w dx

(1−α)/s

s

B

α

(7.2.37)

σB

for all balls B with σB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. First, we assume that 0 < α < 1. Let t = s/(1−α). From Caccioppolitype estimates for the solutions of the A-harmonic equation, there exists a constant C1 , independent of u, such that −1

dut,B ≤ C1 diam(B)

ut,σB

(7.2.38)

for any solution u of the A-harmonic equation in Ω and all balls or cubes B with σB ⊂ Ω, where σ > 1. Now let m = s/(1 + α(r − 1)). Using the H¨ older inequality, (7.2.38), Lemma 3.1.1, and the basic estimate d(T u)s,B ≤ us,B + C2 diam(B)dus,B , we have

(7.2.39)

7.2 Some basic estimates

235

1/s |d(T u)|s wα dx

s 1/s = B |d(T u)|wα/s dx

B

≤ d(T u)t,B

wtα/(t−s) dx

α/s = d(T u)t,B B wdx

(t−s)/st

B

(7.2.40)

α/s ≤ (ut,B + C3 diam(B)dut,B ) B wdx

α/s ≤ (ut,B + C4 ut,σ1 B ) B wdx

α/s ≤ C5 ut,σ1 B B wdx

α/s ≤ C6 |B|(m−t)/mt um,σ2 B B wdx , where σ2 > σ1 > 1. Using the H¨older inequality again, we obtain um,σ2 B

m 1/m = σ2 B |u|wα/s w−α/s dx ≤

σ2 B

|u|s wα dx

1/s

1 1/(r−1) σ2 B

w

(7.2.41)

α(r−1)/s dx

for all balls B with σ2 B ⊂ Ω. Substituting (7.2.41) into (7.2.40), and then using the condition of Ar (Ω)-weights, we ﬁnd that

1/s

|d(T u)|s wα dx

≤ C6 |B|(1−α)/s

B

1/s |u|s wα dx

σ2 B

which ﬁnishes the proof of Theorem 7.2.4 for the case 0 < α < 1. For the case α = 1, the proof is similar to that of Theorem 7.2.1. Now, we shall prove the following global Ar (Ω)-weighted imbedding inequality in a bounded domain Ω for A-harmonic tensors. Theorem 7.2.5. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an Aharmonic tensor in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that 1/s 1/s |d(T u)|s wα dx ≤C |u|s wα dx (7.2.42) Ω

for any real number α with 0 < α ≤ 1.

Ω

236

7 Inequalities for operators

Proof. Using (7.2.37) and the covering lemma, we have

1/s |d(T u)|s wα dx 1/s (1−α)/s s α ≤ Q∈V C1 |Q| |u| w dx ρQ

Ω

≤ C1 |Ω|(1−α)/s

Q∈V

|u|s wα dx ρQ

1/s

1/s ≤ C1 |Ω|(1−α)/s Q∈V Ω |u|s wα dx

1/s ≤ C3 Ω |u|s wα dx , which implies that (7.2.42) holds.

7.3 Compositions of operators We prove the local Sobolev–Poincar´e imbedding theorems for the compositions of the gradient operator ∇, the homotopy operator T , the Laplace– Beltrami operator Δ, and Green’s operator G acting on a smooth form.

7.3.1 Estimates for ∇ ◦ T ◦ G and T ◦ Δ ◦ G Lemma 7.3.1. Let u ∈ C ∞ (∧l B), l = 1, 2, . . . , n, and 1 < p < ∞. Then, ∇(T (G(u))) ∈ Lp (∧l B) and T (Δ(G(u))) ∈ W 1,p (B, ∧l ). Moreover, there exists a constant C, independent of u, such that ∇(T (G(u)))p,B ≤ C|B|up,B

(7.3.1)

T (Δ(G(u)))W 1,p (B) ≤ C|B|up,B

(7.3.2)

and for all balls B with B ⊂ R . n

Proof. We only need to prove (7.3.1) and (7.3.2). In fact, from these two inequalities, the remaining part of the theorem follows. From inequality (1.5.5), we have ∇(T (ω))p,B ≤ C|B|ωp,B for any ω ∈

Lploc (∧l B).

(7.3.3)

Now, setting ω = G(u) in (7.3.3), we ﬁnd that

∇(T (G(u)))p,B ≤ C|B|G(u)p,B .

(7.3.4)

7.3 Compositions of operators

237

Using (3.3.3) and (7.3.4), we obtain (7.3.1). Now, applying (3.3.14), (1.5.5), (1.5.6), and Theorem 3.3.2, we obtain T (Δ(G(u)))W 1,p (B) = diam(B)−1 T (Δ(G(u)))p,B + ∇(T (Δ(G(u))))p,B ≤ diam(B)−1 · C1 |B|diam(B)Δ(G(u))p,B + C2 |B|Δ(G(u))p,B ≤ C3 |B|Δ(G(u))p,B ≤ C3 |B|up,B . Thus, inequality (7.3.2) holds. Theorem 7.3.2. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, T (Δ(G(u))) ∈ Lp (∧l B, w1α ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))p,B,w1α ≤ C|B|diam(B)up,ρB,w2α

(7.3.5)

for all balls B with ρB ⊂ Ω and any real number α with α < λ. Proof. Similar to the proof of Lemma 7.3.1, we only need to prove (7.3.5). We choose t = λp/(λ − α), then the H¨ older inequality yields T (Δ(G(u)))p,B,w1α p 1/p α/p = B |T (Δ(G(u)))|w1 dx ≤ T (Δ(G(u)))t,B = T (Δ(G(u)))t,B

tα/(t−p)

w B 1

B

w1λ dx

(t−p)/pt

(7.3.6)

dx

α/λp

.

Now, applying (1.5.6) and Theorem 3.3.2, we obtain T (Δ(G(u)))t,B ≤ C1 |B|diam(B)Δ(G(u))t,B

(7.3.7)

≤ C2 |B|diam(B)ut,B . Choose m = λp/(λ + α(r − 1)), so that m < p. By the weak reverse H¨older inequality, we have ut,B ≤ C3 |B|(m−t)/mt um,ρB .

(7.3.8)

238

7 Inequalities for operators

Combining (7.3.6), (7.3.7), and (7.3.8), we ﬁnd that T (Δ(G(u)))p,B,w1α ≤ C4 |B|diam(B)|B|(m−t)/mt um,ρB

B

w1λ dx

α/λp

.

(7.3.9)

Using the H¨ older inequality again with 1/m = 1/p + (p − m)/pm, it follows that um,ρB 1/m = ρB |u|m dx = ≤

ρB

α/p

|u|w2

−α/p

m

w2

|u|p w2α dx ρB

1/m

(7.3.10)

dx

1/p

ρB

1 w2

α(r−1)/λp

λ/(r−1) dx

for all balls B with ρB ⊂ M . Substituting (7.3.10) into (7.3.9), we obtain T (Δ(G(u)))p,B,w1α ≤ C4 |B|diam(B)|B|(m−t)/mt up,ρB,w2α α(r−1)/λp

α/λp 1 λ/(r−1) λ × B w1 dx dx . ρB w2

(7.3.11)

From the condition (w1 , w2 ) ∈ Ar,λ (M ), it is easy to ﬁnd the estimate

α/λp wλ dx B 1

≤ C5 |B|αr/λp ×

1 |ρB|

ρB

ρB

1 |ρB|

1 w2

1 w2

α(r−1)/λp

λ/(r−1) dx

wλ dx ρB 1

α/λp α(r−1)/λp

λ/(r−1)

(7.3.12)

dx

≤ C6 |B|αr/λp . Finally, substituting (7.3.12) into (7.3.11) and using (m − t)/mt = −αr/λp, we obtain T (Δ(G(u)))p,B,w1α ≤ C|B|diam(B)up,ρB,w2α . Using the same method developed in the proof of Theorem 7.3.2, we shall prove the following two-weight estimate for the composition of operators.

7.3 Compositions of operators

239

Theorem 7.3.3. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, ∇(T (Δ(G(u)))) ∈ Lp (∧l B, w1α ). Moreover, there exists a constant C, independent of u, such that ∇(T (Δ(G(u))))p,B,w1α ≤ C|B|up,ρB,w2α

(7.3.13)

for all balls B with ρB ⊂ Ω and any real number α with α < λ. Proof. We only need to prove (7.3.13). Choose t = λp/(λ − α), then from the H¨ older inequality, we ﬁnd that ∇(T (Δ(G(u))))p,B,w1α p 1/p α/p = B |∇(T (Δ(G(u))))|w1 dx ≤ ∇(T (Δ(G(u))))t,B = ∇(T (Δ(G(u))))t,B

tα/(t−p)

B

w1

B

w1λ dx

(t−p)/pt

(7.3.14)

dx

α/λp

.

From inequality (1.5.6) and Theorem 3.3.2, we obtain ∇(T (Δ(G(u))))t,B ≤ C1 |B|Δ(G(u))t,B ≤ C2 |B|ut,B .

(7.3.15)

Substituting (7.3.15) into (7.3.14), we ﬁnd that ∇(T (Δ(G(u))))

p,B,w1α

≤ C3 |B|ut,B

α/λp w1λ dx

.

(7.3.16)

B

Choosing m = λp/(λ+α(r−1)) and using the weak reverse H¨ older inequality, we have (7.3.17) ut,B ≤ C4 |B|(m−t)/mt um,ρB . Combining (7.3.16) and (7.3.17), it follows that ∇(T (Δ(G(u))))p,B,w1α ≤ C5 |B||B|(m−t)/mt um,ρB

B

w1λ dx

α/λp

.

(7.3.18)

Now, it is easy to see that inequality (7.3.18) plays the same role as inequality (7.3.9) in the proof of Theorem 7.3.2. Using the same strategy, we establish (7.3.13). Now, we are ready to prove one of our main results, the local two-weight Sobolev–Poincar´e imbedding theorem for the composition of operators applied to A-harmonic tensors in a bounded and convex domain.

240

7 Inequalities for operators

Theorem 7.3.4. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, T (Δ(G(u))) ∈ W 1,p (B, ∧l , w1α ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))W 1,p (B),w1α ≤ C|B|up,ρB,w2α

(7.3.19)

for all balls B with ρB ⊂ Ω and any real number α with α < λ. Proof. We only need to prove (7.3.19). From the deﬁnition of the · W 1,p (B),w1α norm, inequalities (7.3.5) and (7.3.13), we have T (Δ(G(u)))W 1,p (B),w1α = diam(B)−1 T (Δ(G(u)))p,B,w1α + ∇(T (Δ(G(u))))p,B,w1α ≤ diam(B)−1 · C1 |B|diam(B)up,ρB,w2α + C2 |B|up,ρB,w2α ≤ C1 |B|up,ρB,w2α + C2 |B|up,ρB,w2α ≤ C3 |B|up,ρB,w2α . Hence, inequality (7.3.19) follows. Note that we have proved rather general versions of two-weighted Lp estimates in Theorems 7.3.2, 7.3.3, and 7.3.4, where the parameters λ and α are any real numbers with λ ≥ 1 and α < λ. Thus, for example, choosing α = 1 in these results, we obtain Corollaries 7.3.5, ,7.3.6, and 7.3.7, respectively. Corollary 7.3.5. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ > 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, there exists a constant C, independent of u, such that T (Δ(G(u)))p,B,w1 ≤ C|B|diam(B)up,ρB,w2

(7.3.20)

for all balls B with ρB ⊂ Ω. Corollary 7.3.6. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ > 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, there exists a constant C, independent of u, such

7.3 Compositions of operators

241

that ∇(T (Δ(G(u))))p,B,w1 ≤ C|B|up,ρB,w2

(7.3.21)

for all balls B with ρB ⊂ Ω. Corollary 7.3.7. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ > 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, there exists a constant C, independent of u, such that (7.3.22) T (Δ(G(u)))W 1,p (B),w1 ≤ C|B|up,ρB,w2 for all balls B with ρB ⊂ Ω. If we let w1 (x) = w2 (x) = w(x) in (7.3.5), (7.3.13), and (7.3.19), respectively, we obtain T (Δ(G(u)))p,B,wα ≤ C|B|diam(B)up,ρB,wα , ∇(T (Δ(G(u))))p,B,wα ≤ C|B|up,ρB,wα , T (Δ(G(u)))W 1,p (B),wα ≤ C|B|up,ρB,wα , where the weight w(x) satisﬁes sup B⊂E

1/λr

1 |B|

(w)λ dx B

1 |B|

λr /r 1/λr 1 dx < ∞, w B

(7.3.23)

which is a generalization of the usual Ar -weights. It is easy to see that if w1 (x) = w2 (x) = w(x) and λ = 1, then the pair of weights (w1 , w2 ) satisfy sup B⊂E

1 |B|

1/r

wdx B

1 |B|

r /r 1/r 1 dx < ∞, w B

that is ⎛

1 sup ⎝ |B| B⊂E

wdx B

1 |B|

⎞ 1/(r−1) r−1 1/r 1 ⎠ dx <∞ w B

since r /r = 1/(r − 1), and hence the Ar,λ (Ω)-weight reduces to the usual Ar (Ω)-weight. Hence, setting w1 (x) = w2 (x) = w(x) and λ = 1 in Theorem 7.3.4, we obtain the following local Ar (Ω)-weighted imbedding theorem.

242

7 Inequalities for operators

Theorem 7.3.8. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 < p < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (Δ(G(u))) ∈ W 1,p (B, ∧l , wα ). Moreover, there exists a constant C, independent of u, such that (7.3.24) T (Δ(G(u)))W 1,p (B),wα ≤ C|B|up,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with α < 1. Here ρ > 1 is some constant.

7.3.2 Global estimates on manifolds Now, we shall prove the following global Ar,λ (Ω)-weighted Lp -estimates for operators in a bounded, convex domain M . Theorem 7.3.9. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, be an A-harmonic form in a bounded, convex domain Ω. Assume that 1 < p < ∞ and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1) and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, T (Δ(G(u))) ∈ Lp (∧l Ω, w1α ) and ∇(T (Δ(G(u)))) ∈ Lp (∧l Ω, w1α ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))p,Ω,w1α ≤ C|Ω|diam(Ω)up,Ω,w2α

(7.3.25)

and ∇(T (Δ(G(u))))p,Ω,w1α ≤ C|Ω|up,Ω,w2α

(7.3.26)

for any real number α with α < λ. Proof. Applying Theorems 7.3.2 and 1.5.3, we ﬁnd that T (Δ(G(u)))p,Ω,w1α ≤ B∈V T (Δ(G(u)))p,B,w1α

≤ B∈V C1 |B|diam(B)up,ρB,w2α

(7.3.27)

≤ C1 |Ω|diam(Ω)N up,Ω,w2α ≤ C2 |Ω|diam(Ω)up,Ω,w2α , which is equivalent to inequality (7.3.25). Similarly, we can prove (7.3.26). Now, we are ready to prove the following global two-weight Sobolev– Poincar´e imbedding theorem for the composition of operators applied to Aharmonic tensors in a bounded and convex domain.

7.3 Compositions of operators

243

Theorem 7.3.10. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 < p < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω. Assume that 1 < p < ∞, T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1), and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, T (Δ(G(u))) ∈ W 1,p (Ω, ∧l , w1α ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))W 1,p (Ω),w1α ≤ Cup,Ω,w2α

(7.3.28)

for any real number α with α < λ. Proof. Applying the deﬁnition of the · W 1,p (Ω),w1α norm, (7.3.25), and (7.3.26), we have T (Δ(G(u)))W 1,p (Ω),w1α = diam(Ω)−1 T (Δ(G(u)))p,Ω,w1α + ∇(T (Δ(G(u))))p,Ω,w1α ≤ C1 diam(Ω)−1 · |Ω|diam(Ω)up,Ω,w2α + C2 |Ω|up,Ω,w2α

(7.3.29)

≤ C1 |Ω|up,Ω,w2α + C2 |Ω|up,Ω,w2α ≤ C3 |Ω|up,Ω,w2α . Thus, inequality (7.3.28) holds. Choosing α = 1 in Theorem 7.3.10, we have the following version of the imbedding theorem. Theorem 7.3.11. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 < p < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω. Assume that 1 1 and 1 < r < ∞ with 1/r + 1/r = 1. Then, T (Δ(G(u))) ∈ W 1,p (B, ∧l , w1 ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))W 1,p (Ω),w1 ≤ Cup,Ω,w2 .

(7.3.30)

If we select w1 = w2 = w and λ = 1 in Theorem 7.3.10, we obtain the following Ar (Ω)-weighted Sobolev–Poincar´e imbedding theorem. Theorem 7.3.12. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 < p < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω. Assume that 1 < p < ∞ and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1) and w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (Δ(G(u))) ∈ W 1,p (Ω, ∧l , wα ). Moreover, there exists a constant C, independent of u, such

244

7 Inequalities for operators

that T (Δ(G(u)))W 1,p (Ω),wα ≤ Cup,Ω,wα

(7.3.31)

for any real number α with α < 1. Similarly, if we choose w1 = w2 = w and λ = 1 in Theorem 7.3.9, we obtain the following Ar (Ω)-weighted Lp -estimates which are very useful. Theorem 7.3.13. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n, 1 < p < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω. Assume that 1 < p < ∞ and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1), and w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (Δ(G(u))) ∈ Lp (∧l Ω, wα ) and ∇(T (Δ(G(u)))) ∈ Lp (∧l Ω, wα ). Moreover, there exists a constant C, independent of u, such that T (Δ(G(u)))p,Ω,wα ≤ C|Ω|diam(Ω)up,Ω,wα , ∇(T (Δ(G(u))))p,Ω,wα ≤ C|Ω|up,Ω,wα

(7.3.32)

for any real number α with α < 1. Remark. Similar to the local case, for diﬀerent choices of α and λ in Theorems 7.3.9 and 7.3.10, we have diﬀerent versions of the global Ar,λ (Ω) Sobolev–Poincar´e imbedding theorems.

7.3.3 Ls-estimates for T ◦ G Now, we prove the following local estimate for the composition of operators T and G acting on a diﬀerential form in a domain Ω, which can be used to develop the imbedding theorems for the composition of operators. Theorem 7.3.14. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (G(u)) ∈ Lsloc (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that T (G(u))s,B,wα ≤ C|B|diam(B)us,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

(7.3.33)

7.3 Compositions of operators

245

Proof. We only need to prove that inequality (7.3.33) holds. From inequalities (1.5.6) and (7.2.1), we have T (G(u))s,B ≤ C1 |B|diam(B)G(u)s,B

(7.3.34)

≤ C2 |B|diam(B)us,B . We ﬁrst show that (7.3.33) holds for 0 < α < 1. Let t = s/(1 − α). Using the H¨older inequality, we obtain T (G(u))s,B,wα =

B

|T (G(u))|wα/s

≤ T (G(u))t,B = T (G(u))t,B

s

1/s dx

B

wtα/(t−s) dx

B

wdx

α/s

(t−s)/st

(7.3.35)

.

By (7.3.34), we have T (G(u))t,B ≤ C1 |B|diam(B)ut,B .

(7.3.36)

Choose m = s/(1 + α(r − 1)), so that m < s. Substituting (7.3.36) into (7.3.35) and using Lemma 3.1.1, we ﬁnd that T (G(u))s,B,wα ≤ C1 |B|diam(B)ut,B

B

wdx

α/s

≤ C2 |B|diam(B)|B|(m−t)/mt um,ρB

B

wdx

α/s

(7.3.37) .

Now using H¨ older’s inequality with 1/m = 1/s + (s − m)/sm, we obtain um,ρB 1/m = ρB |u|m dx = ≤

ρB

|u|wα/s w−α/s

|u|s wα dx ρB

m

1/m

(7.3.38)

dx

1/s

1 1/(r−1) ρB

w

α(r−1)/s dx

246

7 Inequalities for operators

for all balls B with ρB ⊂ Ω. Substituting (7.3.38) into (7.3.37), we have T (G(u))s,B,wα ≤ C2 |B|diam(B)|B|(m−t)/mt ×

B

wdx

α/s

ρB

1 1/(r−1) ρB

w

|u|s wα dx

1/s (7.3.39)

α(r−1)/s dx

.

Since w ∈ Ar (Ω), it follows that α/s

α/s

w1,B · 1/w1/(r−1),ρB ≤

wdx

ρB

=

|ρB|r

1/(r−1)

ρB

1 |ρB|

(1/w)

wdx ρB

1 |ρB|

r−1 α/s dx

ρB

1 1/(r−1) w

r−1 α/s

(7.3.40)

dx

≤ C3 |B|αr/s .

Combining (7.3.40) and (7.3.39), we ﬁnd that T (G(u))s,B,wα ≤ C4 |B|diam(B)us,ρB,wα

(7.3.41)

for all balls B with ρB ⊂ Ω. Thus, (7.3.33) holds if 0 < α < 1. Next, we prove (7.3.33) for α = 1, that is, we need to show that T (G(u))s,B,w ≤ C|B|diam(B)us,ρB,w .

(7.3.42)

By Lemma 1.4.7, there exist constants β > 1 and C5 > 0, such that w β,B ≤ C5 |B|(1−β)/β w 1,B .

(7.3.43)

Choose t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Since 1/s = 1/t + (t − s)/st, by H¨ older’s inequality, (7.3.34), and (7.3.43), we have

7.3 Compositions of operators

|T (G(u))|s wdx

B

= ≤

B

B

247

1/s

|T (G(u))|w1/s

|T (G(u))|t dx

1/s

s

dx

1/t B

w1/s

st/(t−s)

(t−s)/st dx

1/s

≤ C6 T (G(u))t,B · wβ,B

(7.3.44) 1/s

≤ C6 |B|diam(B)ut,B · wβ,B 1/s

≤ C7 |B|diam(B)|B|(1−β)/βs w1,B · ut,B ≤ C7 |B|diam(B)|B|−1/t w1,B · ut,B . 1/s

Let m = s/r. From Lemma 3.1.1, we ﬁnd that ut,B ≤ C8 |B|(m−t)/mt um,ρB .

(7.3.45)

Now, using H¨older’s inequality again, we obtain um,ρB

m 1/m = ρB |u|w1/s w−1/s dx ≤

ρB

|u|s wdx

1/s

1 1/(r−1) w

ρB

(7.3.46)

(r−1)/s dx

.

Next, since w ∈ Ar (Ω), it follows that 1/s

1/s

w1,B · 1/w1/(r−1),ρB ≤ =

wdx ρB

|ρB|r

1 |ρB|

(1/w)1/(r−1) dx ρB

wdx ρB

1 |ρB|

ρB

r−1 1/s 1 1/(r−1) w

r−1 1/s dx

≤ C9 |B|r/s . Combining (7.3.44), (7.3.45), (7.3.46), and (7.3.47), we ﬁnd that

(7.3.47)

248

7 Inequalities for operators

T (G(u))s,B,w ≤ C10 |B|diam(B)|B|−1/t w1,B |B|(m−t)/mt um,ρB 1/s

≤ C10 |B|diam(B)|B|−1/m w1,B · 1/w1/(r−1),ρB us,ρB,w 1/s

1/s

≤ C11 |B|diam(B)us,ρB,w for all balls B with ρB ⊂ Ω. Hence, inequality (7.3.42) holds.

7.3.4 Local imbedding theorems for T ◦ G First, we prove an unweighted imbedding inequality for T ◦ G, and then extend it into the weighted case. Theorem 7.3.15. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (G(u))W 1,s (B) ≤ C|B|us,B

(7.3.48)

for all balls B with B ⊂ Ω. Proof. Using the deﬁnition of the · W 1,s (B) -norm, inequalities (1.5.5) and (1.5.6), and Lemma 3.3.1, we ﬁnd that T (G(u))W 1,s (B) = diam(B)−1 T (G(u))s,B + ∇(T (G(u)))s,B ≤ diam(B)−1 · C1 |B|diam(B)G(u)s,B + C2 |B|G(u)s,B ≤ C3 |B|G(u)s,B ≤ C4 |B|us,B . Theorem 7.3.16. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, ∇(T (G(u))) ∈ Lsloc (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that

7.3 Compositions of operators

∇(T (G(u)))s,B,w1α ≤ C|B|us,ρB,w2α

249

(7.3.49)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Proof. If (7.3.49) holds, then ∇(T (G(u))) ∈ Lsloc (Ω, ∧l ) follows. Hence, we only need to prove (7.3.49). From (1.5.5) and Lemma 3.3.1, we know that ∇(T (G(u)))s,B ≤ C|B|G(u)s,B ≤ C|B|us,B . Using the method developed in the proof of Theorem 5.7.3, we can extend the above inequality to the two-weighted case ∇(T (G(u)))s,B,w1α ≤ C|B|us,ρB,w2α . Choosing w1 (x) = w2 (x) = w(x) and setting λ = 1 in Theorem 7.3.16, we obtain the following Ar (Ω)-weighted inequality: ∇(T (G(u)))s,B,wα ≤ C|B|us,ρB,wα ,

(7.3.50)

where w ∈ Ar (Ω) and 0 < α < 1. Theorem 7.3.17 Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (G(u))W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.3.51)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. For 0 < α < 1, using inequalities (3.3.15), (7.3.50), and Theorem 7.3.14, we obtain T (G(u))W 1,s (B),wα = diam(B)−1 T (G(u))s,B,wα + ∇(T (G(u)))s,B,wα ≤ diam(B)−1 C1 diam(B)|B|us,ρ1 B,wα + C2 |B|us,ρ2 B,wα

(7.3.52)

≤ C3 |B|us,ρB,wα , where ρ = max{ρ1 , ρ2 }. For the case α = 1, using the method similar to the proof of Theorem 3.2.10, we can also prove that (7.3.51) holds.

250

7 Inequalities for operators

7.3.5 Global imbedding theorems for T ◦ G Next, we prove the global Ar (Ω)-weighted estimates for the composition of operators and the Ar (Ω)-weighted imbedding theorem for T ◦ G. Theorem 7.3.18. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that (7.3.53) T (G(u))s,Ω,wα ≤ C|Ω|diam(Ω)us,Ω,wα for any real number α with 0 < α ≤ 1. Proof. Let V = {Qi } be a Whitney cover of Ω. Applying Theorems 1.5.3 and 7.3.14, we obtain T (G(u))s,Ω,wα ≤ Q∈V T (G(u))s,Q,wα ≤ Q∈V (C1 |Q|diam(Q)us,ρQ,wα ) ≤ C2 |Ω|diam(Ω)N us,Ω,wα ≤ C3 |Ω|diam(Ω)us,Ω,wα .

Similar to the proof of Theorem 7.3.18, using Theorems 1.5.3 and 7.3.16, we obtain the following global weighted estimate for the composition of the operators ∇, T , and G. Theorem 7.3.19. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that ∇(T (G(u)))s,Ω,w1α ≤ C|Ω|us,Ω,w2α for any real number α with 0 < α < λ. Setting w1 (x) = w2 (x) = w(x) and λ = 1 in Theorem 7.3.19, we have the following Ar (Ω)-weighted inequality: ∇(T (G(u)))s,Ω,wα ≤ C|Ω|us,Ω,wα , where w ∈ Ar (Ω).

(7.3.54)

7.3 Compositions of operators

251

Now, we are ready to prove the following global Ar (Ω)-weighted Sobolev– Poincar´e imbedding theorem for the composition of operators applied to Aharmonic tensors. Theorem 7.3.20. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω. Assume that T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1) and w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (G(u)) ∈ W 1,s (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that T (G(u))W 1,s (Ω),wα ≤ C|Ω|us,Ω,wα

(7.3.55)

for any real number α with 0 < α ≤ 1. Proof. Applying (3.3.15), (7.3.54), and Theorem 7.3.18, we ﬁnd that T (G(u))W 1,s (Ω),wα = diam(Ω)−1 T (G(u))s,Ω,wα + ∇(T (G(u)))s,Ω,wα ≤ C1 diam(Ω)−1 · |Ω|diam(Ω)us,Ω,wα + C2 |Ω|us,Ω,wα ≤ C1 |Ω|us,Ω,wα + C2 |Ω|us,Ω,wα ≤ C3 |Ω|us,Ω,wα .

7.3.6 Some special cases We should notice that the parameter α in the above theorems makes our results more applicable and powerful. In fact, particular values of α lead to several interesting inequalities. For example, choosing α = 1 in Theorems 7.3.18 and 7.3.20, and (7.3.54), we have the following corollaries, respectively. Corollary 7.3.21. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned by (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that (7.3.56) T (G(u))s,Ω,w ≤ C|Ω|diam(Ω)us,Ω,w . Corollary 7.3.22. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that

252

7 Inequalities for operators

∇(T (G(u)))s,Ω,w ≤ C|Ω|us,Ω,w .

(7.3.57)

Corollary 7.3.23. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be an A-harmonic tensor in a bounded, convex domain Ω. Assume that T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1) and w ∈ Ar (Ω) for some 1 < r < ∞. Then, T (G(u)) ∈ W 1,s (Ω, ∧l ). Moreover, there exists a constant C, independent of u, such that T (G(u))W 1,s (Ω),w ≤ C|Ω|us,Ω,w .

(7.3.58)

Next, since 1 < s < ∞, letting α = 1/s in (7.3.53), (7.3.54), and (7.3.55), respectively, we obtain T (G(u))s,Ω,w1/s ≤ C|Ω|diam(Ω)us,Ω,w1/s , ∇(T (G(u)))s,Ω,w1/s ≤ C|Ω|us,Ω,w1/s , T (G(u))W 1,s (Ω),w1/s ≤ C|Ω|us,Ω,w1/s .

Similarly, selecting α = s/(r + s) in (7.3.53), (7.3.54), and (7.3.55), respectively, we ﬁnd that T (G(u))s,Ω,ws/(r+s) ≤ C|Ω|diam(Ω)us,Ω,ws/(r+s) , ∇(T (G(u)))s,Ω,ws/(r+s) ≤ C|Ω|us,Ω,ws/(r+s) , T (G(u))W 1,s (Ω),ws/(r+s) ≤ C|Ω|us,Ω,ws/(r+s) .

Finally, from Theorems 7.3.18 and 7.3.19, we have the following property for the compositions of operators. Corollary 7.3.24. The compositions T ◦ G and ∇ ◦ T ◦ G are bounded operators if Ω is a bounded, convex domain.

7.3.7 Ls-estimates for Δ ◦ G ◦ d In the following theorem, we study the composition of the operators Δ, G, and d. Theorem 7.3.25. Let u ∈ C ∞ (∧l Ω), l = 1, 2, . . . , n − 1, be an A-harmonic tensor on a bounded, convex Ω. Assume that ρ > 1, 1 < s < ∞, and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u,

7.3 Compositions of operators

253

such that Δ(G(du))s,B,wα ≤ Cdus,ρB,wα

(7.3.59)

for any ball B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. From inequality (7.2.1), for any smooth l-form ω, we have dd G(ω)s,B + d dG(ω)s,B ≤ C1 ωs,B ,

(7.3.60)

where 1 < s < ∞ and C1 is a constant. Choosing ω = du, we ﬁnd that dd G(du)s,B + d dG(du)s,B ≤ C1 dus,B .

(7.3.61)

By the deﬁnition of the Laplace–Beltrami operator Δ, Minkowski’s inequality, and (7.3.61), we obtain ΔG(du)s,B = (d d + dd )G(du)s,B ≤ d dG(du)s,B + dd G(du)s,B

(7.3.62)

≤ C1 dus,B . We shall ﬁrst show that (7.3.59) holds for 0 < α < 1. Let t = s/(1 − α). Using the H¨ older inequality and (7.3.62), we have Δ(G(du))s,B,wα =

B

|Δ(G(du))|wα/s

≤ Δ(G(du))t,B = Δ(G(du))t,B ≤ C1 dut,B

B

B

B

wdx

1/s

s

dx

wtα/(t−s) dx wdx

α/s

(t−s)/st

(7.3.63)

α/s

.

Let m = s/(1 + α(r − 1)), so that m < s. Applying Theorem 6.2.12, we ﬁnd that dut,B ≤ C2 |B|(m−t)/mt dum,ρB .

(7.3.64)

Now substituting (7.3.64) into (7.3.63), we have Δ(G(du))s,B,wα ≤ C3 |B|(m−t)/mt dum,ρB

α/s wdx

.

(7.3.65)

B

Using the H¨ older inequality again with 1/m = 1/s + (s − m)/sm, we obtain

254

7 Inequalities for operators

dum,ρB = =

|du|m dx

ρB

1/m

|du|wα/s w−α/s

ρB

≤ dus,ρB,wα

m

1/m dx

(7.3.66)

1 1/(r−1) w

ρB

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Next, substitution of (7.3.66) into (7.3.65) gives Δ(G(du))s,B,wα ≤ C3 |B|(m−t)/mt dus,ρB,wα

×

1 1/(r−1) w

ρB

B

wdx

α/s

α(r−1)/s dx

(7.3.67)

.

Since w ∈ Ar (Ω), it follows that α/s

α/s

w1,B · 1/w1/(r−1),ρB ≤

wdx ρB

=

|ρB|

r

1 1/(r−1) ρB

1 |ρB|

ρB

w

wdx

1 |ρB|

r−1 α/s dx ρB

1 1/(r−1) w

r−1 α/s

(7.3.68)

dx

≤ C4 |B|αr/s . Finally, combining (7.3.68) and (7.3.67), we obtain Δ(G(du))s,B,wα ≤ C5 dus,ρB,wα

(7.3.69)

for all balls B with ρB ⊂ Ω. Hence (7.3.59) holds if 0 < α < 1. Next, we prove (7.3.59) for α = 1, that is, we show that Δ(G(du))s,B,w ≤ Cdus,ρB,w .

(7.3.70)

By Lemma 1.4.7, there exist constants β > 1 and C6 > 0, such that w β,B ≤ C6 |B|(1−β)/β w 1,B

(7.3.71)

for any cube or ball B ⊂ Rn . Set t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Note that 1/s = 1/t + (t − s)/st. Using Theorem 1.1.4, (7.3.62), and (7.3.71), we have

7.3 Compositions of operators

B

= ≤

|Δ(G(du))|s wdx B

255

1/s

|Δ(G(du))|w1/s

|Δ(G(du))|t dx B

s

1/s dx

1/t B

w1/s

st/(t−s)

(t−s)/st dx

1/s

≤ C7 Δ(G(du))t,B · wβ,B

(7.3.72)

1/s

≤ C7 dut,B · wβ,B 1/s

≤ C8 |B|(1−β)/βs w1,B · dut,B ≤ C8 |B|−1/t w1,B · dut,B . 1/s

Next, choose m = s/r. Then, from Theorem 6.2.12, it follows that dut,B ≤ C9 |B|(m−t)/mt dum,ρB .

(7.3.73)

Thus, Theorem 1.1.4 yields dum,ρB

m 1/m = ρB |du|w1/s w−1/s dx ≤

ρB

|du|s wdx

1/s ρB

1 1/(r−1) w

(r−1)/s

(7.3.74)

dx

for all balls B with ρB ⊂ Ω. Now, since w ∈ Ar (Ω), (7.3.68) with α = 1 gives 1/s 1/s (7.3.74) w1,B · 1/w1/(r−1),ρB ≤ C10 |B|r/s . Finally, combining (7.3.72), (7.3.73), (7.3.74), and (7.3.74) , we have Δ(G(du))s,B,w ≤ C11 |B|−1/t w1,B |B|(m−t)/mt dum,ρB 1/s

≤ C11 |B|−1/m w1,B · 1/w1/(r−1),ρB dus,ρB,w 1/s

1/s

≤ C12 dus,ρB,w for all balls B with ρB ⊂ Ω. Hence, (7.3.70) holds.

256

7 Inequalities for operators

7.4 Poincar´ e-type inequalities for operators Recently, S. Ding, Y. Xing, and their co-workers investigated various estimates for diﬀerent compositions of operators, such as the homotopy operator, Green’s operator, the projection operator, and the maximal operator. We present some of their contributions in the remaining sections of this chapter. These results have applications in the Lp -theory of diﬀerential forms and related areas. We ﬁrst present the Poincar´e-type estimates for the compositions of operators, including T ◦ G, G ◦ T , and T ◦ H.

7.4.1 Poincar´ e-type inequalities for T ◦ G We begin with the Poincar´e-type estimates for the composition T ◦ G of the homotopy operator T and Green’s operator G. We will discuss single as well as two-weighted cases. Our ﬁrst result provides the local estimate in a bounded, convex domain Ω. Theorem 7.4.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))B s,B ≤ C|B|diam(B)us,B

(7.4.1)

for all balls B ⊂ Ω. Note that since B ⊂ Ω and Ω is bounded, inequality (7.4.1) can be written as T (G(u)) − (T (G(u)))B s,B ≤ Cus,B .

(7.4.1)

However, we include the factor |B|diam(B) on the right-hand side of (7.4.1), so that it is more ﬂexible, particularly, it helps in establishing the global estimates. Proof. It is well known that for any diﬀerential form u, we have T us,B ≤ C1 |B|diam(B)us,B

(7.4.2)

for all balls B ⊂ Ω. Recall the decomposition u = d(T u) + T (du) = uB + T (du)

(7.4.3)

for any diﬀerential form u, where uB = d(T u). Since Ω is bounded, there exists a constant C2 such that |B|diam(B) ≤ C2 . From (7.4.2) and (7.4.3), it follows that

7.4 Poincar´e-type inequalities for operators

257

d(T u)s,B = u − T (du)s,B ≤ us,B + T (du)s,B

(7.4.4)

≤ us,B + C3 |B|diam(B)dus,B . Now from Lemma 3.3.1, we ﬁnd that G(u)s,B ≤ C4 us,B and d(G(u))s,B ≤ C4 us,B

(7.4.5)

for some constant C4 > 0. Next, from (7.4.2), (7.4.3), (7.4.4), and (7.4.5), we obtain T (G(u)) − (T (G(u)))B s,B = T d(T (G(u)))s,B ≤ C5 |B|diam(B)dT (G(u))s,B ≤ C6 |B|diam(B)(G(u)s,B + C7 |B|diam(B)d(G(u))s,B )

(7.4.6)

≤ C6 |B|diam(B)(G(u)s,B + C8 d(G(u))s,B ) ≤ C6 |B|diam(B)(us,B + C9 us,B ) ≤ C10 |B|diam(B)us,B , and hence inequality (7.4.1) holds. Theorem 7.4.2. Suppose that u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, is a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wα

(7.4.7)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. The above Ls -norm inequality can also be written in the integral form as

1/s |T (G(u)) − (T (G(u)))B |s wα dx

B

1/s

≤C

|u|s wα dx

. (7.4.7)

ρB

Proof. First, we prove (7.4.7) for 0 < α < 1. Set t = s/(1 − α). Using the H¨ older inequality and (7.4.1), we have

258

7 Inequalities for operators

T (G(u)) − (T (G(u)))B s,B,wα =

B

|T G(u) − (T G(u))B |wα/s

≤ T G(u) − (T G(u))B t,B = T G(u) − (T G(u))B t,B ≤ C3 |B|diam(B)ut,B

B

dx

B

wtα/(t−s) dx

B

wdx

1/s

s

wdx

(t−s)/st (7.4.8)

α/s

α/s

.

Select m = s/(1 + α(r − 1)), so that m < s. Recalling the weak reverse H¨older inequality (7.4.9) ut,B ≤ C4 |B|(m−t)/mt um,ρB and using (7.4.8), we obtain T G(u) − (T G(u))B s,B,wα ≤ C5 |B|1+(m−t)/mt diam(B)um,ρB

B

wdx

α/s

.

(7.4.10)

Now using the H¨ older inequality again with 1/m = 1/s + (s − m)/sm, we have 1/m um,ρB = ρB |u|m dx =

|u|wα/s w−α/s

ρB

≤ us,ρB,wα

m

1/m dx

(7.4.11)

1 1/(r−1) w

ρB

α(r−1)/s dx

for all balls B with ρB ⊂ Ω. Substituting (7.4.11) into (7.4.10), we ﬁnd that T G(u) − (T G(u))B s,B,wα ≤ C6 |B|1+ ×

B

m−t mt

wdx

diam(B)us,ρB,wα

α/s

ρB

1 1/(r−1) w

Next, since w ∈ Ar (Ω), it follows that

(7.4.12) α(r−1)/s

dx

.

7.4 Poincar´e-type inequalities for operators α/s

259

α/s

w1,B · 1/w1/(r−1),ρB ≤

wdx ρB

=

|ρB|

r

1 |ρB|

r−1 α/s

1 1/(r−1) ρB

ρB

dx

w

wdx

1 |ρB|

1 1/(r−1)

ρB

w

r−1 α/s

(7.4.13)

dx

≤ C7 |B|αr/s . Combining (7.4.13) and (7.4.12), we obtain T G(u) − (T G(u))B s,B,wα ≤ C8 |B|diam(B)us,ρB,wα

(7.4.14)

for all balls B with ρB ⊂ Ω. Thus, (7.4.7) holds if 0 < α < 1. Next, we shall prove (7.4.7) for α = 1, that is, we will show that T G(u) − (T G(u))B s,B,w ≤ C|B|diam(B)us,ρB,w .

(7.4.15)

By Lemma 1.4.7, there exist constants β > 1 and C9 > 0, such that w β,B ≤ C9 |B|(1−β)/β w 1,B

(7.4.16)

for any cube or ball B ⊂ Rn . Set t = sβ/(β − 1), so that 1 < s < t and β = t/(t − s). Note that 1/s = 1/t + (t − s)/st. Using Theorem 1.1.4, (7.4.1), and (7.4.16), we ﬁnd that B

= ≤

|T G(u) − (T G(u))B |s wdx

1/s

B

B

|T G(u) − (T G(u))B |w1/s

|T G(u) − (T G(u))B |t dx

s

1/s dx

1/t B

w1/s

st/(t−s)

(t−s)/st dx

1/s

≤ C10 T G(u) − (T G(u))B t,B · wβ,B 1/s

≤ C11 |B|diam(B)ut,B · wβ,B 1/s

≤ C12 |B|1+(1−β)/βs diam(B)w1,B · ut,B 1/s

≤ C12 |B|1−1/t diam(B)w1,B · ut,B . (7.4.17) Next, choose m = s/r. Using the weak reverse H¨older inequality again for the new exponent, we have

260

7 Inequalities for operators

ut,B ≤ C13 |B|(m−t)/mt um,ρB .

(7.4.18)

Now, from Lemma 1.1.4, we ﬁnd that um,ρB = ≤

ρB

|u|w1/s w−1/s

|u|s wdx ρB

m

1/m dx

1/s

1 1/(r−1) w

ρB

(r−1)/s

(7.4.19)

dx

for all balls B with ρB ⊂ Ω. Finally, since w ∈ Ar (Ω), as for (7.4.13), we have 1/s 1/s (7.4.19) w1,B · 1/w1/(r−1),ρB ≤ C14 |B|r/s . Combining (7.4.17), (7.4.18), (7.4.19), and (7.4.19) , we obtain T G(u) − (T G(u))B s,B,w 1/s

≤ C15 |B|1−1/t diam(B)w1,B |B|(m−t)/mt um,ρB 1/s

1/s

≤ C15 |B|1−1/m diam(B)w1,B · 1/w1/(r−1),ρB us,ρB,w ≤ C16 |B|diam(B)us,ρB,w for all balls B with ρB ⊂ Ω. Hence, (7.4.15) follows. Theorem 7.4.2 for α = 1/s gives the inequality 1/s

|T (G(u)) − (T (G(u)))B | w s

1/s

dx

1/s

≤C

|u| w s

B

1/s

dx

.

ρB

Let α = 1 − t, 0 < t < 1 and the measure μ be deﬁned by dμ = w(x)dx. Then, from Theorem 7.4.2, we have

|T (G(u)) − (T (G(u)))B |s w−t dμ

1/s

|u|s w−t dμ

≤C

B

1/s .

ρB

If we choose α = 1 − 1/s and dμ = w(x)dx, then from Theorem 7.4.2, we obtain 1/s 1/s s −1/s s −1/s |T (G(u)) − (T (G(u)))B | w dμ ≤C |u| w dμ . B

Let α = 1 in Theorem 7.4.2. Then, we ﬁnd that

ρB

7.4 Poincar´e-type inequalities for operators

261

1/s |T (G(u)) − (T (G(u)))B |s wdx

1/s

≤C

|u|s wdx

B

.

ρB

Next, we extend our local weighted Poincar´e-type inequality to the global case.

Theorem 7.4.3. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous Aharmonic equation (1.2.10), T : C ∞ (Ω, ∧1 ) → C ∞ (Ω, ∧0 ) be the homotopy operator deﬁned in (1.5.1), and G be Green’s operator. Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that

1/s |T (G(u)) − (T (G(u)))Q0 | wdx s

1/s

≤C

Ω

|u| wdx s

(7.4.20)

Ω

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. Proof. Letting α = 1 in (7.4.7). Then, we can write (7.4.7) as s |T G(u) − (T G(u))Q | dμ(x) ≤ C1 |Q|diam(Q) |u|s dμ(x), Q

(7.4.21)

σQ

where the measure μ(x) is deﬁned by dμ(x) = w(x)dx. We use the notation and the covering V described in Theorem 1.5.3 (covering lemma) and the properties of the measure μ(x): if w ∈ Ar , then μ(N Q) ≤ M N nr μ(Q)

(7.4.22)

for each cube Q with N Q ⊂ Rn , and max(μ(Qi ), μ(Qi+1 )) ≤ M N nr μ(Qi ∩ Qi+1 )

(7.4.23)

for the sequence of cubes Qi , Qi+1 , i = 0, 1, . . . , k −1. Now, by the elementary inequality |a + b|s ≤ 2s (|a|s + |b|s ), s > 0, we ﬁnd that Ω

|T G(u) − (T G(u))Q0 |s wdx

=

Ω

≤ 2s

|T G(u) − (T G(u))Q0 |s dμ(x)

+2s

Q∈V

Q∈V

Q

|T G(u) − (T G(u))Q |s dμ(x)

Q

|(T G(u))Q0 − (T G(u))Q |s dμ(x).

(7.4.24)

262

7 Inequalities for operators

From (7.4.21) and the covering lemma, we estimate the ﬁrst sum as follows: s Q∈V Q |T G(u) − (T G(u))Q | dμ(x) ≤ C1 Q∈V |Q|diam(Q) σQ |u|s wdx ≤ C1

|u|s wdx σQ

Q∈V

≤ C1 N

Ω

(7.4.25)

|u|s wdx .

Next, we shall estimate the second sum in (7.4.24). Fix a cube Q ∈ V and let Q0 , Q1 , . . . , Qk = Q be the chain in the covering lemma. Clearly, |(T G(u))Q0 − (T G(u))Q | ≤

k−1

|(T G(u))Qi − (T G(u))Qi+1 |.

(7.4.26)

i=0

Using (7.4.21) and (7.4.23), we obtain |(T G(u))Qi − (T G(u))Qi+1 |s 1 = μ(Qi ∩Q |(T G(u))Qi − (T G(u))Qi+1 |s dμ(x) Qi ∩Qi+1 i+1 ) ≤

M N nr max(μ(Qi ),μ(Qi+1 ))

≤ C2 ≤ C3

i+1

1 j=i μ(Qj )

i+1 j=i

Qi ∩Qi+1

Qj

diam(Qj ) μ(Qj )

|(T G(u))Qi − (T G(u))Qi+1 |s dμ(x)

|T G(u) − (T G(u))Qj |s dμ(x)

σQj

|u|s wdx.

Now since Q ⊂ N Qj for j = i, i + 1, 0 ≤ i ≤ k − 1 (see the covering lemma), it follows that |(T G(u))Qi − (T G(u))Qi+1 |s χQ (x) i+1 χN Qj (x)diam(Ω) s |u| wdx . ≤ C3 j=i μ(Qj ) σQj By (7.4.26) and diam(Ω) < ∞, and |a + b|1/s ≤ 21/s (|a|1/s + |b|1/s ), we obtain

|(T G(u))Q0 − (T G(u))Q |χQ (x)

1/s 1 s ≤ C4 R∈V μ(R) |u| wdx · χN R (x) σR

7.4 Poincar´e-type inequalities for operators

263

for every x ∈ Rn . Hence, we have

Q∈V

Q

≤ C5

|(T G(u))Q0 − (T G(u))Q |s dμ(x)

s

1/s 1 s |u| wdx χN R (x) dμ(x). Rn R∈V μ(R) σR

(7.4.27)

From (7.4.27) and Theorem 1.4.15, it follows that

Q∈V

Q

|(T G(u))Q0 − (T G(u))Q |s dμ(x)

s

1/s 1 s dμ(x). |u| wdx χ (x) ≤ C6 Rn R∈V μ(R) R σR Finally, since

χR (x) ≤

R∈V

χσR (x) ≤ N χΩ (x),

R∈V

by the elementary inequality |

N

ti |s ≤ N s−1

i=1

N

|ti |s ,

i=1

and the covering lemma, we ﬁnd that

Q∈V

Q

|(T G(u))Q0 − (T G(u))Q |s dμ(x)

≤ C7 Rn R∈V = C7 ≤ C8

R∈V

Ω

1 μ(R)

s |u| wdx χ (x) dμ(x) R σR

|u|s wdx σR

(7.4.28)

|u|s wdx .

Combination of (7.4.24), (7.4.25), and (7.4.28) immediately gives (7.4.20). Now, we are ready to prove the local and global imbedding inequalities. Theorem 7.4.4. Suppose that u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, is a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists

264

7 Inequalities for operators

a constant C, independent of u, such that T (G(u)) − (T (G(u)))B W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.4.29)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Proof. For any diﬀerential form, we have the following basic inequality: d(T (u))s,B ≤ us,B + C1 diam(B)dus,B . Replacing u by G(u) in above inequality yields d(T (G(u)))s,B ≤ G(u)s,B + C1 diam(B)dG(u)s,B . Since Ω is bounded, using inequality (7.2.1), we have d(T (G(u)))s,B ≤ C2 us,B + C3 diam(B)us,B ≤ C4 us,B . From (1.5.5), (1.5.6), (7.2.1), and the above inequality, we have ∇((T (G(u))B )s,B = ∇((T (G(u)) − T d(T (G(u))))s,B ≤ ∇(T (G(u))s,B + ∇(T d(T (G(u))))s,B ≤ C6 |B|T (G(u)s,B + C7 |B|d(T (G(u)))s,B ≤ C8 diam(B)|B|2 G(u)s,B + C9 |B|us,B ≤ C10 diam(B)|B|2 us,B + C9 |B|us,B ≤ C11 |B|us,B since |B| ⊂ Ω and Ω is bounded. We can extend the above inequality into the weighted version ∇((T (G(u))B )s,B,wα ≤ C12 |B|us,ρB,wα .

(7.4.30)

Applying (3.3.15), (7.4.30), and Theorems 7.4.2 and 5.6.3, we obtain

7.4 Poincar´e-type inequalities for operators

265

T (G(u)) − (T (G(u)))B W 1,s (B),wα = diam(B)−1 T (G(u)) − (T (G(u)))B s,B,wα +∇(T (G(u)) − (T (G(u)))B )s,B,wα = diam(B)−1 T (G(u)) − (T (G(u)))B s,B,wα +∇(T (G(u))) − ∇((T (G(u)))B )s,B,wα ≤ diam(B)−1 T (G(u)) − (T (G(u)))B s,B,wα +∇(T (G(u)))s,B,wα + ∇((T (G(u)))B )s,B,wα ≤ C13 |B|us,ρB,wα . We should notice that all steps before inequality (7.4.30) in the proof of Theorem 7.4.4 will still hold if the ball B is replaced by a bounded, convex domain Ω ⊂ Rn . Thus, we have the following global inequality without weights. Theorem 7.4.4 . Suppose that u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, is a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))Ω W 1,s (Ω) ≤ C|Ω|us,Ω . Next, using Theorem 7.4.3, we prove the following global weighted imbedding inequality.

Theorem 7.4.5. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous Aharmonic equation (1.2.10), T : C ∞ (Ω, ∧1 ) → C ∞ (Ω, ∧0 ) be the homotopy operator deﬁned in (1.5.1), and G be Green’s operator. Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.4.31)

for any bounded δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. Proof. Since u is a 1-form, then (T (G(u)))Q0 is a closed 0-form (function). Thus, we have ∇((T (G(u)))Q0 )s,Ω,w = d((T (G(u)))Q0 )s,Ω,w = 0.

(7.4.32)

266

7 Inequalities for operators

Applying (3.3.15), (7.4.32), and Theorems 7.4.3 and 7.3.18, we obtain T (G(u)) − (T (G(u)))Q0 W 1,s (Ω),w = diam(Ω)−1 T (G(u)) − (T (G(u)))Q0 s,Ω,w +∇(T (G(u)) − (T (G(u)))Q0 )s,Ω,w = diam(Ω)−1 T (G(u)) − (T (G(u)))Q0 s,Ω,w +∇(T (G(u))) − ∇((T (G(u)))Q0 )s,Ω,w ≤ diam(Ω)−1 T (G(u)) − (T (G(u)))Q0 s,Ω,w +∇(T (G(u)))s,Ω,w + ∇((T (G(u)))Q0 )s,Ω,w = diam(Ω)−1 T (G(u)) − (T (G(u)))Q0 s,Ω,w + ∇(T (G(u)))s,Ω,w ≤ diam(Ω)−1 C1 diam(Ω)|Ω|us,Ω,w + C2 |Ω|us,Ω,w ≤ C3 |Ω|us,Ω,w ≤ C4 us,Ω,w , that is, T (G(u)) − (T (G(u)))Q0 W 1,s (Ω),w ≤ C4 us,Ω,w . In our earlier results, we have developed the Ar (Ω)-weighted Poincar´etype estimates for the composition T ◦ G. Now, we shall present estimates with diﬀerent weights, such as Ar (λ, Ω)-weights, Aλr (Ω)-weights. The proof of the following inequalities with Ar (λ, Ω)-weights is similar to the proof of Theorem 7.4.4. Theorem 7.4.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))B s,B,wαλ ≤ C|B|diam(B)us,ρB,wα

(7.4.33)

7.4 Poincar´e-type inequalities for operators

267

and T (G(u)) − (T (G(u)))B W 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(7.4.34)

for all balls B with ρB ⊂ Ω and arbitrary real number α with 0 < α < 1. Here ρ > 1 is some constant. Note that inequality (7.4.33) can be written as 1s

|T (G(u)) − (T (G(u)))B | w s

αλ

dx

≤ C|B|diam(B)

B

|u| w dx s

α

1s .

ρB

Similarly, we have the analog of inequalities (7.4.33) and (7.4.34) with Aλr (Ω)-weights. Theorem 7.4.7. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (G(u)) − (T (G(u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ

(7.4.35)

and T (G(u)) − (T (G(u)))B W 1,s (B),

wα

≤ C|B|us,ρB,wαλ

(7.4.36)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Each of the above two inequalities has an integral version. For example, inequality (7.4.35) can be written as |T (G(u)) − (T (G(u)))B |s wα dx B

1s

≤ C|B|diam(B)

|u|s wαλ dx

1s .

ρB

Now, we extend the above estimates into the following two-weight case. Theorem 7.4.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

268

7 Inequalities for operators

|T (G(u)) − (T (G(u)))B |s w1αλ dx

1s

≤ C|B|diam(B)

B

|u|s w2α dx

1s

ρB

(7.4.37) and T (G(u)) − (T (G(u)))B W 1,s (B),

(7.4.37)

≤ C|B|us,ρB,w2α

w1αλ

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Note that (7.4.37) can be written as T (G(u)) − (T (G(u)))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α .

(7.4.37)

Proof. Select t = s/(1−α), so that 1 < s < t. Noticing 1/s = 1/t+(t−s)/st and using the H¨ older inequality, we obtain B

= ≤

|T (G(u)) − (T (G(u)))B |s w1αλ dx

1/s

B

B

×

αλ/s

|T (G(u)) − (T (G(u)))B |w1

|T (G(u)) − (T (G(u)))B |t dx B

αλ/s

1/s dx

1/t (7.4.38)

(t−s)/st

st/(t−s)

w1

s

dx

≤ T (G(u)) − (T (G(u)))B t,B ·

B

w1λ dx

α/s

for all balls B with B ⊂ Ω. Applying Theorem 7.4.1, we have T (G(u)) − (T (G(u)))B t,B ≤ C1 |B|diam(B)ut,B .

(7.4.39)

Choose m = s/(1 + α(r − 1)), so that m < s < t. Substituting (7.4.39) into (7.4.38) and using Lemma 3.1.1, we obtain B

|T (G(u)) − (T (G(u)))B |s w1αλ dx

≤ C1 |B|diam(B)ut,B

B

w1λ dx

1/s

α/s

≤ C2 |B|diam(B)|B|(m−t)/mt um,ρB

(7.4.40) B

w1λ dx

α/s

.

Using the H¨ older inequality with 1/m = 1/s + (s − m)/sm, we ﬁnd that

7.4 Poincar´e-type inequalities for operators

269

um,ρB = = ≤

|u|m dx

ρB

−α/s

α/s

|u|w2 w2

ρB

1/m

|u|s w2α dx ρB

m

1/m

(7.4.41)

dx

1/s

ρB

1 w2

α(r−1)/s

1/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (7.4.41) into (7.4.40), it follows that

1/s |T (G(u)) − (T (G(u)))B |s w1αλ dx B ≤ C3 |B|diam(B)|B|(m−t)/mt ×

w1λ dx

B

α/s ρB

1 w2

|u|s w2α dx ρB

1/s (7.4.42)

α(r−1)/s

1/(r−1) dx

.

Now, using the condition (w1 , w2 ) ∈ Ar (λ, Ω), we obtain

α/s wλ dx B 1 ≤

ρB

wλ dx ρB 1

= |ρB|r

1 |ρB|

1 w2

ρB

dx

1/(r−1)

ρB

α(r−1)/s

1/(r−1)

r−1 α/s

(1/w2 )

dx

w1λ dx

1 |ρB|

(7.4.43) ρB

1 w2

1/(r−1)

r−1 α/s dx

≤ C4 |B|αr/s . Finally, combining (7.4.42) and (7.4.43), we ﬁnd that B

|T (G(u)) − (T (G(u)))B |s w1αλ dx

≤ C|B|diam(B)

ρB

|u|s w2α dx

1s

1s

for all balls B with ρB ⊂ Ω. The proof of inequality (7.4.37) is similar to that of Theorem 7.4.4. Theorem 7.4.9. Suppose that u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, is a diﬀerential form satisfying (1.2.10) in a bounded, convex domain

270

7 Inequalities for operators

Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) is a homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1 |T (G(u)) − (T (G(u)))B |s w1α dx s B ≤ C|B|diam(B)

|u|s w2α dx ρB

1s

(7.4.44)

and T (G(u)) − (T (G(u)))B W 1,s (B),

w1α

≤ C|B|diam(B)us,ρB,w2αλ (7.4.44)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that (7.4.44) can be written as T (G(u)) − (T (G(u)))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α .

(7.4.44)

Proof. Choose t = λs/(λ − α). Since 1/s = 1/t + (t − s)/st, from the H¨older inequality, we obtain B

= ≤

|T (G(u)) − (T (G(u)))B |s w1α dx

1/s

B

B

×

α/s

|T (G(u)) − (T (G(u)))B |w1

|T (G(u)) − (T (G(u)))B |t dx B

α/s

w1

s

1/s dx

1/t (7.4.45)

(t−s)/st

st/(t−s) dx

≤ T (G(u)) − (T (G(u)))B t,B ·

B

w1λ dx

α/λs

for all balls B with B ⊂ Ω. Using Theorem 7.4.1, we ﬁnd that T (G(u)) − (T (G(u)))B t,B ≤ C1 |B|diam(B)ut,B .

(7.4.46)

Set m = λs/(λ + α(r − 1)), so that m < s < t. Substituting (7.4.46) into (7.4.45) and using Lemma 3.1.1, we obtain

7.4 Poincar´e-type inequalities for operators

271

1/s |T (G(u)) − (T (G(u)))B |s w1α dx

α/λs ≤ C1 |B|diam(B)ut,B B w1λ dx

α/λs ≤ C2 |B|diam(B)|B|(m−t)/mt um,ρB B w1λ dx .

B

(7.4.47)

Now, using Theorem 1.1.4 with 1/m = 1/s + (s − m)/sm, we have um,ρB 1/m = ρB |u|m dx = ≤

ρB

m

−α/s

α/s

|u|w2 w2

|u|s w2α dx ρB

1/m

(7.4.48)

dx

1/s

ρB

1 w2

α(r−1)/λs

λ/(r−1) dx

for all balls B with ρB ⊂ Ω. Substituting (7.4.48) into (7.4.47), we obtain B

|T (G(u)) − (T (G(u)))B |s w1α dx

≤ C3 |B||B|(m−t)/mt ×

|u|s w2α dx ρB

B

w1λ dx

1/s

α/λs ρB

1 w2

1/s (7.4.49)

α(r−1)/λs

λ/(r−1) dx

.

Next, since (w1 , w2 ) ∈ Ar,λ (Ω), it follows that

wλ dx B 1 ≤

α/λs ρB

wλ dx ρB 1

= |ρB|r

1 |ρB|

ρB

1 w2

α(r−1)/λs

λ/(r−1) dx

(1/w2 )λ/(r−1) dx ρB w1λ dx

1 |ρB|

ρB

r−1 α/λs (7.4.50) 1 w2λ

1 r−1

r−1 α/λs dx

≤ C4 |B|αr/λs . Combining (7.4.49) and (7.4.50), we have |T (G(u)) − (T (G(u)))B |

s

B

w1α dx

1s

≤ C|B|diam(B)

|u|

s

ρB

w2α dx

1s

272

7 Inequalities for operators

for all balls B with ρB ⊂ Ω. This completes the proof of (7.4.44). The proof of (7.4.44) is similar to that of Theorem 7.4.4. Next, we shall discuss the following version of two-weight Poincar´e inequalities for the composition of the operators acting on diﬀerential forms. Theorem 7.4.10. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1s s α |T (G(u)) − (T (G(u))) | w dx B 1 B

1 ≤ C|B|diam(B) σB |u|s w2αλ dx s

(7.4.51)

and T (G(u)) − (T (G(u)))B W 1,s (B),

w1α

≤ C|B|us,σB,w2αλ

(7.4.51)

for all balls B with σB ⊂ Ω. Proof. Set t = s/(1 − α), so that 1 < s < t. Noticing that 1/s = 1/t + (t − s)/st, by the H¨ older inequality, we ﬁnd that B

≤ ≤

|T (G(u)) − (T (G(u)))B |s w1α dx

α/s

B

(|T (G(u)) − (T (G(u)))B |w1 )s dx

B

|T (G(u)) − (T (G(u)))B |t dx

×

1/s

αt/(t−s)

B

w1

1/s

1/t

(7.4.52)

(t−s)/st dx

= T (G(u)) − (T (G(u)))B t,B Next, choose m=

B

w1 dx

α/s

.

s . αλ(r − 1) + 1

Using Theorem 7.4.1 and Lemma 3.1.1, we obtain T (G(u)) − (T (G(u)))B t,B ≤ C1 |B|diam(B)ut,B ≤ C2 |B|diam(B)|B|(m−t)/mt um,σB

(7.4.53)

7.4 Poincar´e-type inequalities for operators

273

for all balls B with σB ⊂ Ω. Now since 1/m = 1/s + (s − m)/sm, by the H¨ older inequality again, we ﬁnd that um,σB = ≤

=

αλ/s

|u|w2

σB

|u|

1/s w2αλ dx

|u|

1/s w2αλ dx

s

σB

s

σB

−αλ/s

m

w2

1/m dx

σB

σB

1 w2

1 w2

(s−m)/sm

αλm/(s−m) dx

(7.4.54)

αλ(r−1)/s

1/(r−1) dx

.

From (7.4.52), (7.4.53), and (7.4.54), we have B

|T (G(u)) − (T (G(u)))B |s w1α dx

≤ C2 |B|(m−t)/tm |B|diam(B)

×

σB

1 w2

1/s

B

w1 dx

αλ(r−1)/s

1/(r−1) dx

α/s (7.4.55)

σB

|u|s w2αλ dx

1/s

.

Since (w1 , w2 ) ∈ Aλr (Ω), it follows that

w dx B 1 =

α/s

B

σB

w1 dx

1 w2

σB

αλ(r−1)/s

1/(r−1) dx

1 w2

λ(r−1) α/s

1/(r−1) dx

# 1 ≤ |σB|λ(r−1)+1 |σB| w dx σB 1 ×

1 |σB|

σB

1 w2

λ(r−1)

1/(r−1)

(7.4.56) α/s

dx

≤ C3 |σB|αλ(r−1)/s+α/s ≤ C4 |B|αλ(r−1)/s+α/s . Finally, substituting (7.4.56) into (7.4.55) and using −α αλ(r − 1) m−t = − , mt s s

274

7 Inequalities for operators

we obtain 1s |T (G(u)) − (T (G(u)))B |s w1α dx ≤ C|B|diam(B) B

|u|s w2αλ dx

1s

σB

Thus, inequality (7.4.51) holds. By the same method used in the proof of Theorem 7.4.4, we can prove inequality (7.4.51) . For λ = 1/α, Theorem 7.4.10 reduces to the following corollary. Corollary 7.4.11. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1 |T (G(u)) − (T (G(u)))B |s w1α dx s

1 ≤ C|B|diam(B) σB |u|s w2 dx s

B

(7.4.57)

and T (G(u)) − (T (G(u)))B W 1,s (B),

w1α

≤ C|B|us,σB,w2

for all balls B with σB ⊂ Ω. Selecting α = 1/s in Theorem 7.4.10, we have the following two-weighted Poincar´e inequalities. Corollary 7.4.12. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that B

1

|T (G(u)) − (T (G(u)))B |s w1s dx

≤ C|B|diam(B)

λ/s

1/s

|u|s w2 dx σB

1/s

(7.4.58)

and T (G(u)) − (T (G(u)))B W 1,s (B),

1/s

w1

≤ C|B|us,σB,wλ/s 2

for all balls B with σB ⊂ Ω. The choice λ = 1 in Corollary 7.4.12 gives the following symmetric twoweighted inequalities.

.

7.4 Poincar´e-type inequalities for operators

275

Corollary 7.4.13. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1

|T (G(u)) − (T (G(u)))B |s w1s dx B

≤ C|B|diam(B)

|u|

s

σB

1/s w2 dx

1/s

1/s

(7.4.59)

and T (G(u)) − (T (G(u)))B W 1,s (B),

1/s

w1

≤ C|B|us,σB,w1/s 2

for all balls B with σB ⊂ Ω. Finally, when λ = s in Theorem 7.4.10, we obtain the following twoweighted inequalities. Corollary 7.4.14. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1 |T (G(u)) − (T (G(u)))B |s w1α dx s

1 ≤ C|B|diam(B) σB |u|s w2αs dx s

B

(7.4.60)

and T (G(u)) − (T (G(u)))B W 1,s (B),

w1α

≤ C|B|us,σB,w2αs

for all balls B with σB ⊂ Ω.

7.4.2 Poincar´ e-type inequalities for G ◦ T In the previous section, we have obtained various estimates for the composition T ◦G. In this section, we present local and global Poincar´e-type estimates for G ◦ T with single weight and two-weights. We begin with the following basic Poincar´e-type estimates for G ◦ T applied to the smooth solutions of the nonhomogeneous A-harmonic equation. Theorem 7.4.15. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1).

276

7 Inequalities for operators

Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))B s,B ≤ C|B|diam(B)us,B

(7.4.61)

for all balls B ⊂ Ω. As in Theorem 7.4.1, Poincar´e-type estimate (7.4.61) can also be written as G(T (u)) − (G(T (u)))B s,B ≤ Cus,B ,

(7.4.61)

but because of the same reason as in (7.4.1), we keep the factor |B|diam(B) on the right-hand side of (7.4.61). Proof. Replacing u by T u in the second inequality in (7.4.5) and using (7.4.2), we have d(G(T u))s,B ≤ C1 T us,B ≤ C2 |B|diam(B)us,B

(7.4.62)

≤ C3 us,B . From (7.4.2) and replacing u by G(u) in (7.4.4), then using (7.4.62), we obtain G(T (u)) − (G(T (u)))B s,B = T d(G(T (u)))s,B ≤ C4 |B|diam(B)d(G(T (u)))s,B

(7.4.63)

≤ C5 B|diam(B)us,B . Thus, inequality (7.4.61) holds. Theorem 7.4.16. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wα

(7.4.64)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. The above Ls -norm inequality can also be written in the integral form as

7.4 Poincar´e-type inequalities for operators

277

1/s |G(T (u)) − (G(T (u)))B |s wα dx

1/s

≤C

|u|s wα dx

B

ρB

. (7.4.64)

Theorem 7.4.17. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10), T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1), and G be Green’s operator. Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that

1/s |G(T (u)) − (G(T (u)))Q0 |s wdx

Ω

1/s

≤C

|u|s wdx

(7.4.65)

Ω

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. Next, we present the local and global imbedding inequalities. Theorem 7.4.18. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))B W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.4.66)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

Theorem 7.4.19. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10), T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1), and G be Green’s operator. Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.4.67)

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. We have presented above the Ar (Ω)-weighted Poincar´e-type estimates for the composition G ◦ T . Now, we present some other estimates with diﬀerent weights, such as Ar (λ, Ω)-weights, Aλr (Ω)-weights. The proofs of the following results are similar to those provided in earlier chapters, and hence are omitted.

278

7 Inequalities for operators

Theorem 7.4.20. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))B s,B,wαλ ≤ C|B|diam(B)us,ρB,wα

(7.4.68)

and G(T (u)) − (G(T (u)))B W 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(7.4.69)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Here ρ > 1 is some constant. Note that inequality (7.4.68) can be written as 1s

|G(T (u)) − (G(T (u)))B | w s

αλ

dx

≤ C|B|diam(B)

B

|u| w dx s

α

1s .

ρB

Theorem 7.4.21. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be a homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that G(T (u)) − (G(T (u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ

(7.4.70)

and G(T (u)) − (G(T (u)))B W 1,s (B),

wα

≤ C|B|diam(B)us,ρB,wαλ (7.4.71)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. It should be noticed that each of the above two inequalities has an integral version. For example, inequality (7.4.70) can be written as |G(T (u)) − (G(T (u)))B |s wα dx B

1s

≤ C|B|diam(B)

|u|s wαλ dx

1s .

ρB

We also notice that the above estimates can be extended into the following two-weight case. Theorem 7.4.22. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded,

7.4 Poincar´e-type inequalities for operators

279

convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |G(T (u)) − (G(T (u)))B |

s

w1αλ dx

1s

≤ C|B|diam(B)

B

|u|

s

w2α dx

1s

ρB

(7.4.72) and (7.4.72)

G(T (u)) − (G(T (u)))B W 1,s (B),w1αλ ≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Note that (7.4.72) can be written as G(T (u)) − (G(T (u)))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α .

(7.4.72)

Theorem 7.4.23. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1 |G(T (u)) − (G(T (u)))B |s w1α dx s 1s ≤ C|B|diam(B) ρB |u|s w2α dx

B

(7.4.73)

and (7.4.73)

G(T (u)) − (G(T (u)))B W 1,s (B),w1α ≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that (7.4.73) can be written as G(T (u)) − (G(T (u)))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α .

(7.4.73)

Next, we discuss the following version of two-weight Poincar´e inequalities for the compositions of the operators acting on diﬀerential forms. Theorem 7.4.24. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1).

280

7 Inequalities for operators

Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1 |G(T (u)) − (G(T (u)))B |s w1α dx s

1 ≤ C|B|diam(B) σB |u|s w2αλ dx s

B

(7.4.74)

and G(T (u)) − (G(T (u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αλ

(7.4.74)

for all balls B with σB ⊂ Ω. If we choose λ = 1/α in Theorem 7.4.24, we have the following version of the Aλr -weighted Poincar´e inequality. Corollary 7.4.25. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1s s α |G(T (u)) − (G(T (u))) | w dx B 1 B

1 ≤ C|B|diam(B) σB |u|s w2 dx s

(7.4.75)

for all balls B with σB ⊂ Ω. Choosing α = 1/s in Theorem 7.4.24, we have the following two-weighted Poincar´e inequality. Corollary 7.4.26. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that

1

|G(T (u)) − (G(T (u)))B |s w1s dx B

≤ C|B|diam(B)

λ/s

1s

|u|s w2 dx σB

1s

(7.4.76)

for all balls B with σB ⊂ Ω. When λ = 1 in Corollary 7.4.26, we obtain the following symmetric twoweight inequality.

7.5 The homotopy operator

281

Corollary 7.4.27. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1

|G(T (u)) − (G(T (u)))B |s w1s dx B

≤ C|B|diam(B)

1

|u|s w2s dx σB

1s

1s

(7.4.77)

for all balls B with σB ⊂ Ω. Finally, choose λ = s in Theorem 7.4.24, to obtain the following two-weight inequalities. Corollary 7.4.28. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1 |G(T (u)) − (G(T (u)))B |s w1α dx s

1 ≤ C|B|diam(B) σB |u|s w2αs dx s

B

(7.4.78)

and G(T (u)) − (G(T (u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αs

(7.4.78)

for all balls B with σB ⊂ Ω.

7.5 The homotopy operator We have developed various Poincar´e-type estimates for the compositions T ◦ G and G ◦ T in the previous sections. Now, we obtain Poincar´e-type estimates for the homotopy operator T applied to the diﬀerential forms satisfying the nonhomogeneous A-harmonic equation.

7.5.1 Basic estimates for T First, we prove the following basic local Poincar´e-type estimate for the homotopy operator T . Then, we extend it to the diﬀerent weighted cases and the global cases. We also obtain some weighted imbedding theorems.

282

7 Inequalities for operators

Theorem 7.5.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (u) − (T (u))B s,B ≤ C|B|diam(B)us,ρB

(7.5.1)

for all balls B with ρB ⊂ Ω, where ρ > 1 is a constant. Proof. Choosing the closed form c = 0 in the Caccioppoli inequality, we have dus,B ≤ C1 (diam(B))−1 us,ρB

(7.5.2)

for any solution u of the A-harmonic equation. Using the decomposition for the form T (u) yields T (u) = d(T (T (u))) + T (d(T (u))) = (T (u))B + T (d(T (u))).

(7.5.3)

Now, since B ⊂ Ω and Ω is bounded, we have |B| ≤ K for some constant K > 0. Thus, from (7.4.2), (7.4.4), and (7.5.2), it follows that T (u) − (T (u))B s,B = T d(T (u))s,B ≤ C2 |B|diam(B)d(T (u))s,B ≤ C2 |B|diam(B)(us,B + C3 |B|diam(B)dus,B )

(7.5.4)

≤ C2 |B|diam(B)(us,B + C4 |B|us,ρB ) ≤ C2 |B|diam(B)(us,B + C5 us,ρB ) ≤ C6 |B|diam(B)us,ρB , that is T (u) − (T (u))B s,B ≤ C7 |B|diam(B)us,ρB .

(7.5.5)

From the proof of Theorem 7.5.1, it is easy to see that the following result still holds if the condition that u is a solution of the A-harmonic equation is dropped. We state it as the following theorem. Theorem 7.5.1 . Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω and T :

7.5 The homotopy operator

283

C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (u) − (T (u))B s,B ≤ C|B|diam(B)(us,B + dus,B )

(7.5.5)

for all balls B ⊂ Ω.

7.5.2 Ar (Ω)-weighted estimates for T Based on the basic Poincar´e-type estimate for the homotopy operator T established above, we can state the following Ar (Ω)-weighted inequality. Theorem 7.5.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (u) − (T (u))B s,B,wα ≤ C|B|diam(B)us,ρB,wα

(7.5.6)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. The above Ls -norm inequality can also be written in the integral form as follows: 1/s 1/s s α s α |T (u) − (T (u))B | w dx ≤C |u| w dx . (7.5.6) B

ρB

Also, using the procedure developed to extend the local inequalities into the John domains, we have the following global Poincar´e-type inequality.

Theorem 7.5.3. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that

1/s |T (u) − (T (u))Q0 | wdx s

Ω

1/s

≤C

|u| wdx s

(7.5.7)

Ω

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube.

284

7 Inequalities for operators

7.5.3 Poincar´ e-type imbedding for T By the same method used to prove the imbedding inequalities, we can prove the following local and global imbedding inequalities, Theorems 7.5.4 and 7.5.5, respectively. Theorem 7.5.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w(x) ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (u) − (T (u))B W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.5.8)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

Theorem 7.5.5. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that T (u) − (T (u))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.5.9)

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. So far, we have presented the Ar (Ω)-weighted Poincar´e-type estimates for the homotopy operator T . Now, we state other estimates with diﬀerent weights, such as Ar (λ, Ω)-weights, Aλr (Ω)-weights. Theorem 7.5.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (u) − (T (u))B s,B,wαλ ≤ C|B|diam(B)us,ρB,wα

(7.5.10)

and T (u) − (T (u))B W 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(7.5.11)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Here ρ > 1 is some constant. Note that inequality (7.5.10) can be written as

7.5 The homotopy operator

285

|T (u) − (T (u))B |s wαλ dx

1s

≤ C|B|diam(B)

|u|s wα dx

B

1s .

ρB

Theorem 7.5.7. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (u) − (T (u))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ

(7.5.12)

and T (u) − (T (u))B W 1,s (B),

wα

≤ C|B|us,ρB,wαλ

(7.5.13)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. The above inequalities have integral representations, for example, inequality (7.5.12) can be written as |T (u) − (T (u))B | w dx s

α

1s

≤ C|B|diam(B)

B

|u| w s

αλ

1s dx

.

ρB

7.5.4 Two-weight Poincar´ e-type imbedding for T The above estimates can be extended into the following two-weight case. Theorem 7.5.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |T (u) − (T (u))B |

s

w1αλ dx

1s

≤ C|B|diam(B)

B

|u|

s

w2α dx

1s

ρB

(7.5.14) and

(7.5.14)

T (u) − (T (u))B W 1,s (B),w1αλ ≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Note that inequality (7.5.14) can be written as T (u) − (T (u))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α .

(7.5.14)

286

7 Inequalities for operators

In Theorem 7.5.8, we have assumed that (w1 , w2 ) ∈ Ar (λ, Ω). If the weights w1 and w2 satisfy some other condition, say (w1 , w2 ) ∈ Ar,λ (Ω), we have the following version of Poincar´e-type inequality. Theorem 7.5.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |T (u) − (T (u))B |

s

w1α dx

1s

≤ C|B|diam(B)

B

|u|

s

w2α dx

1s

ρB

(7.5.15) and T (u) − (T (u))B W 1,s (B),w1α ≤ C|B|us,ρB,w2α

(7.5.16)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that inequality (7.5.15) can be written as T (u) − (T (u))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α .

(7.5.16)

Similarly, if (w1 , w2 ) ∈ Aλr (Ω), we have the following version of two-weight Poincar´e inequality for diﬀerential forms. Theorem 7.5.10. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that |T (u) − (T (u))B |s w1α dx B

1s

≤ C|B|diam(B)

|u|s w2αλ dx

1s

σB

(7.5.17) and T (u) − (T (u))B W 1,s (B),w1α ≤ C|B|us,σB,w2αλ

(7.5.18)

for all balls B with σB ⊂ Ω. If we choose λ = 1/α in Theorem 7.5.10, we have the following version of 1/α the Poincar´e inequality with (w1 , w2 ) ∈ Ar (Ω). Corollary 7.5.11. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

7.5 The homotopy operator

287

|T (u) − (T (u))B |s w1α dx

1s

≤ C|B|diam(B)

|u|s w2 dx

B

1s

σB

(7.5.19) for all balls B with σB ⊂ Ω. Choosing α = 1/s in Theorem 7.5.10, we obtain the following two-weighted Poincar´e inequality. Corollary 7.5.12. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that

1 s

|T (u) − (T (u))B | w1 dx s

1s

≤ C|B|diam(B)

B

|u|

s

λ/s w2 dx

1s

σB

(7.5.20) for all balls B with σB ⊂ Ω. Letting λ = 1 in Corollary 7.5.12, we ﬁnd the following symmetric twoweighted inequality. Corollary 7.5.13. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1 s

|T (u) − (T (u))B | w1 dx s

B

1s

≤ C|B|diam(B)

1 s

|u| w2 dx s

1s

σB

(7.5.21) for all balls B with σB ⊂ Ω. Finally, when λ = s in Theorem 7.5.10, we have the following two-weighted inequality. Corollary 7.5.14. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that 1s 1s s α s αs |T (u) − (T (u))B | w1 dx ≤ C|B|diam(B) |u| w2 dx B

σB

(7.5.22)

288

7 Inequalities for operators

and T (u) − (T (u))B W 1,s (B),w1α ≤ C|B|us,σB,w2αs for all balls B with σB ⊂ Ω.

7.6 Homotopy and projection operators In the previous sections, we have discussed Poincar´e-type estimates for some compositions of operators. In this section, we shall develop Poincar´e-type estimates for the composition of the homotopy operator T and the projection operator H which are applied to the diﬀerential forms or the solutions of the nonhomogeneous A-harmonic equation.

7.6.1 Basic estimates for T ◦ H As we have done in Section 7.5, here also, ﬁrst, we shall prove the following basic local Poincar´e-type estimate for the composition T ◦ H. Then, we shall discuss the diﬀerent weighted cases, the local cases and the global cases. As applications of the Ls -norm inequalities, we will also develop some weighted imbedding inequalities. Theorem 7.6.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))B s,B ≤ C|B|diam(B)us,ρB

(7.6.1)

for all balls B with ρB ⊂ Ω, where ρ > 1 is a constant. Proof. Let H be the projection operator and T be the homotopy operator. Then, for any diﬀerential form u, we have H(u) = u − ΔG(u).

(7.6.2)

Replacing u by du in (7.6.2), we obtain H(du) = du − ΔG(du).

(7.6.3)

Now, since the diﬀerential operator d commutes with Δ and G, using (7.6.3) we ﬁnd that

7.6 Homotopy and projection operators

289

d(H(u)) = d(u − ΔG(u)) = du − dΔG(u) = du − ΔG(du)

(7.6.4)

= H(du). Thus, H commutes with the diﬀerential operator d. Next, from (7.6.3) and Theorem 3.3.2, we have d(H(u))s,B = H(du)s,B = du − ΔG(du)s,B ≤ dus,B + ΔG(du)s,B

(7.6.5)

≤ dus,B + C1 dus,B ≤ C2 dus,B . Using (7.4.2), (7.4.4), and (7.6.5), we ﬁnd that T (H(u)) − (T (H(u)))B s,B = T d(T (H(u)))s,B ≤ C3 |B|diam(B)d(T (H(u)))s,B ≤ C3 |B|diam(B)(H(u)s,B +C4 |B|diam(B)d(H(u))s,B )

(7.6.6)

≤ C3 |B|diam(B)(H(u)s,B +C5 diam(B)d(H(u))s,B ) ≤ C3 |B|diam(B)(H(u)s,B + C5 diam(B)dus,B ) ≤ C3 |B|diam(B)(H(u)s,B + C6 us,ρB ). Applying Theorem 3.3.2 and (7.6.2), it follows that H(u)s,B = u − ΔG(u)s,B ≤ us,B + ΔG(u)s,B ≤ us,B + C7 us,B ≤ C8 us,B .

(7.6.7)

290

7 Inequalities for operators

Finally, substituting (7.6.7) into (7.6.6), we obtain T (H(u)) − (T (H(u)))B s,B ≤ C3 |B|diam(B)(H(u)s,B + C6 us,ρB ) ≤ C9 |B|diam(B)us,ρB which is equivalent to inequality (7.6.1). Note that in the proof of Theorem 7.6.1, we have used Caccioppoli inequality to obtain (7.6.6), which requires that u to be a solution of the A-harmonic equation. If we drop this condition, then the above result can be stated as follows. Theorem 7.6.1 . Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that (7.6.1)

T (H(u)) − (T (H(u)))B s,B ≤ C|B|diam(B)(us,B + dus,B ) for all balls B ⊂ Ω.

7.6.2 Ar (Ω)-weighted inequalities for T ◦ H Using the unweighted Poincar´e-type inequality obtained above, we can easily establish the Ar (Ω)-weighted inequalities for T ◦ H. We state these results in the following theorems. Theorem 7.6.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wα

(7.6.8)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Note that the above Ls -norm inequality can also be written as

1/s |T (H(u)) − (T (H(u)))B | w dx s

B

α

1/s

≤C

|u| w dx s

ρB

α

. (7.6.8)

7.6 Homotopy and projection operators

291

Theorem 7.6.3. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that

1/s |T (H(u)) − (T (H(u)))Q0 | wdx s

1/s

≤C

Ω

|u| wdx s

(7.6.9)

Ω

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. Using Theorem 7.6.2 and the deﬁnition of the · W 1,s (E),wα -norm, we can prove the following local and global imbedding inequalities. Theorem 7.6.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))B W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.6.10)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

Theorem 7.6.5. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.6.11)

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube.

7.6.3 Other single weighted cases In the previous several theorems, we have presented the Ar (Ω)-weighted Poincar´e-type estimates for the composition T ◦ H. Now, we state results with diﬀerent weights, namely Ar (λ, Ω)-weights and Aλr (Ω)-weights.

292

7 Inequalities for operators

Theorem 7.6.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))B s,B,wαλ ≤ C|B|diam(B)us,ρB,wα

(7.6.12)

and T (H(u)) − (T (H(u)))B W 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(7.6.13)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Here ρ > 1 is some constant. The above inequality (7.6.12) can be written as 1s

|T (H(u)) − (T (H(u)))B | w s

αλ

dx

≤ C|B|diam(B)

B

|u| w dx s

α

1s .

ρB

Theorem 7.6.7. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (H(u)) − (T (H(u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ

(7.6.14)

and T (H(u)) − (T (H(u)))B W 1,s (B),

wα

≤ C|B|us,ρB,wαλ

(7.6.15)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. The integral version of inequality (7.6.14) can be written as |T (H(u)) − (T (H(u)))B |s wα dx B

1s

≤ C|B|diam(B)

|u|s wαλ dx

1s

ρB

7.6.4 Inequalities with two-weights in Ar (λ, Ω) Now, we present the following versions of two-weight Poincar´e inequalities for diﬀerential forms satisfying the nonhomogeneous A-harmonic equation.

.

7.6 Homotopy and projection operators

293

Theorem 7.6.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |T (H(u)) − (T (H(u)))B |

s

w1αλ dx

1s

≤ C|B|diam(B)

B

|u|

s

w2α dx

1s

ρB

(7.6.16) and T (H(u)) − (T (H(u)))B W 1,s (B),w1αλ ≤ C|B|us,ρB,w2α

(7.6.17)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Inequality (7.6.16) can be written as T (H(u)) − (T (H(u)))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α .

(7.6.17)

Theorem 7.6.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |T (H(u)) − (T (H(u)))B |s w1α dx B

1s

≤ C|B|diam(B)

|u|s w2α dx

1s

ρB

(7.6.18) and T (H(u)) − (T (H(u)))B W 1,s (B),w1α ≤ C|B|us,ρB,w2α

(7.6.19)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that (7.6.18) is equivalent to T (H(u)) − (T (H(u)))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α .

(7.6.19)

7.6.5 Inequalities with two-weights in Aλ r (Ω) In the following results, we will state other two-weight cases. Theorem 7.6.10. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and

294

7 Inequalities for operators

T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|T (H(u)) − (T (H(u)))B |s w1α dx

≤ C|B|diam(B)

σB

|u|s w2αλ dx

1/s

1/s

(7.6.20)

and T (H(u)) − (T (H(u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αλ

(7.6.21)

for all balls B with σB ⊂ Ω. If we choose λ = 1/α in Theorem 7.6.10, we have the following version of the Aλr -weighted Poincar´e inequality. Corollary 7.6.11. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|T (H(u)) − (T (H(u)))B |s w1α dx

≤ C|B|diam(B)

|u|s w2 dx σB

1/s

1/s

(7.6.22)

for all balls B with σB ⊂ Ω. Choosing α = 1/s in Theorem 7.6.10, we ﬁnd the following two-weight Poincar´e inequality. Corollary 7.6.12. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that

1

|T (H(u)) − (T (H(u)))B |s w1s dx B

≤ C|B|diam(B) for all balls B with σB ⊂ Ω.

λ/s

σB

|u|s w2 dx

1/s

1/s

(7.6.23)

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295

Letting λ = 1 in Corollary 7.6.12, we obtain the following symmetric twoweight inequality. Corollary 7.6.13. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1

|T (H(u)) − (T (H(u)))B |s w1s dx B

≤ C|B|diam(B)

1

|u|s w2s dx σB

1/s

1/s

(7.6.24)

for all balls B with σB ⊂ Ω. Finally, when λ = s in Theorem 7.6.10, we have the following two-weighted inequality. Corollary 7.6.14. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1/s |T (H(u)) − (T (H(u)))B |s w1α dx

1/s ≤ C|B|diam(B) σB |u|s w2αs dx

B

(7.6.25)

and T (H(u)) − (T (H(u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αs

(7.6.26)

for all balls B with σB ⊂ Ω.

7.6.6 Basic estimates for H ◦ T Similar to the case of T ◦ H, Poincar´e-type inequalities also hold for H ◦ T . We list them as follows. Theorem 7.6.15. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be

296

7 Inequalities for operators

the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B s,B ≤ C|B|diam(B)us,ρB

(7.6.27)

for all balls B with ρB ⊂ Ω, where ρ > 1 is a constant. Proof. For any diﬀerential form ω, we have ω − (ω)B s,B = T d(ω)s,B

(7.6.28)

for any ball B. Now, replacing ω by H(T (u)) in (7.6.28), and using (7.4.2), we obtain H(T (u)) − (H(T (u)))B s,B = T d(H(T (u)))s,B

(7.6.29)

≤ C1 |B|diam(B)d(H(T (u)))s,B . In the proof of Theorem 7.6.1, we have shown that H commutes with d. Thus, inequality (7.6.29) can be written as H(T (u)) − (H(T (u)))B s,B ≤ C1 |B|diam(B)H(d(T (u)))s,B .

(7.6.30)

Replacing u by d(T (u)) in (7.6.2), we ﬁnd that H(d(T (u))) = d(T (u)) − ΔG(d(T (u))).

(7.6.31)

For any diﬀerential form u (note that u need not be a solution of the Aharmonic equation), using Theorem 3.3.2, we have ΔG(u)s,B ≤ C2 us,B .

(7.6.32)

Next, replacing u by d(T (u)) in (7.6.32), we ﬁnd that ΔG(d(T (u)))s,B ≤ C2 d(T (u))s,B .

(7.6.33)

Finally, applying inequality (3.3.8) and Caccioppoli’s inequality, and noticing that |B| ≤ |Ω| ≤ C3 since Ω is bounded, we obtain d(T (u))s,B ≤ us,B + C4 |B|diam(B)dus,B ≤ us,B + C5 |B|u − cs,ρB ≤ us,B + C6 |B|us,ρB ≤ us,B + C7 us,ρB ≤ C8 us,ρB

(7.6.34)

7.6 Homotopy and projection operators

297

for some ρ > 1, where we have chosen c = 0 when we used Caccioppoli’s inequality. Combination of (7.6.33) and (7.6.34) yields ΔG(d(T (u)))s,B ≤ C9 us,ρB .

(7.6.35)

Hence, from (7.6.30), (7.6.31), (7.6.34), and (7.6.35), it follows that H(T (u)) − (H(T (u)))B s,B ≤ C1 |B|diam(B)H(d(T (u)))s,B ≤ C1 |B|diam(B)d(T (u)) − ΔG(d(T (u)))s,B ≤ C1 |B|diam(B) (d(T (u))s,B + ΔG(d(T (u)))s,B )

(7.6.36)

≤ C1 |B|diam(B) (C8 us,B + C9 us,ρB ) ≤ C10 |B|diam(B)us,ρB , that is H(T (u)) − (H(T (u)))B s,B ≤ C10 |B|diam(B)us,ρB . The weak version of Theorem 7.6.15 can be stated as follows. Theorem 7.6.15 . Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B s,B ≤ C|B|diam(B)(us,B + dus,B ) (7.6.36) for all balls B ⊂ Ω.

7.6.7 Weighted inequalities for H ◦ T Theorem 7.6.16. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

(7.6.37)

298

7 Inequalities for operators

Note that the above Ls -norm inequality can also be written as 1/s

|H(T (u)) − (H(T (u)))B | w dx s

α

1/s

≤C

|u| w dx s

B

α

ρB

. (7.6.37)

Theorem 7.6.17. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that 1/s

|H(T (u)) − (H(T (u)))Q0 | wdx s

Ω

1/s

≤C

|u| wdx s

(7.6.38)

Ω

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. From Theorem 7.6.16 and the deﬁnition of the · W 1,s (E),wα -norm, we can prove the following local and global imbedding inequalities. Theorem 7.6.18. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B W 1,s (B),wα ≤ C|B|us,ρB,wα

(7.6.39)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

Theorem 7.6.19. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.6.40)

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube. In the above discussion, we have presented the Ar (Ω)-weighted Poincar´etype estimates for the composition H ◦T . Next, we state results with diﬀerent weights, namely Ar (λ, Ω)-weights and Aλr (Ω)-weights.

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299

Theorem 7.6.20. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B s,B,wαλ ≤ C|B|diam(B)us,ρB,wα

(7.6.41)

and H(T (u)) − (H(T (u)))B W 1,s (B),

wαλ

≤ C|B|us,ρB,wα

(7.6.42)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Here ρ > 1 is some constant. The above inequality (7.6.41) can be written as 1s

|H(T (u)) − (H(T (u)))B | w s

αλ

dx

≤ C|B|diam(B)

B

|u| w dx s

α

1s .

ρB

Theorem 7.6.21. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ

(7.6.43)

and H(T (u)) − (H(T (u)))B W 1,s (B),

wα

≤ C|B|us,ρB,wαλ

(7.6.44)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. The integral version of inequality (7.6.43) can be written as |H(T (u)) − (H(T (u)))B | w dx s

α

1s

≤ C|B|diam(B)

B

|u| w s

αλ

1s dx

ρB

7.6.8 Two-weight inequalities for H ◦ T In the following theorems, we state the two-weight Poincar´e-type inequalities for H ◦ T . We begin with the following estimate with weights (w1 , w2 ) ∈ Ar (λ, Ω).

.

300

7 Inequalities for operators

Theorem 7.6.22. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |H(T (u)) − (H(T (u)))B |s w1αλ dx B 1/s (7.6.45) ≤ C|B|diam(B) ρB |u|s w2α dx and H(T (u)) − (H(T (u)))B W 1,s (B),w1αλ ≤ C|B|us,ρB,w2α

(7.6.46)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Inequality (7.6.45) can be written as H(T (u)) − (H(T (u)))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α .

(7.6.46)

Theorem 7.6.23. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

1/s |H(T (u)) − (H(T (u)))B |s w1α dx 1/s ≤ C|B|diam(B) ρB |u|s w2α dx

B

(7.6.47)

and H(T (u)) − (H(T (u)))B W 1,s (B),w1α ≤ C|B|us,ρB,w2α

(7.6.48)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that inequality (7.6.47) is equivalent to H(T (u)) − (H(T (u)))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α .

(7.6.48)

Now, we discuss the following version of two-weight Poincar´e inequalities for diﬀerential forms. Theorem 7.6.24. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and

7.6 Homotopy and projection operators

301

T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|H(T (u)) − (H(T (u)))B |s w1α dx

≤ C|B|diam(B)

|u|s w2αλ dx σB

1s

1s

(7.6.49)

and H(T (u)) − (H(T (u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αλ

(7.6.50)

for all balls B with σB ⊂ Ω. Selecting λ = 1/α in Theorem 7.6.24, we ﬁnd the following version of the Aλr -weighted Poincar´e inequality. Corollary 7.6.25. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|H(T (u)) − (H(T (u)))B |s w1α dx

≤ C|B|diam(B)

|u|s w2 dx σB

1s

1s

(7.6.51)

for all balls B with σB ⊂ Ω. Now setting α = 1/s in Theorem 7.6.24, we obtain the following two-weight Poincar´e inequality. Corollary 7.6.26. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that

1

|H(T (u)) − (H(T (u)))B |s w1s dx B

≤ C|B|diam(B) for all balls B with σB ⊂ Ω.

λ/s

|u|s w2 dx σB

1s

1s

(7.6.52)

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7 Inequalities for operators

Letting λ = 1 in Corollary 7.6.26, we have the following symmetric two-weight inequality. Corollary 7.6.27. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1

|H(T (u)) − (H(T (u)))B |s w1s dx B

≤ C|B|diam(B)

1

|u|s w2s dx σB

1s

1s

(7.6.53)

for all balls B with σB ⊂ Ω. Finally, when λ = s in Theorem 7.6.24, we ﬁnd the following two-weight inequality. Corollary 7.6.28. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

1 |H(T (u)) − (H(T (u)))B |s w1α dx s

1 ≤ C|B|diam(B) σB |u|s w2αs dx s

B

(7.6.54)

and H(T (u)) − (H(T (u)))B W 1,s (B),w1α ≤ C|B|us,σB,w2αs

(7.6.55)

for all balls B with σB ⊂ Ω.

7.6.9 Some global inequalities From the covering lemma, we can extend the local inequalities stated in this section to the global cases. For example, Theorems 7.6.20, 7.6.21, 7.6.22, and 7.6.23 can be extended to the following Theorems 7.6.29, 7.6.30, 7.6.31, and 7.6.32, respectively. Theorem 7.6.29. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a diﬀerential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded

7.6 Homotopy and projection operators

303

domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that (7.6.56) H(T (u)) − (H(T (u)))B0 s,Ω,wαλ ≤ Cus,Ω,wα and H(T (u)) − (H(T (u)))B0 W 1,s (Ω),

wαλ

≤ Cus,Ω,wα

(7.6.57)

for all bounded, convex domains Ω and any real number α with 0 < α < 1. Here B0 ⊂ Ω is a ﬁxed ball. Theorem 7.6.30. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that H(T (u)) − (H(T (u)))B0 s,Ω,wα ≤ Cus,Ω,wαλ

(7.6.58)

and H(T (u)) − (H(T (u)))B0 W 1,s (Ω),

wα

≤ Cus,Ω,wαλ

(7.6.59)

for all bounded domains Ω and any real number α with 0 < α < 1. Here B0 ⊂ Ω is a ﬁxed ball. Theorem 7.6.31. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar (λ, Ω) for some λ > 0 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that |H(T (u)) − (H(T (u)))B0 |s w1αλ dx Ω

1s

≤C

|u|s w2α dx

1s (7.6.60)

Ω

and H(T (u)) − (H(T (u)))B0 W 1,s (Ω),w1αλ ≤ Cus,Ω,w2α

(7.6.61)

for all bounded domains Ω and any real number α with 0 < α < 1. Here B0 ⊂ Ω is a ﬁxed ball. Theorem 7.6.32. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and

304

7 Inequalities for operators

1 < r < ∞. Then, there exists a constant C, independent of u, such that |H(T (u)) − (H(T (u)))B0 |

s

w1α dx

1s

≤C

Ω

|u|

s

w2α dx

1s (7.6.62)

Ω

and H(T (u)) − (H(T (u)))B0 W 1,s (Ω),w1α ≤ Cus,Ω,w2α for all bounded domains Ω and any real number α with 0 < α < 1. Here B0 ⊂ Ω is a ﬁxed ball.

7.7 Compositions of three operators In this section, we discuss the diﬀerent compositions generated by three basic operators, the homotopy operator T , the diﬀerential operator d, and the projection operator H. For a smooth diﬀerential form u, it is easy to see that d(H(T (u))) and d(T (H(u))) are closed. Thus, the Poincar´e-type estimates for these compositions are trivial and hence we will not discuss them here.

7.7.1 Basic estimates for T ◦ d ◦ H First, we prove the following unweighted Poincar´e-type inequality for T ◦ d ◦ H. This elementary result forms the bases for the weighted Poincar´e inequalities for T ◦ d ◦ H which will be developed later in this chapter. Theorem 7.7.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n − 1, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (d(H(u))) − (T (d(H(u))))B s,B ≤ C|B|diam(B)dus,B

(7.7.1)

for all balls B with B ⊂ Ω. Proof. From (7.6.4) and Theorem 3.3.2, we have dΔG(u)s,B ≤ C1 dus,B .

(7.7.2)

Also, for any diﬀerential form u, d(du) = 0, and hence d(d(H(u))) = 0. Thus, from (7.6.4) and (7.7.2), we ﬁnd that

7.7 Compositions of three operators

305

T (d(H(u))) − (T (d(H(u))))B s,B = T d(T dH(u))s,B ≤ C2 |B|diam(B)dT dH(u)s,B ≤ C2 |B|diam(B)(dH(u)s,B +C3 |B|diam(B)d(d(H(u)))s,B ) ≤ C2 |B|diam(B)(dH(u)s,B + 0) ≤ C2 |B|diam(B)dH(u)s,B

(7.7.3)

≤ C2 |B|diam(B)H(du)s,B = C2 |B|diam(B)du − ΔG(du)s,B ≤ C2 |B|diam(B)(dus,B + ΔG(du)s,B ) ≤ C2 |B|diam(B)(dus,B + C3 dus,B ) ≤ C4 |B|diam(B)dus,B , which implies that inequality (7.7.1) holds. Note that the diﬀerential form u in Theorem 7.7.1 is not necessarily a solution of the A-harmonic equation. If u is a solution of the A-harmonic equation, then we can employ the Caccioppoli inequality, to obtain the following corollary. Corollary 7.7.2. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (d(H(u))) − (T (d(H(u))))B s,B ≤ C|B|diam(B)us,ρB

(7.7.4)

for all balls B with ρB ⊂ Ω, where ρ > 1 is a constant. Since d is commutative with G and H, respectively, it follows that HdT (u) and GdT (u) are closed. Speciﬁcally, we have the following corollary. Corollary 7.7.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, be any smooth differential form in a bounded, convex domain Ω, H be the projection operator, G be Green’s operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, HdT (u) and GdT (u) are closed forms.

306

7 Inequalities for operators

7.7.2 Ar (Ω)-weighted inequalities for T ◦ d ◦ H Based on the elementary inequality established above, we can now state the following Ar (Ω)-weighted Poincar´e inequalities. Theorem 7.7.4. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that T (d(H(u)) − (T (d(H(u))))B s,B,wα ≤ C|B|diam(B)us,ρB,wα

(7.7.5)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1. Note that the above Ls -norm inequality can also be written as

1/s |T (d(H(u))) − (T (d(H(u))))B | w dx s

α

1/s

≤C

B

|u| w dx s

α

.

ρB

(7.7.6)

Theorem 7.7.5. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10). Then, there exists a constant C, independent of u, such that 1/s

|T (d(H(u))) − (T (d(H(u))))Q0 |s wdx

1/s

≤C

Ω

|u|s wdx Ω

(7.7.7) for any bounded, convex δ-John domain Ω ⊂ R . Here Q0 ⊂ Ω is a ﬁxed cube. n

Using Theorem 7.7.4 and the deﬁnition of the · W 1,s (E),wα -norm, we can prove the following local and global imbedding inequalities. Theorem 7.7.6. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Ar (Ω) for some 1 < r < ∞. Then, there exists a constant C, independent of u, such that

7.7 Compositions of three operators

307

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),wα (7.7.8)

≤ C|B|us,ρB,wα for all balls B with ρB ⊂ Ω and any real number α with 0 < α ≤ 1.

Theorem 7.7.7. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (Ω) for some 1 < r < ∞ and s is a ﬁxed exponent associated with the A-harmonic equation (1.2.10), r < s < ∞. Then, there exists a constant C, independent of u, such that T (d(H(u))) − (T (d(H(u))))Q0 W 1,s (Ω),w ≤ Cus,Ω,w

(7.7.9)

for any bounded, convex δ-John domain Ω ⊂ Rn . Here Q0 ⊂ Ω is a ﬁxed cube.

7.7.3 Cases of other weights So far, we have presented the Ar (Ω)-weighted Poincar´e-type estimates for the composition T ◦ d ◦ H. Now, we state other estimates with diﬀerent weights, namely Ar (λ, Ω)-weights and Aλr (Ω)-weights. Theorem 7.7.8. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a differential form satisfying the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that w ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (d(H(u))) − (T (d(H(u))))B s,B,wαλ (7.7.10) ≤ C|B|diam(B)us,ρB,wα and T (d(H(u))) − (T (d(H(u))))B W 1,s (B), ≤ C|B|us,ρB,wα

wαλ

(7.7.11)

for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Here ρ > 1 is some constant. The above inequality (7.6.10) can be written as

308

7 Inequalities for operators

B

|T (d(H(u))) − (T (d(H(u))))B |s wαλ dx

≤ C|B|diam(B)

|u|s wα dx ρB

1s

1s (7.7.12)

.

Theorem 7.7.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and w ∈ Aλr (Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that T (d(H(u))) − (T (d(H(u))))B s,B,wα ≤ C|B|diam(B)us,ρB,wαλ (7.7.13) and

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),

wα

(7.7.14)

≤ C|B|us,ρB,wαλ for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. The integral version of inequality (7.7.13) can be written as

|T (d(H(u))) − (T (d(H(u))))B |s wα dx 1s ≤ C|B|diam(B) ρB |u|s wαλ dx .

1s

B

(7.7.15)

7.7.4 Cases of two-weights In our next several theorems, we provide the two-weight estimates for this composition of the operators. First, we state the following estimates with weights (w1 , w2 ) ∈ Ar (λ, Ω). Theorem 7.7.10. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Assume that ρ > 1 and (w1 , w2 ) ∈ Ar (λ, Ω) for some r > 1 and λ > 0. Then, there exists a constant C, independent of u, such that

1/s |T (d(H(u))) − (T (d(H(u))))B |s w1αλ dx B 1/s (7.7.16) ≤ C|B|diam(B) ρB |u|s w2α dx

7.7 Compositions of three operators

and

309

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),w1αλ (7.7.17)

≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < 1. Inequality (7.7.16) can be written as

T (d(H(u))) − (T (d(H(u))))B s,B,w1αλ ≤ C|B|diam(B)us,ρB,w2α . (7.7.17) Theorem 7.7.11. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that ρ > 1 and (w1 , w2 ) ∈ Ar,λ (Ω) for some λ ≥ 1 and 1 < r < ∞. Then, there exists a constant C, independent of u, such that

|T (d(H(u))) − (T (d(H(u))))B |s w1α dx 1s ≤ C|B|diam(B) ρB |u|s w2α dx

1s

B

and

(7.7.18)

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),w1α (7.7.19)

≤ C|B|us,ρB,w2α for all balls B with ρB ⊂ Ω and any real number α with 0 < α < λ. Note that (7.7.18) is equivalent to

T (d(H(u))) − (T (d(H(u))))B s,B,w1α ≤ C|B|diam(B)us,ρB,w2α . (7.7.19) Now, we state the following versions of two-weight Poincar´e inequalities for diﬀerential forms. Theorem 7.7.12. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1 and λ > 0. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|T (d(H(u))) − (T (d(H(u))))B |s w1α dx

≤ C|B|diam(B)

σB

|u|s w2αλ dx

1s

1s (7.7.20)

310

7 Inequalities for operators

and

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),w1α (7.7.21)

≤ C|B|us,σB,w2αλ for all balls B with σB ⊂ Ω.

If we choose λ = 1/α in Theorem 7.7.12, we have the following version of the Aλr -weighted Poincar´e inequality. Corollary 7.7.13. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). 1/α Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that B

|T (d(H(u))) − (T (d(H(u))))B |s w1α dx

≤ C|B|diam(B)

σB

|u|s w2 dx

1s (7.7.22)

1s

for all balls B with σB ⊂ Ω. Choosing α = 1/s in Theorem 7.7.12, we ﬁnd the following two-weight Poincar´e inequality. Corollary 7.7.14. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Aλr (Ω) for some r > 1, λ > 0, and σ > 1, then there exists a constant C, independent of u, such that

1

|T (d(H(u))) − (T (d(H(u))))B |s w1s dx B

≤ C|B|diam(B)

λ/s

|u|s w2 dx σB

1s

1s (7.7.23)

for all balls B with σB ⊂ Ω. Let λ = 1 in Corollary 7.7.14, to obtain the following symmetric two-weight inequality. Corollary 7.7.15. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1).

7.7 Compositions of three operators

311

Suppose that (w1 , w2 ) ∈ Ar (Ω) for some r > 1 and σ > 1, then there exists a constant C, independent of u, such that

1

|T (d(H(u))) − (T (d(H(u))))B |s w1s dx B

≤ C|B|diam(B)

1

|u|s w2s dx σB

1s

1s

(7.7.24)

for all balls B with σB ⊂ Ω. Finally, when λ = s in Theorem 7.6.12, we obtain the following two-weight inequality. Corollary 7.7.16. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω ⊂ Rn and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Suppose that (w1 , w2 ) ∈ Asr (Ω) for some r > 1. If 0 < α < 1 and σ > 1, then there exists a constant C, independent of u, such that

|T (d(H(u))) − (T (d(H(u))))B |s w1α dx

1 ≤ C|B|diam(B) σB |u|s w2αs dx s

1s

B

and

(7.7.25)

T (d(H(u))) − (T (d(H(u))))B W 1,s (B),w1α ≤ C|B|us,σB,w2αs

(7.7.26)

for all balls B with σB ⊂ Ω.

7.7.5 Estimates for T ◦ H ◦ d Theorem 7.7.17. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n − 1, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (H(du)) − (T (H(du)))B s,B ≤ C|B|diam(B)dus,B

(7.7.27)

for all balls B with B ⊂ Ω. Proof. Let H be the projection operator and T be the homotopy operator. Then, for any diﬀerential form u, we have

312

7 Inequalities for operators

T (H(du)) − (T (H(du)))B s,B = T d(T Hd(u))s,B ≤ C1 |B|diam(B)d(T Hd(u))s,B ≤ C1 |B|diam(B)(Hd(u)s,B +C2 |B|diam(B)d(Hd(u))s,B )

(7.7.28)

≤ C1 |B|diam(B)(Hd(u)s,B + 0) ≤ C1 |B|diam(B)Hd(u)s,B ≤ C2 |B|diam(B)dus,B , and hence inequality (7.7.27) holds. Using the Caccioppoli inequality and Theorem 7.7.17, we have the following corollary immediately. Corollary 7.7.18. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (H(du)) − (T (H(du)))B s,B ≤ C|B|u − cs,B

(7.7.29)

for all balls B with B ⊂ Ω, where c is any closed form.

7.7.6 Estimates for H ◦ T ◦ d Theorem 7.7.19. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that HT (du) − (HT (du))B s,B ≤ C|B|diam(B)dus,B for all balls B with B ⊂ Ω. Proof. By the decomposition theorem and Theorem 3.3.2, we have

(7.7.30)

7.8 The maximal operators

313

HT (du) − (HT (du))B s,B = T d(HT (du))s,B ≤ C1 |B|diam(B)d(HT (du))s,B ≤ C1 |B|diam(B)HdT (du)s,B ≤ C1 |B|diam(B)dT (du) − ΔGdT (du)s,B

(7.7.31)

≤ C1 |B|diam(B)(dT (du)s,B + ΔGdT (du)s,B ) ≤ C1 |B|diam(B)(dT (du)s,B + dT (du)s,B ) ≤ C2 |B|diam(B)dT (du)s,B . Also, from (3.3.8), we have dT (du)s,B ≤ dus,B + C3 |B|diam(B)d(du)s,B ≤ dus,B + 0

(7.7.32)

≤ dus,B . Combining (7.7.31) and (7.7.32), we obtain HT (du) − (HT (du))B s,B ≤ C2 |B|diam(B)dus,B .

7.8 The maximal operators In this section, we ﬁrst develop some estimates related to the Hardy– Littlewood maximal operator IMs and the sharp maximal operator IMs , and then study the relationship between IMs s,Ω and IMs s,Ω . We also discuss the compositions of these operators with other operators.

7.8.1 Global Ls-estimates We begin this section with the following global estimate for the composition of the sharp maximal operator and the homotopy operator. Theorem 7.8.1. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω, IMs be the sharp maximal operator deﬁned in (7.1.2), and T be the homotopy operator deﬁned in (1.5.1). Then,

314

7 Inequalities for operators

IMs (T u)s,Ω ≤ Cus,Ω

(7.8.1)

for some constant C, independent of u. Proof. Using Theorem 7.5.1 over the ball B(x, r), we have

1 |B(x,r)|

|T (u) − (T (u))B(x,r) |s dy B(x,r)

≤ C1 |B(x, r)|1−1/s diam(B(x, r)) ≤ C2 |Ω|

1−1/s

1/s

|u|s dy ρB(x,r)

1/s (7.8.2)

diam(Ω)us,Ω

≤ C3 us,Ω since 1 − 1/s > 0 and Ω is bounded. Thus, it follows that sup r>0

1 |B(x, r)|

1/s

|T (u) − (T (u))B(x,r) | dy s

≤ C3 us,Ω .

(7.8.3)

B(x,r)

Now, from (7.8.2) and (7.8.3), and using the deﬁnition of IMs , we obtain IMs (T u)s,Ω 1/s = Ω |IMs (T u)|s dx 1/s s 1/s 1 s supr>0 dx |B(x,r)| B(x,r) |T (u) − (T (u))B(x,r) | dy Ω

= ≤

Ω

s

|C3 us,Ω | dx

1/s

≤ C4 us,Ω , and hence IMs (T u)s,Ω ≤ C4 us,Ω . Using Poincar´e-type estimates obtained in the previous sections, we can obtain some other similar estimates. For example, applying Theorem 7.4.1 and the same method developed in the proof of Theorem 7.8.1, we have IMs (T (G(u)))s,Ω ≤ Cus,Ω .

(7.8.4)

From [204], we know that if u ∈ Ls (Ω, ∧l ), 1 < s < ∞, then IM(u) ∈ Ls (Ω); speciﬁcally, we have the following theorem.

7.8 The maximal operators

315

Theorem 7.8.2. Let u ∈ Ls (Ω, ∧l ), l = 0, 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form in a bounded, convex domain Ω, IM be the Hardy–Littlewood maximal operator deﬁned in (7.1.1), and T be the homotopy operator deﬁned in (1.5.1). Then, (7.8.5) IMus,Ω ≤ Cus,Ω for some constant C, independent of u. Note that (7.8.5) holds for any diﬀerential form in a bounded, convex domain Ω. Thus, we can replace u in (7.8.5) by G(u), H(u), and T (u), respectively. Then, using the basic estimates for these operators, we ﬁnd that IMG(u)s,Ω ≤ C1 us,Ω ,

(7.8.6)

IMH(u)s,Ω ≤ C2 us,Ω ,

(7.8.7)

IMT (u)s,Ω ≤ C3 us,Ω

(7.8.8)

and hold for any bounded domain Ω.

7.8.2 The norm comparison theorem Theorem 7.8.3. Let u ∈ Lsloc (Ω, ∧l ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a smooth diﬀerential form in a bounded domain Ω, IM be the Hardy–Littlewood maximal operator deﬁned in (7.1.1), and IMs be the sharp maximal operator deﬁned in (7.1.2). Then, IMs us,Ω ≤ CIMs dus,Ω

(7.8.9)

for some constant C, independent of u. Proof. For any ball B(x, r) ⊂ Ω with radius r, centered at x ∈ Ω, using the basic Poincar´e inequality (Theorem 3.2.3), we have

1/s

|u(y) − (u(y))B(x,r) | dy s

≤ C1 |B(x, r)|

B(x,r)

1/s |du| dy

1/n

s

B(x,r)

that is,

1 |B(x,r)|

B(x,r)

|u(y) − (u(y))B(x,r) |s dy

≤ C1 |B(x, r)|1/n

1 |B(x,r)|

1/s

|du|s dy B(x,r)

1/s

(7.8.10) .

316

7 Inequalities for operators

Now, by deﬁnitions of the Hardy–Littlewood maximal operator IMs and the sharp maximal operator IMs , and (7.8.10), we obtain IMs (u)s,Ω 1/s = Ω |IMs (u)|s dx 1/s s 1/s 1 s supr>0 dx |B(x,r)| B(x,r) |u(y) − (u(y))B(x,r) | dy Ω

=

1/s s 1/s 1 s supr>0 C1 |B(x, r)|1/n dx |B(x,r)| B(x,r) |du(y)| dy Ω

≤

1/s s 1/s 1 s supr>0 C1 |Ω|1/n dx |du(y)| dy |B(x,r)| B(x,r) Ω

≤

1/s s 1/s 1 s supr>0 dx |B(x,r)| B(x,r) |du(y)| dy Ω

≤ C2 ≤ C2

s

Ω

|IMs du(x)| dx

1/s

= C3 IMs dus,Ω , which is equivalent to IMs us,Ω ≤ C3 IMs dus,Ω .

7.8.3 The fractional maximal operator Next, following the idea given in [126], we introduce the fractional maximal operator of order α. Let u(y) be a locally Ls -integrable form, 1 ≤ s < ∞, and α be a real number. We deﬁne the fractional maximal operator IMs,α of order α by IMs,α u(x) = sup r>0

1 |B(x, r)|1+α/n

1/s

|u(y)| dy s

.

(7.8.11)

B(x,r)

Clearly, (7.8.11) for α = 0 reduces to the Hardy–Littlewood maximal operator, and hence, we write IMs (u) = IMs,0 (u). Theorem 7.8.4. Let u ∈ Ls (Ω, ∧l ), l = 0, 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying equation (1.2.10) in a bounded domain Ω, IMs be the Hardy–Littlewood maximal operator deﬁned in (7.1.1), and IMs,α be the fractional maximal operator of order α. Then,

7.8 The maximal operators

317

IMs du(x)s,Ω ≤ CIMs,α (u(x) − c)s,Ω

(7.8.12)

for some constant C, independent of u, where α = s and c is any closed form. Proof. Let B(x, r) ⊂ Ω be a ball with radius r and center at x ∈ Ω. From Theorem 4.3.1 (Caccioppoli inequality), we have 1/s

|du(y)| dy s

−1/n

≤ C1 |B(x, r)|

B(x,r)

1/s |u(y) − c| dy s

,

B(x,r)

that is,

1 |B(x,r)|

|du(y)|s dy B(x,r)

≤ C1 |B(x, r)|−1/n

1/s

1 |B(x,r)|

|u(y) − c|s dy B(x,r)

1/s

(7.8.13) .

Now, by deﬁnitions of the maximal operators and (7.8.13), we obtain IMs du(x)s,Ω

1/s = Ω |IMs du(x)|s dx 1/s s 1/s 1 s dx = Ω supr>0 |B(x,r)| B(x,r) |du(y)| dy 1/s s 1/s 1 s supr>0 C1 |B(x, r)|−1/n dx |B(x,r)| B(x,r) |u(y) − c| dy Ω

≤

1/s s 1/s 1 s supr>0 C1 dx |u(y) − c| dy Ω |B(x,r)|1+s/n B(x,r)

≤

≤ C2

Ω

s

|IMs,α (u(x) − c)| dx

1/s

= C2 IMs,α (u(x) − c)s,Ω , which can be written as IMs du(x)s,Ω ≤ C2 IMs,α (u(x) − c)s,Ω . Note that in Theorem 7.8.4, c is any closed form. Thus, we can choose c = 0 in Theorem 7.8.4, to obtain the following corollary. Corollary 7.8.5. Let u ∈ Ls (Ω, ∧l ), l = 0, 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form satisfying equation (1.2.10) in a bounded domain Ω,

318

7 Inequalities for operators

IMs be the Hardy–Littlewood maximal operator deﬁned in (7.1.1), and IMs,α be the fractional maximal operator of order α. Then, IMs du(x)s,Ω ≤ CIMs,α u(x)s,Ω for some constant C, independent of u. We conclude this chapter with the following estimate for the composition of the sharp maximal operator, Green’s operator, and the homotopy operator. Theorem 7.8.6. Let u ∈ Ls (Ω, ∧l ), l = 0, 1, 2, . . . , n, 1 < s < ∞, be a diﬀerential form in a bounded, convex domain Ω, IMs be the sharp maximal operator deﬁned in (7.1.2), G be Green’s operator, and T be the homotopy operator deﬁned in (1.5.1). Then, IMs (T G(u))s,Ω ≤ Cus,Ω

(7.8.14)

for some constant C, independent of u.

7.9 Singular integrals The purpose of this section is to estimate the singular integral of the composition of the homotopy operator T and the projection operator H. We can also discuss the integrals of other composite operators with a singular factor using the similar method. The consideration was motivated from physics. For instance, when calculating an electric ﬁeld, we will deal with the integral r−x 1 ρ(x) dv, E(r) = 4π0 D r − x3 where ρ(x) is a charge density and x is the integral variable. It is singular if r ∈ D. Obviously, the singular integrals are more interesting to us because of their wide applications in diﬀerent ﬁelds of mathematics and physics. Theorem 7.9.1. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation in a bounded and convex domain Ω, H be the projection operator, and T be the homotopy operator. Then, there exists a constant C, independent of u, such that

|T (H(u)) − (T (H(u)))B |s |x−x1B |α dx B

≤ C|B|γ

|u|s |x−x1 B |λ dx ρB

1/s

1/s (7.9.1)

7.9 Singular integrals

319

for all balls B with ρB ⊂ Ω and any real number α and λ with α > λ ≥ 0, where γ = 1 + n1 − α−λ ns and xB is the center of ball B. Proof. Let ε ∈ (0, 1) be small enough such that εn < α − λ and B ⊂ Ω be any ball with center xB and radius rB . Choose t = s/(1−ε) and β = t/(t − s), then t > s. Using the H¨ older inequality and Theorem 7.6.1, we have

(|T H(u) − (T H(u))B |) B

=

s

1 |x−xB |α dx

1/s

|T H(u) − (T H(u))B | |x−x1B |α/s

B

≤ T H(u) − (T H(u))B t,B = T H(u) − (T H(u))B t,B

B

B

s

1 |x−xB |

1/s dx (t−s)/st

tα/(t−s) dx

|x − xB |−αβ dx

(7.9.2)

1/βs

≤ C1 |B|diam(B)ut,B |x − xB |−α β,B . 1/s

We may assume that xB = 0. Otherwise, we can move the center to the origin by a simple transformation. Then for any x ∈ B, |x − xB | ≥ |x| − |xB | = |x|. By using the polar coordinate substitution, we have rB C (rB )n−αβ . |x − xB |−αβ dx ≤ C ρ−αβ ρn−1 dρ ≤ (7.9.3) n − αβ B 0 Choose m = nst/(ns + αt − λt), then 0 < m < s. By the reverse H¨ older inequality, we ﬁnd that ut,B ≤ C2 |B|

m−t mt

um,σB .

(7.9.4)

By the H¨older inequality again, we obtain um,σB

m 1/m = σB |u||x − xB |−λ/s |x − xB |λ/s dx ≤ ≤

|u||x − xB |−λ/s

σB

σB

≤ C4

s

|u|s |x − xB |−λ dx

σB

dx

1/s

|u|s |x − xB |−λ dx

1/s

σB

|x − xB |λ/s

ms

s−m

dx

s−m ms

C3 (σrB )λ/s+n(s−m)/ms

1/s

(rB )λ/s+n(s−m)/ms . (7.9.5)

Note that diam(B) · |B|1+ t − m = |B|1+ n + t − 1

1

1

1

ns+αt−λt nst

= |B|1+ n − 1

α−λ ns

.

(7.9.6)

320

7 Inequalities for operators

Substituting (7.9.3), (7.9.4), and (7.9.5) into (7.9.2) and using (7.9.6), we have 1/s s 1 (|T H(u) − (T H(u)) |) dx α B |x−xB | B ≤ C5 |B|γ

σB

|u|s |x − xB |−λ dx

1/s

.

Remark. 1) Replacing α by 2α and λ by α in Theorem 7.9.1, we have

|T (H(u)) − (T (H(u)))B |s |x−x1B |2α dx B

≤ C|B|γ

ρB

|u|s |x−x1B |α dx

1/s

1/s .

2) If λ = 0, inequality (7.9.1) reduces to

|T (H(u)) − (T (H(u)))B |s |x−x1B |α dx B

≤ C|B|γ

|u|s dx ρB

1/s

1/s

(7.9.7) ,

which does not contain a singular factor in the integral on the right-hand side of the inequality.

Theorem 7.9.2. Let u ∈ D (Ω, ∧1 ) be a solution of the nonhomogeneous A-harmonic equation, H be the projection operator, and T be the homotopy operator. Assume that s is a ﬁxed exponent associated with the nonhomogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

1 |T (H(u)) − (T (H(u)))B0 |s ) d(x,∂Ω) α dx Ω

≤C

1 |u|s d(x,∂Ω) λ dx Ω

1/s

1/s (7.9.8)

for any bounded and convex Ls (μ)-averaging domain Ω. Here B0 ⊂ Ω is a ﬁxed ball and α and λ are constants with 0 ≤ λ < α < min{n, s+λ+n(s−1)}. Proof. Let rB be the radius of a ball B ⊂ Ω. We may assume the center of B 1 is 0. Then, d(x, ∂Ω) ≥ rB −|x| for any x ∈ B. Therefore, d−1 (x, ∂Ω) ≤ rB −|x| for any x ∈ B. Similar to the proof of Theorem 7.9.1, we have

7.9 Singular integrals

321

B

1 |T H(u) − (T H(u))B |s d(x,∂Ω) α dx

≤ C1 |B|γ

1 |u|s d(x,∂Ω) λ dx ρB

1/s

1/s

(7.9.9)

for all balls B with ρB ⊂ Ω and any real number α and λ with α > λ ≥ 0 1 where γ = 1 + n1 − α−λ ns . Write dμ = d(x,∂Ω)α dx. Then,

μ(B) =

dμ = B

B

1 α dx ≥ d(x, ∂Ω)

B

1 dx = C1 |B|, (diam(Ω))α

C2 1 and hence μ(B) ≤ |B| . Since Ω is an Ls (μ)-averaging domain, using (7.9.9) and noticing that γ − 1/s = (1 − 1/s) + (s + λ − α)/ns > 0, we have

1 μ(Ω)

=

1 |T (H(u)) − (T (H(u)))B0 |s ) d(x,∂Ω) α dx Ω

1 μ(Ω)

Ω

|T (H(u)) − (T (H(u)))B0 |s )dμ

≤ C3 sup4B⊂Ω ≤ C4 sup4B⊂Ω

1 μ(B) 1 |B|

≤ C6

Ω

1/s

|T (H(u)) − (T (H(u)))B |s dμ B

|T (H(u)) − (T (H(u)))B |s dμ B

≤ C5 sup4B⊂Ω |B|γ−1/s ≤ C5 |Ω|γ−1/s

1/s

1 |u|s d(x,∂Ω) λ dx ρB

1 |u|s d(x,∂Ω) λ dx

1 |u|s d(x,∂Ω) λ dx Ω

1/s

1/s

1/s ,

which is equivalent to

1 |T (H(u)) − (T (H(u)))B0 |s ) d(x,∂Ω) α dx Ω

≤C

Ω

1 |u|s d(x,∂Ω) λ dx

1/s .

1/s

1/s

1/s

Chapter 8

Estimates for Jacobians

In this chapter, we ﬁrst present some integral inequalities related to a strictly increasing convex function on [0, ∞) and then improve the H¨ older inequalp α ity with L (log L) (Ω)-norms to the case 0 < p, q < ∞. Next, we prove Lp (log L)α (Ω)-integrability of Jacobians. We also show that the integrability exponents described in Theorem 8.2.7 are the best possible.

8.1 Introduction Although, in Section 3.6 we have introduced some basic concepts and notations about Jacobians, for convenience, we recollect some of those here. We assume that Ω is an open subset of Rn , n ≥ 2, and f : Ω →Rn , 1,p f = (f 1 , . . . , f n ), is a mapping of Sobolev class Wloc (Ω, Rn ), 1 ≤ p < ∞, i whose distributional diﬀerential Df = [∂f /∂xj ] : Ω → GL(n) is a locally integrable function on Ω with values in the space GL(n) of all n×n-matrices. 1,n A homeomorphism f : Ω →Rn of Sobolev class Wloc (Ω, Rn ) is said to be K-quasiconformal, 1 ≤ K < ∞, if its diﬀerential matrix Df (x) and the Jacobian determinant J = J(x, f ) = detDf (x) 1 fx1 fx12 fx13 · · · 2 fx1 fx22 fx23 · · · = .. .. .. .. . . . . fn fn fn · · · x1 x2 x3

fx1n fx2n .. . fxnn

R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 8,

323

324

8 Estimates for Jacobians

satisfy |Df (x)|n ≤ KJ(x, f ), where |Df (x)| = max{|Df (x)h| : |h| = 1} denotes the norm of the Jacobi matrix Df (x). Now, since the Jacobian J(x, f ) is an n-form, speciﬁcally, J(x, f )dx = df 1 ∧ · · · ∧ df n , where dx = dx1 ∧ dx2 ∧ · · · ∧ dxn , many results about diﬀerential forms carry over for Jacobians. Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations for evaluating multiple integrals and solving differential equations. Since then, the Jacobian has played a crucial role in multidimensional analysis and related ﬁelds, including nonlinear elasticity, weakly diﬀerentiable mappings, continuum mechanics, nonlinear PDEs, and calculus of variations. The integrability of Jacobians has become a rather important topic, see [310, 311], for example. In 1973, F.W. Gehring [6] ﬁrst proved the higher integrability properties of the Jacobians; he invented reverse H¨ older inequalities and used these inequalities to obtain the L1+ε -integrability of the Jacobians of quasiconformal mappings, ε > 0. In 1990, S. M¨ uller [147] 1,n proved that the Jacobian of an orientation-preserving mapping f ∈ Wloc (Ω, n R ) belongs to the Zygmund class L log L(E) for each compact E ⊂ Ω. This result can be stated as the following theorem [312]. 1,n Theorem 8.1.1. Let Ω ⊂Rn be a bounded domain. If f ∈ Wloc (Ω, Rn ) is an orientation-preserving mapping and |Df | ∈ Ln (Ω), then J(x, f ) ∈ L log L(E) and J(x, f ) dx ≤ C J(x, f ) log e + |Df (x)|n dx (8.1.1) J(y, f )dy E Ω Ω

for each compact E ⊂ Ω.

8.2 Global integrability In this section, we shall develop some basic estimates and the weighted estimates for Jacobians of the orientation-preserving mappings. These estimates will provide the higher integrability of Jacobians.

8.2 Global integrability

325

8.2.1 Preliminary lemmas We ﬁrst establish some preliminary integral inequalities which will be used to study the Lϕ (Ω)-integrability of diﬀerential forms or functions. These results may also ﬁnd applications in the study of the Ls -theory and related topics, such as Lϕ (μ)-averaging domains and Orlicz norm estimates. Lemma 8.2.1. Let ϕ be a strictly increasing convex function on [0, ∞) with ϕ(0) = 0 and D be a domain in Rn . Assume that u is a function in D such that ϕ(|u|) ∈ L1 (D; μ) and μ({x ∈ D : |u − c| > 0}) > 0 for any constant c. Then, for any positive constant a, we have a ϕ( |u − uD,μ |)dμ ≤ ϕ(a|u|)dμ. (8.2.1) 2 D D Proof. First, we note that 1 1 |uD,μ | = udμ ≤ |u|dμ. μ(D) D μ(D) D

(8.2.2)

Applying ϕ to the both sides of (8.2.2), and then using Jensen inequality, we obtain 1 ϕ(a|uD,μ |) ≤ ϕ(a|u|)dμ, μ(D) D where a is any positive constant. This inequality is equivalent to ϕ(a|uD,μ |)dμ ≤ ϕ(a|u|)dμ. D

(8.2.3)

D

Now, by the fact that ϕ is increasing and convex, from (8.2.3), we have

ϕ a2 |u − uD,μ | dμ ≤ D ϕ a2 |u| + a2 |uD,μ | dμ D ≤ 12 D ϕ(a|u|)dμ + 12 D ϕ(a|uD,μ |)dμ ≤ D ϕ(a|u|)dμ. Thus, inequality (8.2.1) holds. Lemma 8.2.2. Let ϕ be a strictly increasing convex function on [0, ∞) with ϕ(0) = 0 and D be a domain in Rn . Assume that u is a function in D such that ϕ(|u|) ∈ L1 (D; μ) and μ({x ∈ D : |u − c| > 0}) > 0 for any constant c. Then, for any positive constant a, we have

326

8 Estimates for Jacobians

ϕ(a|u|)dμ ≤ C

D

ϕ(2a|u − c|)dμ,

(8.2.4)

D

where C is a positive constant. Proof. For any constant c, let C1 = ϕ(2a|c|)dμ = ϕ(2a|c|)μ(D). D

Note that μ({x ∈ D : 2a|u − c| > 0}) = μ({x ∈ D : |u − c| > 0}) > 0. Then, there exists a constant C2 such that ϕ(2a|u − c|)dμ, C1 ≤ C2 D

that is

ϕ(2a|c|)dμ ≤ C2 D

ϕ(2a|u − c|)dμ. D

Now, since ϕ is an increasing convex function, we obtain D

− c|) + 12 (2a|c|) dμ ≤ 12 D ϕ(2a|u − c|)dμ + 12 D ϕ(2a|c|)dμ ≤ 12 D ϕ(2a|u − c|)dμ + C22 D ϕ(2a|u − c|)dμ ≤ C3 D ϕ(2a|u − c|)dμ.

ϕ(a|u|)dμ ≤

D

ϕ

1

2 (2a|u

(8.2.5)

Hence, inequality (8.2.4) holds. Since c is any constant in Lemma 8.2.2, we may select c = uD,μ . Thus, combining Lemmas 8.2.1 and 8.2.2, we obtain the following corollary immediately. Corollary 8.2.3. Assume that ϕ is as above and u is a function in D ⊂ Rn such that ϕ(|u|) ∈ L1 (D; μ) and μ({x ∈ D : |u − uD,μ | > 0}) > 0. Then, for any positive constant a, we have 1 a|u − uD,μ | dμ ≤ ϕ ϕ(a|u|)dμ ≤ C ϕ(2a|u − uD,μ |)dμ, (8.2.6) 2 D D D where C is a positive constant.

8.2 Global integrability

327

8.2.2 Lp(log L)α(Ω)-integrability The purpose of this section is to study the Lp (log L)α (Ω)-integrability of the Jacobian of a composite mapping. For this, we keep using the symbols and notations introduced in Section 1.9. From Theorem 1.9.1, we ﬁnd that the Orlicz space Lψ (Ω) with ψ(t) = tp logα (e + t) can be denoted by Lp (log L)α (Ω) and the corresponding norm can also be written as [f ]Lp (log L)α (Ω) . The following version of H¨ older’s inequality appears in [312, Proposition 2.2]. Theorem 8.2.4. Let 1 < p, q < ∞, α, β > 0, p1 + 1q = 1r , αp + βq = γr , and f ∈ Lp (log L)α (Ω), g ∈ Lq (log L)β (Ω). Then, f g ∈ Lr (log L)γ (Ω) and ||f g||Lr logγ L ≤ C||f ||Lp logα L ||g||Lq logβ L .

(8.2.7)

In a recent paper [194], we have improved the condition 1 < p, q < ∞ to 0 < p, q < ∞. We state and prove this result in the following theorem. Theorem 8.2.5. Let m, n, α, β > 0, 1/s = 1/m + 1/n, α/m + β/n = γ/s. Assume that f ∈ Lm (log L)α (Ω) and g ∈ Ln (log L)β (Ω). Then, f g ∈ Ls (log L)γ (Ω) and

γ s e+ |f g| log Ω

≤C

|f g| ||f g||s

α m e+ |f | log Ω

1/s dx

|f | ||f ||m

dx

1/m

β n e+ |g| log Ω

|g| ||g||n

1/n dx

,

(8.2.8) where C is a positive constant. Note that inequality (8.2.8) can be written as [f g]Ls (log L)γ (Ω) ≤ C [f ]Lm (log L)α (Ω) [g]Ln (log L)β (Ω) . Proof. Using the elementary inequality log(e + xa ) ≤ log(e + x)a+1 for a > 0, x > 0, we obtain

(8.2.9)

328

8 Estimates for Jacobians

log e +

(|f |s |g|s )1/s 1/s |||f |s |g|s ||1

≤ log e +

|f |s |g|s |||f |s |g|s ||1

= (1/s + 1) log e +

1/s+1

(8.2.10)

|f |s |g|s |||f |s |g|s ||1

and |f | s |f | s+1 |f | ≤ log e + , (8.2.11) ≤ (s + 1) log e + log e + ||f ||m ||f ||m ||f ||m |g| s |g| s+1 |g| log e + ≤ log e + . (8.2.12) ≤ (s + 1) log e + ||g||n ||g||n ||g||n 1 1 Now, from the H¨ older inequality (8.2.7) with 1 = m/s + n/s (note that m/s > 1, n/s > 1 since 1/s = 1/m + 1/n) and (8.2.10), (8.2.11), and (8.2.12), we have |f g| γ s e + |f g| log ||f g||s dx Ω

=

γ s s e+ (|f | |g| ) log Ω

≤ C1 ≤ C2 ×

β s n/s e+ (|g| ) log Ω

dx dx

|f |s |||f |s ||m/s

|g|s |||g|s ||n/s

s/m dx

s/n dx

s s/m |f | α m e + dx |f | log ||f || Ω m

Ω

|f |s |g|s |||f |s |g|s ||1

α s m/s e+ (|f | ) log Ω

≤ C3 ×

γ s s e+ (|f | |g| ) log Ω

= C2 ×

||f |s |g|s |1/s 1/s |||f |s |g|s ||1

s s/n |g| dx |g|n logβ e + ||g|| n

α m e+ |f | log Ω

β n e+ |g| log Ω

|f | ||f ||m

|g| ||g||n

s/m dx

s/n dx

.

(8.2.13)

8.2 Global integrability

329

Hence, it follows that

γ s e+ |f g| log Ω

≤ C4 ×

|f g| ||f g||s

α m e+ |f | log Ω

Ω

|g|n logβ e +

1/s dx |f | ||f ||m

|g| ||g||n

1/m dx

(8.2.14)

1/n dx

,

which shows that inequality (8.2.8) is true. From Theorem 8.2.5, we have the following general result immediately. Corollary 8.2.6. Let pi > 0, αi > 0 for i = 1, 2, . . . , k, 1 1 1 1 + + ··· + = p1 p2 pk p

and

α1 α2 αk α + + ··· + = . p1 p2 pk p

Assume that fi ∈ Lpi (log L)αi (Ω) for i = 1, 2, . . . , k. Then, f1 f2 · · · fk ∈ Lp (log L)α (Ω) and [f1 f2 · · · fk ]Lp (log L)α (Ω) ≤ C [f1 ]Lp1 (log L)α1 (Ω) [f2 ]Lp2 (log L)α2 (Ω) · · · [fk ]Lpk (log L)αk (Ω) ,

(8.2.15)

where C is a positive constant and the norms [f1 f2 · · · fk ]Lp (log L)α (Ω) and [fi ]Lpi (log L)αi (Ω) , i = 1, 2, . . . , k, are deﬁned in Section 1.9.

8.2.3 Applications Next, we explore some applications of the new version of the H¨ older inequality established above. Speciﬁcally, we study the integrability of the Jacobian of the composition of mappings f : Ω → Rn deﬁned by f = (f 1 (u1 , u2 , . . . , un ), f 2 (u1 , u2 , . . . , un ), . . . , f n (u1 , u2 , . . . , un )) 1,p of Sobolev class Wloc (Ω, Rn ), where

ui = ui (x1 , x2 , . . . , xn ), i = 1, 2, . . . , n,

330

8 Estimates for Jacobians

are functions of x = (x1 , x2 , . . . , xn ) ∈ Ω with continuous partial deriva∂ui tives ∂x , j = 1, 2, . . . , n. For this, we assume that distributional diﬀerentials j Df (u) = [∂f i /∂uj ] and Du(x) = [∂ui /∂xj ] are locally integrable functions with values in the space GL(n) of all n × n-matrices. As usual, we will write ∂(f 1 · · · f n ) , ∂(x1 · · · xn )

J(x, f ) = detDf (u(x)) =

(8.2.16)

J(u, f ) = detDf (u) =

∂(f 1 · · · f n ) , ∂(u1 · · · un )

(8.2.17)

J(x, u) = detDu(x) =

∂(u1 · · · un ) , ∂(x1 · · · xn )

(8.2.18)

and

respectively. From Theorem 8.2.5, we have the following integrability result for the Jacobian of the composition of mappings. Theorem 8.2.7. Let s, t, β, γ > 0, with p1 = 1s + 1t and βs + γt = αp . Assume that J(x, f ), J(u, f ), and J(x, u) are Jacobians deﬁned in (8.2.16), (8.2.17), and(8.2.18), respectively. If J(u(x), f ) ∈ Ls (log L)β (Ω) and J(x, u) ∈ Lt (log L)γ (Ω), then J(x, f ) ∈ Lp (log L)α (Ω) and

α p e+ |J(x, f )| log Ω

≤C ×

|J(x,f )| ||J(x,f )||p

1/p dx

Ω

|J(u, f )|s logβ e +

|J(u,f )| ||J(u,f )||s

Ω

|J(x, u)|t logγ e +

|J(x,u)| ||J(x,u)||t

1/s dx

(8.2.19)

1/t dx

,

where C is a positive constant. Proof. Note that the Jacobian of the composition of f and u can be expressed as 1 ···f n ) J(x, f ) = ∂(f ∂(x1 ···xn ) =

∂(f 1 ···f n ) ∂(u1 ···un )

·

∂(u1 ···un ) ∂(x1 ···xn )

= J(u, f ) · J(x, u). From Theorem 8.2.5 and (8.2.20), we ﬁnd that

(8.2.20)

8.2 Global integrability

Ω

=

331

|J(x, f )|p logα e +

≤C ×

|J(x,f )| ||J(x,f )||p

1/p dx

α p e+ |J(u, f ) · J(x, u)| log Ω β s e+ |J(u, f )| log Ω

|J(u,f )| ||J(u,f )||s

γ t e+ |J(x, u)| log Ω

|J(x,u)| ||J(x,u)||t

|J(u,f )·J(x,u)| ||J(u,f )·J(x,u)||p

1/p dx

1/s dx

(8.2.21)

1/t dx

<∞ since J(u(x), f ) ∈ Ls (log L)β (Ω) and J(x, u) ∈ Lt (log L)γ (Ω). Thus, J(x, f ) ∈ Lp (log L)α (Ω). Now, as an application of the H¨ older inequality with Lp -norm f g s,E ≤ f α,E · g β,E ,

(8.2.22)

where 0 < α, β < ∞, s−1 = α−1 + β −1 , and f and g are any measurable functions on a measurable set E ⊂ Rn , we have the following Lp -integrability theorem for the Jacobian of a composite mapping. Theorem 8.2.8. Let J(x, f ), J(u, f ), and J(x, u) be Jacobians deﬁned in (8.2.16), (8.2.17), and(8.2.18), respectively. If J(u(x), f ) ∈ Ls (Ω) and J(x, u) ∈ Lt (Ω), s, t > 0, then J(x, f ) ∈ Lp (Ω) and ||J(x, f )||Lp (Ω) ≤ C||J(u(x), f )||Ls (Ω) ||J(x, u)||Lt (Ω) , where C is a positive constant and the integrability exponent p of J(x, f ) determined by p1 = 1s + 1t is the best possible.

8.2.4 Examples The following example shows that the integrability exponent p of J(x, f ) cannot be improved further.

332

8 Estimates for Jacobians

Example 8.2.9. For any (x, y) ∈ D = {(x, y) : 0 < x2 + y 2 ≤ ρ2 }, we consider the mappings deﬁned by x y 1 2 , , f (x, y) = (f , f ) = (x2 + y 2 )σ (x2 + y 2 )σ and x = r−k cos θ, y = r−k sin θ, (r, θ) ∈ Ω = {(r, θ) : 0 < r < ρ, 0 < θ ≤ 2π}, where σ and ρ are positive constants. After a simple calculation, we obtain the following Jacobians: J1 =

∂(f 1 ,f 2 ) ∂(r,θ)

=

k(2σ−1) , r 4σ+2k+1

J2 =

∂(f 1 ,f 2 ) ∂(x,y)

=

1−2σ r 4σ ,

J3 =

∂(x,y) ∂(r,θ)

=

−k , r 2k+1

(8.2.23) 0 < r < ρ.

It is easy to see that J1 ∈ L1/(4σ+2k+1) (Ω), but J1 ∈ Lp (Ω) for any p > 1 1 1/4σ (Ω), but J2 ∈ Ls (Ω) for any s > 4σ ; and 4σ+2k+1 . Similarly, J2 ∈ L 1 1/(2k+1) t J3 ∈ L (Ω), but J3 ∈ L (Ω) for any t > 2k+1 . Here, the integrability exponent p = 1/(4σ + 2k + 1) of ∂(f 1 , f 2 )/∂(r, θ) is determined by 1 1 1 = 4σ + 2k + 1 = + , p s t where s = 1/4σ and t = 1/(2k + 1) are the integrability exponents of the Jacobians ∂(f 1 , f 2 )/∂(x, y) and ∂(x, y)/∂(r, θ), respectively. Thus, the integrability exponent p of J(x, f ) is the best possible. Example 8.2.10. Let J1 = ∂(f 1 , f 2 )/∂(r, θ), J2 = ∂(f 1 , f 2 )/∂(x, y), and J3 = ∂(x, y)/∂(r, θ) be the Jacobians obtained in Example 8.2.9. For any ε > 0, there exists a constant C1 > 0 such that |J1 | |J1 |p log e + ≤ C1 |J1 |p+ε/(4σ+2k+1) . J1 p From (8.2.23) and (8.2.24), we have

(8.2.24)

8.2 Global integrability

333

Ω

|J1 |p log e +

= 2π

= C2 ≤ C3 ≤ C4

ρ 0

|J1 |

J1 p

1

0

r 4σ+2k+1

0

ρ 0

drdθ

|J1 |p log e +

ρ ρ

p

|J1 |

J1 p

log e +

dr (2σ−1)k | r 4σ+2k+1 (2σ−1)k

4σ+2k+1 p r

|

dr

ε/4σ+2k+1 r−(4σ+2k+1)p r−(4σ+2k+1) dr r−(4σ+2k+1)p−ε dr

= C5 < ∞ for any p satisfying 0
ε 1 − . 4σ + 2k + 1 4σ + 2k + 1

Since ε > 0 is arbitrary, it follows that J1 ∈ Lp log L(Ω) for any p with 0 < 1 1 p < 4σ+2k+1 . Similarly, we have J2 ∈ Ls log L(Ω) for any s with 0 < s < 4σ 1 and J3 ∈ Lt log L(Ω) for any t with 0 < t < 2k+1 . This example shows that 1

2

,f ) the integrability exponent p of ∂(f ∂(r,θ) determined by 1/p = 1/s + 1/t is the best possible when α = β = γ = 1 in Theorem 8.2.7.

8.2.5 The norm comparison In this section, we shall discuss the relationship between norms f Lp logα L and [f ]Lp (log L)α (Ω) , which will provide an alternative method to prove Theorems 8.2.5 and 8.2.7. For this, we shall need the following general inequality which ﬁrst appeared in [312, Theorem A.1]. Theorem 8.2.11. Suppose A, B, C : [0, ∞) → [0, ∞) are continuous, monotone-increasing functions for which there exist positive constants c and d such that (i) B −1 (t)C −1 (t) ≤ cA−1 (t) for all t > 0 and (ii) A( dt ) ≤ 12 A(t) for all t > 0.

334

8 Estimates for Jacobians

Suppose that G is an open subset of Rn , and f ∈ LB (G) and g ∈ LC (G). Then, f g ∈ LA (G) and ||f g||A ≤ cd||f ||B ||g||C . Proof. Note that if f ∈ LB (G), the monotonicity of A and an application of the monotone convergence theorem gives |f (x)| dx ≤ 1. A f A G Hence, from the deﬁnition of the Luxemburg norm, we obtain f (x)g(x) B c f B g C dx G (x)| |g(x)| dx + ≤ G B |f C

f B

g C dx G ≤ 2. We therefore have

A G

f (x)g(x) cdf B gC

dx ≤ 1

and, again by the deﬁnition of the Luxemburg norm, we have the result. Thus, the norm f Lp logα L is equivalent to the norm [f ]Lp (log L)α (Ω) for 1 < p < ∞. We have also proved in Theorem 1.9.1 that the norm f Lp logα L is equivalent to the norm [f ]Lp (log L)α (Ω) , that is, for each f ∈ Lp (log L)α (Ω), 0 < p < ∞ and α ≥ 0, f p ≤ f Lp logα L ≤ [f ]Lp (log L)α (Ω) ≤ Cf Lp logα L ,

(8.2.25)

α 1/p α is a constant independent of f . where C = 2α/p 1 + ep Hence, for any 0 < p < ∞ and α ≥ 0, the Luxemburg norm f Lp logα L is equivalent to the norm [f ]Lp (log L)α (Ω) deﬁned in Section 1.9. Therefore, Theorems 8.2.5 and 8.2.7 can also be proved by employing Theorem 8.2.11 with suitable choices of functions A(t), B(t), and C(t). The following norm comparison theorem appeared in [312] whose proof is available in [204, p. 23]. Theorem 8.2.12. Let f be supported in a ball B ⊂ Rn and let IM be the Hardy–Littlewood maximal operator. Then, f ∈ L log L(B) if and only if

8.2 Global integrability

335

IMf ∈ L1 (B). Furthermore, there exist constants C1 and C2 independent of f for which C1 IMf L1 (B) ≤ f L log L(B) ≤ C2 IMf L1 (B) . Now, for k = 0, 1, . . . , n − 1, consider the subdeterminant of the Jacobian J(x, f ) which is obtained by deleting the k rows and k columns from J(x, f ), i.e., J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k ) i1 fxj1 i2 fxj1 = .. . in−k fxj 1

fxi1j2

fxi1j3

···

fxi1j

n−k

fxi2j2 .. .

fxi2j3 .. .

···

fxi2j

n−k

i

fxn−k j2

i

fxn−k j3

..

.. .

.

···

i

fxn−k jn−k

which is an (n − k) × (n − k) subdeterminant of J(x, f ), {i1 , i2 , . . . , in−k } ⊂ {1, 2, . . . , n} and {j1 , j2 , . . . , jn−k } ⊂ {1, 2, . . . , n}. Let u = J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k )dxj1 ∧ dxj2 ∧ · · · ∧ dxjn−k . (8.2.26) It is clear that u is an (n − k)-form. From Corollary 8.2.3, the following estimate for a subdeterminant of the Jacobian J(x, f ) follows immediately. Theorem 8.2.13. Let u be an (n − k) × (n − k) subdeterminant of J(x, f ) deﬁned in (8.2.26), k = 0, 1, . . . , n − 1, and ϕ be a strictly increasing convex function on [0, ∞) with ϕ(0) = 0, and D be a domain in Rn . If ϕ(|u|) ∈ L1 (D; μ) and μ({x ∈ D : |u − c| > 0}) > 0 for any constant c, then for any positive constant a, it follows that a ϕ( |u − uD,μ |)dμ ≤ ϕ(a|u|)dμ ≤ C ϕ(2a|u − uD,μ |)dμ, (8.2.27) 2 D D D where C is a positive constant. If we choose k = 0 in Theorem 8.2.13, we have the following estimate for the Jacobian J(x, f ). Corollary 8.2.14. Let ϕ be a strictly increasing convex function on [0, ∞) with ϕ(0) = 0, and D be a domain in Rn . If ϕ(|J(x, f )|) ∈ L1 (D; μ) and μ({x ∈ D : |J(x, f ) − c| > 0}) > 0 for any constant c, then for any positive constant a, it follows that

336

8 Estimates for Jacobians

ϕ( a2 |J(x, f ) − J(x, f )D,μ |)dμ ≤ D ϕ(a|J(x, f )|)dμ ≤ C D ϕ(2a|J(x, f ) − J(x, f )D,μ |)dμ,

D

where C is a positive constant. Choosing ϕ(t) = tp logα (e + t), 1 < p < ∞ and α ≥ 0, in Corollary 8.2.14, we obtain the following estimate for the Jacobian:

C1 D |J(x, f ) − J(x, f )D,μ |p logα e + a2 |J(x, f ) − J(x, f )D,μ | dμ ≤ D |J(x, f )|p logα (e + a|J(x, f )|) dμ ≤ C2 D |J(x, f ) − J(x, f )D,μ |p logα (e + 2a|J(x, f ) − J(x, f )D,μ |) dμ, where a ≥ 0 is any constant and C1 and C2 are some positive constants. Similarly, choosing ϕ(t) = tp , 1 < p < ∞, in Corollary 8.2.14, we obtain the following estimate for the Jacobian: C1 D |J(x, f ) − J(x, f )D,μ |p dμ ≤ D |J(x, f )|p dμ ≤ C2 D |J(x, f ) − J(x, f )D,μ |p dμ, where C1 and C2 are some positive constants. In our next result we shall provide an estimate for the sharp maximal operator IMs . Theorem 8.2.15. Let u ∈ Ls (log L)α (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, and α ≥ 0, be a diﬀerential form satisfying (1.2.10) in a bounded, convex domain Ω, IMs be the sharp maximal operator deﬁned in (7.1.2), and T be the homotopy operator deﬁned in (1.5.1), then cIMs (T u)s,Ω ≤ us,Ω ≤ uLs logα L(Ω) ≤ [u]Ls (log L)α (Ω)

(8.2.28)

for some constant c, independent of u. Proof. From Theorem 7.8.1, there exists a constant C1 > 0 such that IMs (T u)s,Ω ≤ C1 us,Ω ,

(8.2.29)

8.2 Global integrability

337

which is equivalent to cIMs (T u)s,Ω ≤ us,Ω ,

(8.2.30)

where c = 1/C1 . On the other hand, from Theorem 1.9.1, we obtain us,Ω ≤ uLs logα L(Ω) ≤ [u]Ls (log L)α (Ω) .

(8.2.31)

Combining (8.2.30) and (8.2.31), we have (8.2.28) immediately. Finally, we state the following corollary, which follows from Theorem 1.9.1. Corollary 8.2.16. Let u ∈ Ls (log L)α (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, and α ≥ 0, be an (n − k) × (n − k) subdeterminant of J(x, f ) deﬁned in (8.2.26), k = 0, 1, . . . , n − 1. Then, us,Ω ≤ uLs logα L(Ω) ≤ [u]Ls (log L)α (Ω) , where Ω is a bounded domain. If k = 0 in Corollary 8.2.16, we obtain the following estimate: J(x, f )s,Ω ≤ J(x, f )Ls logα L(Ω) ≤ [J(x, f )]Ls (log L)α (Ω) , where Ω is a bounded domain.

(8.2.32)

Chapter 9

Lipschitz and BM O norms

In this chapter we provide some norm comparison theorems related to the BM O norms and the Lipschitz norms. We prove that the integrability exponents described in the Lipschitz norm comparison theorem (Theorem 9.2.1) are the best possible. We also develop some norm comparison theorems for the operators.

9.1 Introduction The bounded mean oscillation (BM O) space was introduced by John and Nirenberg in 1961. “Bounded mean oscillation” soon became one of the main concepts in many ﬁelds, such as harmonic analysis, complex analysis, and partial diﬀerential equations. A function f ∈ L1loc (Ω, μ) is said to be in BM O(Ω, μ) if there is a constant C such that 1 |f − fB |dμ ≤ C (9.1.1) μ(B) B for all balls B with σB ⊂ Ω, where σ > 1 is a constant. The least C for which (9.1.1) holds is denoted by f = f ,Ω and called the BM O norm of f . Equivalently, 1 f ,Ω = sup |f − fB |dμ, (9.1.2) σB⊂Ω μ(B) B where σ > 1 is a constant. One of the most useful results for BM O is the John–Nirenberg lemma which was ﬁrst proved by Muckenhoupt and Wheeden in [146]. Abstract versions of the John–Nirenberg lemma are also available, see [313], for example. R.P. Agarwal et al., Inequalities for Diﬀerential Forms, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-68417-8 9,

339

340

9 Lipschitz and BM O norms

For relationship between BM O and quasiconformal mappings, see [70, 71]. Also, see [123] for properties of BM O spaces. Theorem 9.1.1. (John–Nirenberg lemma for doubling weights). Let μ be deﬁned by dμ = w(x)dx and w(x) be a doubling weight. Then, a function f is in BM O(Ω, μ) if and only if μ({x ∈ B : |f (x) − fB | > t}) ≤ c1 e−c2 t μ(B)

(9.1.3)

for each ball B ⊂ Ω and t > 0. Here c1 and c2 are positive constants. The proof of John–Nirenberg lemma is also available in [59] where the Calder´ on–Zygmund decomposition technique is adopted. We state the following corollary from [59] which provides a useful tool to study the averaging domains. Corollary 9.1.2. A function f is in BM O(Ω, μ) if and only if there exist positive constants k and C such that 1 ek|f −fB | dμ ≤ C (9.1.4) μ(B) B for any ball B which is a compact subset of Ω. If (9.1.4) holds, then f ≤ C/k. Conversely, if f is in BM O(Ω, μ), then (9.1.4) holds with C = 3 and k = (log 2)/(8c0 N f ), where N and c0 are constants appearing in the Calder´ on–Zygmund decomposition for doubling weights.

9.2 BMO spaces and Lipschitz classes In this section, we will ﬁrst present some interesting results for the BM O spaces and the Lipschitz spaces of diﬀerential forms, and, then provide an example to show that the integrability exponents in Lipschitz conditions for conjugate A-harmonic tensors are the best possible.

9.2.1 Some recent results Let ω ∈ L1loc (Ω, ∧l ), l = 0, 1, . . . , n. We write ω ∈ locLipk (Ω, ∧l ), 0 ≤ k ≤ 1, if ωlocLipk ,Ω = sup |Q|−(n+k)/n ω − ωQ 1,Q < ∞ (9.2.1) σQ⊂Ω

9.2 BMO spaces and Lipschitz classes

341

for some σ ≥ 1. Further, we write Lipk (Ω, ∧l ) for those forms whose coeﬃcients are in the usual Lipschitz space with exponent k and write ωLipk ,Ω for this norm. Similarly, for ω ∈ L1loc (Ω, ∧l ), l = 0, 1, . . . , n, we write ω ∈ BMO(Ω, ∧l ) if u,Ω = sup |Q|−1 ω − ωQ 1,Q < ∞

(9.2.2)

σQ⊂Ω

for some σ ≥ 1. When ω is a 0-form, (9.2.2) reduces to the classical deﬁnition of BMO(Ω). In 1999, C. Nolder [71] proved the following result. Theorem 9.2.1. If 0 ≤ k, l ≤ 1 satisfy p(k − 1) = q(l − 1), then there exists a constant C such that C −1 uplocLipk ,Ω ≤ vqlocLipl ,Ω ≤ CuplocLipk ,Ω

(9.2.3)

for all conjugate A-harmonic tensors u and v in Ω. Note that a pair of conjugate A-harmonic tensors are solutions of the conjugate A-harmonic equation (1.2.6). Proof. We shall only prove the left inequality. The right inequality follows similarly. From the deﬁnition of · locLipk ,Ω -norm, we have ulocLipk ,Ω = sup |Q|−(n+k)/n u − uQ 1,Q .

(9.2.4)

2Q⊂Ω

Now, using the condition p(k − 1) = q(l − 1) and Theorem 1.5.2, we obtain |Q|−(n+k)/n u − uQ 1,Q ≤ C1 (|Q|−(n+l)/n v − c1,2Q )q/p

(9.2.5)

for all cubes Q with 2Q ⊂ Ω. Next, we choose c such that c = ( v)2Q . The left inequality now follows from (9.2.4) and (9.2.5). The following result which appeared in [71] describes the relationship between a BM O space and a local Lipschitz space. Theorem 9.2.2. Let ω be a solution to the homogeneous A-harmonic equation (1.2.4). Then, the following statements are equivalent: (a) ω ∈ BMO(Ω, ∧); (b) sup{|Q|(p−n)/pn dωp,Q |σQ ⊂ Ω} < ∞ for some σ > 1.

342

9 Lipschitz and BM O norms

Similarly, the following statements are equivalent: (c) ω ∈ locLipk (Ω, ∧); (d) sup{|Q|(p−pk−n)/pn dωp,Q |σQ ⊂ Ω} < ∞ for some σ > 1. We conclude this section with the following global result established in [71]. Theorem 9.2.3. There exists a constant C such that C −1 u − uQ Lipk ,Q ≤ v − cLipk ,Q ≤ Cu − uQ Lipk ,Q for all conjugate A-harmonic tensors u and v in a cube Q ⊂ Rn . Here 0 < k ≤ 1 and c = ( v)Q .

9.2.2 Sharpness of integrability exponents From Section 1.8, we know that f (x) = (f 1 , f 2 , f 3 ) = x|x|β = (x1 |x|β , x2 |x|β , x3 |x|β ) is a K-quasiregular mapping in R3 . Here β = −1 is a real number. Applying Example 1.2.4 with l = 1, we ﬁnd that u = f 1 = x1 |x|β and v = f 2 df 3 = x2 |x|β d(x3 |x|β ) form a pair of conjugate A-harmonic tensors. Now since p = n 3 n−l = 2 , by simple calculation, it follows that v

=

n l

= 3, q =

βx1 x2 x3 |x|2β−2 dx2 ∧ dx3 − βx22 x3 |x|2β−2 dx1 ∧ dx3 +|x|2β−2 x2 (|x|2 + βx23 )dx1 ∧ dx2 .

We also know that |u| = |x1 ||x|β and |v| = |x|2β−1 |x2 |(|x|2 + (β 2 + 2β)x23 )1/2 . Thus, if β ≥ 0 or β ≤ −2, then

9.2 BMO spaces and Lipschitz classes

343

(|x|2 + (β 2 + 2β)x23 )1/2 ≤ (|x|2 + (β 2 + 2β)|x|2 )1/2 = |x|(β 2 + 2β + 1)1/2 = |β + 1||x|. Hence, we have |x2 ||x|2β ≤ |v| ≤ |β + 1||x2 ||x|2β . Now, we select Ω ⊂ R3 so that for any x ∈ Ω, C1 |x1 | ≤ |x| ≤ C2 |x1 | and C1 |x2 | ≤ |x| ≤ C2 |x2 |, where C1 and C2 are constants. For example, when Ω = {(x1 , x2 , x3 ) | 1 ≤ x1 ≤ 2, 1 ≤ x2 ≤ 2, 0 ≤ x3 ≤ 1} , then for any x ∈ Ω,

|x| ≤ (x21 + 4x21 + 1)1/2 ≤ 1+4+

1 |x1 |2

1/2

≤ |x1 |(1 + 4 + 1)1/2 √ = 6|x1 |. Thus, |x1 | ≤ |x| ≤

√

6|x1 |

holds for any x ∈ Ω. Similarly, |x2 | ≤ |x| ≤

√

6|x2 |

holds for any x ∈ Ω. Therefore, |x1 | ∼ |x| and |x2 | ∼ |x| in Ω. So, we have |u| ∼ |x1 ||x|β ∼ |x|β+1 = |x|k

(9.2.6)

and |v| ∼ |x2 ||x|2β ∼ |x|2β+1 = |x|l

(9.2.7)

in Ω. Here k = β + 1 and l = 2β + 1. Now, since p = n/l = 3 and q = n/(n − l) = 3/2, it follows that

344

9 Lipschitz and BM O norms

p(k − 1) = 3((β + 1) − 1) = 3β =

3 ((2β + 1) − 1) = q(l − 1), 2

that is p(k − 1) = q(l − 1). Thus, the integrability exponents in Theorem 9.2.1 are sharp.

9.3 Global integrability In recent years new interest has developed in the study of the global integrability for solutions of the A-harmonic equation. T. Kilpel¨ ainen and P. Koskela have proved the following global integrability theorem in [314]. Theorem 9.3.1. If u is a bounded A-harmonic function (solution to equation (1.2.2)) in a ball B ⊂Rn , then |∇u|(log(2 + |∇u|))−1−ε < ∞ (9.3.1) B

for all ε > 0. We say that Ω satisﬁes a Whitney cube number condition with exponent λ if there is a constant M such that N (j) ≤ M 2λj , where N (j) is the number of Whitney cubes with the side length 2−j . Balls and cubes satisfy a Whitney cube number condition with λ = n − 1; see [204] for more properties of Whitney cubes.

9.3.1 Estimates for du In [72], C. Nolder obtained the following result for more general domains. Theorem 9.3.2. Suppose that Ω satisﬁes a Whitney cube number condition with exponent λ < n. If u is A-harmonic in Ω and u ∈ locLipk (Ω), 0 < k ≤ 1, then |∇u|q < ∞ (9.3.2) Ω

for q < (n − λ)/(1 − k).

9.3 Global integrability

345

He also proved the following global integrability theorem for the solutions of the nonhomogeneous A-harmonic equation (1.2.10). 1,p Theorem 9.3.3. Suppose that the diﬀerential form u ∈ Wloc (Ω, ∧) satisﬁes (1.2.10) and Ω satisﬁes a Whitney cube number condition with exponent λ, 0 ≤ λ < n. If u ∈ BMO(Ω, ∧) and n − λ ≤ p, then |du|n−λ (log(2 + |du|))−1−ε < ∞ (9.3.3) Ω

for all ε > 0. If u ∈ locLipk (Ω, ∧) and (n − λ)/(1 − k) ≤ p, then |du|(n−λ)/(1−k) (log(2 + |du|))−1−ε < ∞

(9.3.4)

Ω

for all ε > 0.

9.3.2 Estimates for du+ Theorem 9.3.3 is a far-reaching extension of Theorem 9.3.1. The following global integrability theorems also appear in [72]. Theorem 9.3.4. Let u be a solution to the nonhomogeneous A-harmonic equation (1.2.10) in Ω and η ∈ C0∞ (Ω) with η ≥ 0. Then, there exists a constant C, depending only on a, b, and p, such that (9.3.5) |du+ |p η p ≥ C |u+ |p |∇η|p + |u+ |p η p . Ω

Ω

Ω

The same is true for u− . Theorem 9.3.5. Let u be a solution to the nonhomogeneous A-harmonic equation (1.2.10) in Ω and η ∈ C0∞ (Ω) with η ≥ 0. For each q ≥ 0, there exists a constant C, depending only on a, b, p, q, and n, such that |u+ |q |du|p η p ≥ C |u+ |p+q (|∇η|p + η p ). (9.3.6) Ω

Ω −

The inequality also holds for u .

346

9 Lipschitz and BM O norms

Similar to Theorem 9.2.2 for the case of homogeneous A-harmonic equation, the equivalent statements hold for the solutions of the nonhomogeneous A-harmonic equation (1.2.10). Theorem 9.3.6. Let ω be a solution to the nonhomogeneous A-harmonic equation (1.2.10). Then, the following statements are equivalent: (a) ω ∈ BMO(Ω, ∧); (b) sup{|Q|(p−n)/pn dωp,Q |σQ ⊂ Ω} < ∞ for some σ > 1. Similarly, the following statements are equivalent: (c) ω ∈ locLipk (Ω, ∧); (d) sup{|Q|(p−pk−n)/pn dωp,Q |σQ ⊂ Ω} < ∞ for some σ > 1.

9.4 Lipschitz and BM O norms In this section, we will establish some estimates for the Lipschitz and BM O norms of the operators applied to some diﬀerential forms. We will also obtain some norm comparison theorems and discuss weighted cases.

9.4.1 Estimates for Lipschitz norms First, we will develop some Lipschitz norm estimates for the operators. We begin with the following results for the homotopy operator T deﬁned in (1.5.1). Theorem 9.4.1. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (u)locLipk ,Ω ≤ Cus,Ω ,

(9.4.1)

where k is a constant with 0 ≤ k ≤ 1. Proof. From Theorem 7.5.1, we have T (u) − (T (u))B s,B ≤ C1 |B|diam(B)us,σB

(9.4.2)

9.4 Lipschitz and BM O norms

347

for all balls B with σB ⊂ Ω, where σ > 1 is a constant. Using the H¨ older inequality with 1 = 1/s + (s − 1)/s, we ﬁnd that T (u) − (T (u))B 1,B = B |T (u) − (T (u))B |dx ≤

B

|T (u) − (T (u))B |s dx

1/s B

1s/(s−1) dx

(s−1)/s

= |B|(s−1)/s T (u) − (T (u))B s,B

(9.4.3)

= |B|1−1/s T (u) − (T (u))B s,B ≤ |B|1−1/s (C1 |B|diam(B)us,σB ) ≤ C2 |B|2−1/s+1/n us,σB . Now, from the deﬁnition of the Lipschitz norm, (9.4.3), and 2 − 1/s + 1/n − 1 − k/n = 1 − 1/s + 1/n − k/n > 0, we obtain T (u)locLipk ,Ω = supσB⊂Ω |B|−(n+k)/n T (u) − (T (u))B 1,B = supσB⊂Ω |B|−1−k/n T (u) − (T (u))B 1,B ≤ supσB⊂Ω |B|−1−k/n C2 |B|2−1/s+1/n us,σB = supσB⊂Ω C2 |B|1−1/s+1/n−k/n us,σB

(9.4.4)

≤ supσB⊂Ω C2 |Ω|1−1/s+1/n−k/n us,σB ≤ C3 supσB⊂Ω us,σB ≤ C3 us,Ω , that is, T (u)locLipk ,Ω ≤ C3 us,Ω . Note that for a bounded domain D, both T (u)s,B ≤ C|B|diam(B)us,B

(9.4.5)

348

9 Lipschitz and BM O norms

and T (u)s,B ≤ Cdiam(B)us,B

(9.4.6)

hold for any ball B with B ⊂ D. Thus, if we use (9.4.5) in the proof of Theorem 3.3.7, instead of (9.4.6), we obtain the following version of the Poincar´etype estimate G(u) − (G(u))B s,B ≤ C|B|diam(B)dus,B .

(9.4.7)

Further, from the above inequality we have the following Poincar´e-type estimate for the projection operator H(u) − (H(u))B s,B ≤ C|B|diam(B)dus,B .

(9.4.8)

From (9.4.7) and (9.4.8) and the method similar to the proof of Theorem 9.4.1, noticing that both of G and H commute with d, we can establish the following Lipschitz norm inequalities for Green’s operator G and the projection operator H. Theorem 9.4.2. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n − 1, 1 < s < ∞, be a smooth diﬀerential form in a bounded domain Ω and G be Green’s operator, and H be the projection operator. Then, there exists a constant C, independent of u, such that G(u)locLipk ,Ω ≤ Cdus,Ω

(9.4.9)

H(u)locLipk ,Ω ≤ Cdus,Ω ,

(9.4.10)

and where k is a constant with 0 ≤ k ≤ 1. Theorem 9.4.3. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded domain Ω and G be Green’s operator. Then, there exists a constant C, independent of u, such that G(u)locLipk ,Ω ≤ Cu,Ω , (9.4.11) where k is a constant with 0 ≤ k ≤ 1. Proof. Using the H¨ older inequality with 1 = 1/s + (s − 1)/s and (9.4.7), we ﬁnd that

9.4 Lipschitz and BM O norms

349

G(u) − (G(u))B 1,B = B |G(u) − (G(u))B |dx ≤

B

|G(u) − (G(u))B |s dx

1/s B

1s/(s−1) dx

(s−1)/s

= |B|(s−1)/s G(u) − (G(u))B s,B

(9.4.12)

= |B|1−1/s G(u) − (G(u))B s,B ≤ |B|1−1/s (C1 |B|diam(B)dus,B ) ≤ C2 |B|2−1/s+1/n dus,B , where we have used diam(B) = C|B|1/n . Now, from the Caccioppoli inequality (Theorem 4.2.1), we have dus,B ≤ C3 |B|−1/n u − cs,σ1 B

(9.4.13)

for any ball B and some constant σ1 > 1, where c is any closed form. Next, choosing c = uB in (9.4.13), we ﬁnd that dus,B ≤ C3 |B|−1/n u − uB s,σ1 B .

(9.4.14)

Combining (9.4.12) and (9.4.14), it follows that G(u) − (G(u))B 1,B ≤ C2 |B|2−1/s+1/n dus,B ≤ C4 |B|

2−1/s

(9.4.15)

u − uB s,σ1 B .

Applying the weak reverse H¨older inequality for the solutions of the nonhomogeneous A-harmonic equation, we obtain u − uB s,σ1 B ≤ C5 |B|(1−s)/s u − uB 1,σ2 B ,

(9.4.16)

where σ2 > σ1 > 1 is a constant. Substituting (9.4.16) into (9.4.15), we have G(u) − (G(u))B 1,B ≤ C6 |B|u − uB 1,σ2 B , which is equivalent to

(9.4.17)

350

9 Lipschitz and BM O norms

|B|−(n+k)/n G(u) − (G(u))B 1,B ≤ C6 |B|1−k/n |B|−1 u − uB 1,σ2 B

(9.4.18)

≤ C7 |B|1−k/n |σ2 B|−1 u − uB 1,σ2 B . Finally, taking the supremum over all balls σ3 B ⊂ Ω with σ3 > σ2 and using the deﬁnitions of the Lipschitz and BM O norms, we obtain G(u)locLipk ,Ω = supσ3 B⊂Ω |B|−(n+k)/n G(u) − (G(u))B 1,B ≤ supσ3 B⊂Ω C6 |B|1−k/n |B|−1 u − uB 1,σ2 B

(9.4.19)

≤ C7 supσ3 B⊂Ω |B|−1 u − uB 1,σ2 B ≤ C8 u,Ω . Thus, inequality (9.4.11) follows.

9.4.2 Lipschitz norms of T ◦ G Now, using Theorem 7.4.1 and the method developed in the proof of Theorem 9.4.1, we have the following estimate for the Lipschitz norm of the composition T ◦ G of the homotopy operator T and Green’s operator G. Theorem 9.4.4. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, G be Green’s operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (G(u))locLipk ,Ω ≤ Cus,Ω ,

(9.4.20)

where k is a constant with 0 ≤ k ≤ 1. Note that in Theorem 9.4.4, u need not be a solution of the A-harmonic equation. Further, if u ∈ Ls (Ω, ∧1 ), then T (G(u)) ∈ Ls (Ω, ∧0 ). Using Lemma 8.2.2 with ϕ(t) = ts and c = uB over the ball B, we have

9.4 Lipschitz and BM O norms

351

us,B ≤ Cu − uB s,B ,

(9.4.21)

where C is a constant. Theorem 9.4.5. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω and G be Green’s operator. Then, there exists a constant C, independent of u, such that T (G(u))locLipk ,Ω ≤ Cu,Ω , (9.4.22) where k is a constant with 0 ≤ k ≤ 1. Proof. From the deﬁnition of the Lipschitz norm, the H¨ older inequality with 1 = 1/s + (s − 1)/s, (7.4.1), and (9.4.21), for any ball B with B ⊂ Ω, we ﬁnd that T (G(u)) − (T (G(u)))B 1,B = B |T (G(u)) − (T (G(u)))B |dx ≤

B

|T (G(u)) − (T (G(u)))B |s dx

1/s

s

B

1 s−1 dx

(s−1)/s

= |B|(s−1)/s T (G(u)) − (T (G(u)))B s,B = |B|1−1/s T (G(u)) − (T (G(u)))B s,B

(9.4.23)

≤ |B|1−1/s (C1 |B|diam(B)us,B ) ≤ C2 |B|2−1/s+1/n us,B ≤ C3 |B|2−1/s+1/n u − uB s,B . Next, from the weak reverse H¨ older inequality for solutions of the nonhomogeneous A-harmonic equation, we have u − uB s,B ≤ C4 |B|(1−s)/s u − uB 1,σ1 B

(9.4.24)

for some constant σ1 > 1. Combination of (9.4.23) and (9.4.24) gives T (G(u)) − (T (G(u)))B 1,B ≤ C3 |B|2−1/s+1/n u − uB s,B ≤ C5 |B|

1+1/n

u − uB 1,σ1 B .

(9.4.25)

352

9 Lipschitz and BM O norms

Hence, we obtain |B|−(n+k)/n T (G(u)) − (T (G(u)))B 1,B ≤ C5 |B|1/n−k/n u − uB 1,σ1 B = C5 |B|1+1/n−k/n |B|−1 u − uB 1,σ1 B (9.4.26)

≤ C6 |B|1+1/n−k/n |σ1 B|−1 u − uB 1,σ1 B ≤ C6 |Ω|1+1/n−k/n |σ1 B|−1 u − uB 1,σ1 B ≤ C7 |σ1 B|−1 u − uB 1,σ1 B .

Thus, taking the supremum on both sides of (9.4.26) over all balls σ2 B ⊂ Ω with σ2 > σ1 and using the deﬁnitions of the Lipschitz and BM O norms, we ﬁnd that T (G(u))locLipk ,Ω = supσ2 B⊂Ω |B|−(n+k)/n T (G(u)) − (T (G(u)))B 1,B ≤ C7 supσ2 B⊂Ω |σ1 B|−1 u − uB 1,σ1 B ≤ C7 u,Ω , that is, T (G(u))locLipk ,Ω ≤ Cu,Ω . Remark. (1) The above global norm comparison inequality has been proved on a bounded, convex domain Ω. We encourage readers to explore similar estimates in other kind of domains, such as Ls -averaging domains and Ls (μ)averaging domains. (2) Let 0 < p < ∞, 0 ≤ k ≤ 1, and u ∈ Lploc (Ω, ∧l , μ), l = 0, 1, . . . , n, where the measure μ is deﬁned by dμ = w(x)dx for some weight w(x). We may deﬁne the generalized Lipschitz norm by ulocLipk ,p,Ω,w = sup

σB⊂Ω

1 (μ(B))(n+k)/n

1/p

|u − uB,μ | dμ p

B

for some σ > 1, where the measure μ is deﬁned by dμ = w(x)dx for some weight w(x). (3) For 0 1, where the measure μ is deﬁned by dμ = w(x)dx for some weight w(x). When u is a 0-form and p = 1, the above BMO norm reduces to the classical BMO norm for functions. (4) Similarly, we can study the relationship between ulocLipk ,p,Ω,w and u,p,Ω,w as we discussed in Theorems 9.4.1, 9.4.2, 9.4.3, 9.4.4, and 9.4.5.

9.4.3 Lipschitz norms of G ◦ T Similarly, using Theorem 7.4.15 and the method developed in the proof of Theorem 9.4.1, we have the following estimate for the Lipschitz norm of the composition G ◦ T . Theorem 9.4.6. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a smooth diﬀerential form in a bounded, convex domain Ω, G be Green’s operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that G(T (u))locLipk ,Ω ≤ Cus,Ω ,

(9.4.27)

where k is a constant with 0 ≤ k ≤ 1. Applying the same method used in the proof of Theorem 9.4.5, we ﬁnd the following inequality for the Lipschitz and BMO norms. Theorem 9.4.7. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a solution of the Aharmonic equation (1.2.10) in a bounded, convex domain Ω and G be Green’s operator. Then, there exists a constant C, independent of u, such that G(T (u))locLipk ,Ω ≤ Cu,Ω , where k is a constant with 0 ≤ k ≤ 1. Now, if in the proof of Lemma 3.7.2, we use T (u)s,B ≤ C|B|diam(B)us,B , instead of (3.7.7), then (3.7.11) will appear as

(9.4.28)

354

9 Lipschitz and BM O norms

ΔG(u) − (ΔG(u))B s,B ≤ C|B|diam(B)dus,B .

(9.4.29)

As an application of (9.4.29), we have the following result, which is analogous to Theorem 9.4.2. Theorem 9.4.8. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n − 1, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded domain Ω, Δ be the Laplace–Beltrami operator, and G be Green’s operator. Then, there exists a constant C, independent of u, such that ΔG(u)locLipk ,Ω ≤ Cdus,Ω , where k is a constant with 0 ≤ k ≤ 1. Finally, following the proof of Theorem 9.4.3, we obtain the following norm inequality. Theorem 9.4.9. Let u ∈ Lsloc (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded domain Ω, Δ be the Laplace–Beltrami operator, and G be Green’s operator. Then, there exists a constant C, independent of u, such that ΔG(u)locLipk ,Ω ≤ Cu,Ω ,

(9.4.30)

where k is a constant with 0 ≤ k ≤ 1.

9.4.4 Lipschitz norms of T ◦ H and H ◦ T From Theorem 7.6.1 and the method developed in the proof of Theorem 9.4.1, we have the following results. Theorem 9.4.10. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (H(u))locLipk ,Ω ≤ Cus,Ω ,

(9.4.31)

where k is a constant with 0 ≤ k ≤ 1. Theorem 9.4.11. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the

9.4 Lipschitz and BM O norms

355

projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T (H(u))locLipk ,Ω ≤ Cu,Ω , (9.4.32) where k is a constant with 0 ≤ k ≤ 1. Note that inequality (9.4.32) implies that the norm T (H(u))locLipk ,Ω of T (H(u)) can be controlled by the norm u,Ω when u is a 1-form. Now, using Theorem 7.6.15 and the same method as in Theorem 9.4.1, we obtain the following Theorems 9.4.12 and 9.4.13. Theorem 9.4.12. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that H(T (u))locLipk ,Ω ≤ Cus,Ω ,

(9.4.33)

where k is a constant with 0 ≤ k ≤ 1. Theorem 9.4.13. Let u ∈ Lsloc (Ω, ∧1 ), 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω, H be the projection operator, and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that H(T (u))locLipk ,Ω ≤ Cu,Ω , (9.4.34) where k is a constant with 0 ≤ k ≤ 1.

9.4.5 Estimates for BM O norms

In Section 9.4.1, we have developed some estimates for the Lipschitz norm · locLipk ,Ω . Now, we will focus on the estimates for the BM O norm · ,Ω . For this, let u ∈ locLipk (Ω, ∧l ), l = 0, 1, . . . , n, 0 ≤ k ≤ 1, and Ω be a bounded domain. Then, from the deﬁnitions of the Lipschitz and BM O norms, we have

356

9 Lipschitz and BM O norms

u,Ω = supσB⊂Ω |B|−1 u − uB 1,B = supσB⊂Ω |B|k/n |B|−(n+k)/n u − uB 1,B ≤ supσB⊂Ω |Ω|k/n |B|−(n+k)/n u − uB 1,B ≤ |Ω|k/n supσB⊂Ω |B|−(n+k)/n u − uB 1,B

(9.4.35)

≤ C1 supσB⊂Ω |B|−(n+k)/n u − uB 1,B ≤ C1 ulocLipk ,Ω , where C1 is a positive constant. Hence, we have proved the following result. Theorem 9.4.14. If a diﬀerential form u ∈ locLipk (Ω, ∧l ), l = 0, 1, . . . , n, 0 ≤ k ≤ 1, in a bounded domain Ω, then u ∈ BMO(Ω, ∧l ) and u,Ω ≤ CulocLipk ,Ω ,

(9.4.36)

where C is a constant. Theorem 9.4.15. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T : C ∞ (Ω, ∧l ) → C ∞ (Ω, ∧l−1 ) be the homotopy operator deﬁned in (1.5.1). Then, there exists a constant C, independent of u, such that T u,Ω ≤ Cus,Ω .

(9.4.37)

Similar to the proof of Theorem 9.4.1, we can prove Theorem 9.4.15 directly by using Theorem 7.5.1 and the H¨ older inequality. Alternatively, using Theorems 9.4.1 and 9.4.14, we have the following simple proof. Proof of Theorem 9.4.15. Since inequality (9.4.36) holds for any diﬀerential form, we may replace u by T u in inequality (9.4.36). Thus, it follows that T u,Ω ≤ C1 T ulocLipk ,Ω , (9.4.38) where k is a constant with 0 ≤ k ≤ 1. On the other hand, from Theorem 9.4.1, we have T (u)locLipk ,Ω ≤ C2 us,Ω . (9.4.39) Combination of (9.4.38) and (9.4.39) yields T u,Ω ≤ C3 us,Ω . As in the proof of Theorem 9.4.15, using inequality (9.4.36) and Theorem 9.4.2, we obtain the following result immediately.

9.4 Lipschitz and BM O norms

357

Theorem 9.4.16. Let u ∈ Ls (Ω, ∧l ), l = 1, 2, . . . , n − 1, 1 < s < ∞, be a solution of the A-harmonic equation (1.2.10) in a bounded domain Ω, G be Green’s operator, and H be the projection operator. Then, there exists a constant C, independent of u, such that G(u),Ω ≤ Cdus,Ω

(9.4.40)

H(u),Ω ≤ Cdus,Ω .

(9.4.41)

and For the compositions of two or three operators, including T ◦ G, G ◦ T , H ◦T , T ◦H, and Δ◦G which have been well studied in the previous sections, we have similar estimates. Some of these we list as follows: T (G(u)),Ω ≤ Cus,Ω ,

(9.4.42)

G(T (u)),Ω ≤ Cus,Ω ,

(9.4.43)

H(T (u)),Ω ≤ Cus,Ω ,

(9.4.44)

T (H(u)),Ω ≤ Cus,Ω ,

(9.4.45)

Δ(G(u)),Ω ≤ Cus,Ω .

(9.4.46)

9.4.6 Weighted norm inequalities So far, we have developed various basic estimates for the Lipschitz and BM O norms of diﬀerential forms and the operators applied to these forms. In this section, we discuss the weighted Lipschitz and BM O norms. For ω ∈ L1loc (Ω, ∧l , wα ), l = 0, 1, . . . , n, we write ω ∈ locLipk (Ω, ∧l , wα ), 0 ≤ k ≤ 1, if ωlocLipk ,Ω,wα = sup (μ(Q))−(n+k)/n ω − ωQ 1,Q,wα < ∞

(9.4.47)

σQ⊂Ω

for some σ > 1, where the measure μ is deﬁned by dμ = w(x)α dx, w is a weight and α is a real number. For convenience, we shall write the simple notation locLipk (Ω, ∧l ) for locLipk (Ω, ∧l , wα ). Similarly, for ω ∈ L1loc (Ω, ∧l , wα ), l = 0, 1, . . . , n, we will write ω ∈ BMO(Ω, ∧l , wα ) if ω,Ω,wα = sup (μ(Q))−1 ω − ωQ 1,Q,wα < ∞ σQ⊂Ω

(9.4.48)

358

9 Lipschitz and BM O norms

for some σ > 1, where the measure μ is deﬁned by dμ = w(x)α dx, w is a weight, and α is a real number. Again, we shall write BMO(Ω, ∧l ) to replace BMO(Ω, ∧l , wα ) when it is clear that the integral is weighted. Theorem 9.4.17. Let u ∈ Ls (Ω, ∧l , μ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T be the homotopy operator deﬁned in (1.5.1), where the measure μ is deﬁned by dμ = wα dx and w ∈ Ar (Ω) for some r > 1 with w(x) ≥ ε > 0 for any x ∈ Ω. Then, there exists a constant C, independent of u, such that T (u)locLipk ,Ω,wα ≤ Cus,Ω,wα , (9.4.49) where k and α are constants with 0 ≤ k ≤ 1 and 0 < α ≤ 1. Proof. First, we note that

wα dx ≥

μ(B) = B

εα dx = C1 |B|,

(9.4.50)

B

which implies that C2 1 ≤ μ(B) |B|

(9.4.51)

for any ball B. From Theorem 7.5.2, we have T (u) − (T (u))B s,B,wα ≤ C3 |B|diam(B)us,σB,wα

(9.4.52)

for all balls B with σB ⊂ Ω, where σ > 1 is a constant. Using the H¨older inequality with 1 = 1/s + (s − 1)/s, we ﬁnd that T (u) − (T (u))B 1,B,wα = B |T (u) − (T (u))B |dμ ≤

B

|T (u) − (T (u))B |s dμ

1/s B

1s/(s−1) dμ

= (μ(B))(s−1)/s T (u) − (T (u))B s,B,wα

(s−1)/s (9.4.53)

= (μ(B))1−1/s T (u) − (T (u))B s,B,wα ≤ (μ(B))1−1/s (C3 |B|diam(B)us,σB,wα ) ≤ C4 (μ(B))1−1/s |B|1+1/n us,σB,wα . Now, using the deﬁnition of the weighted Lipschitz norm, (9.4.53), and (9.4.51), we obtain

9.4 Lipschitz and BM O norms

359

T (u)locLipk ,Ω,wα = supσB⊂Ω (μ(B))−(n+k)/n T (u) − (T (u))B 1,B,wα = supσB⊂Ω (μ(B))−1−k/n T (u) − (T (u))B 1,B,wα ≤ C5 supσB⊂Ω (μ(B))−1/s−k/n |B|1+1/n us,σB,wα ≤ C6 supσB⊂Ω |B|−1/s−k/n+1+1/n us,σB,wα

(9.4.54)

≤ C6 supσB⊂Ω |Ω|−1/s−k/n+1+1/n us,σB,wα ≤ C6 |Ω|−1/s−k/n+1+1/n supσB⊂Ω us,σB,wα ≤ C7 us,Ω,wα since −1/s − k/n + 1 + 1/n > 0 and |Ω| < ∞. Inequality (9.4.54) is the same as (9.4.49). As we have done in the previous section, we now develop the · ,Ω,wα norm estimate. Let u ∈ locLipk (Ω, ∧l ), l = 0, 1, . . . , n, 0 ≤ k ≤ 1, in a bounded domain Ω. From the deﬁnitions of the weighted Lipschitz and the weighted BM O norms, we have u,Ω,wα = supσB⊂Ω (μ(B))−1 u − uB 1,B,wα = supσB⊂Ω (μ(B))k/n (μ(B))−(n+k)/n u − uB 1,B,wα ≤ supσB⊂Ω (μ(Ω))k/n (μ(B))−(n+k)/n u − uB 1,B,wα ≤ (μ(Ω))

k/n

−(n+k)/n

supσB⊂Ω (μ(B))

(9.4.55)

u − uB 1,B,wα

≤ C1 supσB⊂Ω (μ(B))−(n+k)/n u − uB 1,B,wα ≤ C1 ulocLipk ,Ω,wα , where C1 is a positive constant. Thus, we have proved the following result. Theorem 9.4.18. Let u ∈ locLipk (Ω, ∧l , μ), l = 0, 1, . . . , n, 0 ≤ k ≤ 1, be any diﬀerential form in a bounded domain Ω, where w ∈ Ar (Ω) is a weight for some r > 1. Then, u ∈ BMO(Ω, ∧l , wα ) and u,Ω,wα ≤ CulocLipk ,Ω,wα , where C and α are constants with 0 < α ≤ 1.

(9.4.56)

360

9 Lipschitz and BM O norms

Theorem 9.4.19. Let u ∈ Ls (Ω, ∧l , μ), l = 1, 2, . . . , n, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded, convex domain Ω and T be the homotopy operator deﬁned in (1.5.1), where the measure μ is deﬁned by dμ = wα dx and w ∈ Ar (Ω) for some r > 1 with w(x) ≥ ε > 0 for any x ∈ Ω. Then, there exists a constant C, independent of u, such that T u,Ω,wα ≤ Cus,Ω,wα , (9.4.57) where α is a constant with 0 < α ≤ 1. Proof. Replacing u by T u in Theorem 9.4.18, we have T u,Ω,wα ≤ C1 T ulocLipk ,Ω,wα ,

(9.4.58)

where k is a constant with 0 ≤ k ≤ 1. Now, from Theorem 9.4.17, we ﬁnd that T (u)locLipk ,Ω,wα ≤ C2 us,Ω,wα . (9.4.59) Substituting (9.4.59) into (9.4.58), we obtain T u,Ω,wα ≤ C3 us,Ω,wα .

9.4.7 Estimates in averaging domains Next, we discuss the weighted norm inequalities for the solutions of the Aharmonic equation (1.2.2) in the L1 (μ)-averaging domain. From Deﬁnition 1.6.2, we know that a proper subdomain Ω ⊂Rn is called an L1 (μ)-averaging domain if there exists a constant C such that 1 1 |u − uB0 ,μ |dμ ≤ C sup |u − uB,μ |dμ (9.4.60) μ(B0 ) Ω μ(B) B 4B⊂Ω for some ball B0 ⊂ Ω and all u ∈ L1loc (Ω; μ). Here the supremum is taken over all balls B ⊂ Ω with 4B ⊂ Ω. The factor 4 here is for convenience and these domains are independent of this expansion factor, see [183]. Now, recall that in Lemmas 8.2.1 and 8.2.2, we have proved that for any positive constant a, the inequalities a ϕ( |u − uD,μ |)dμ ≤ ϕ(a|u|)dμ (9.4.61) 2 D D and

9.4 Lipschitz and BM O norms

361

ϕ(a|u|)dμ ≤ C

D

ϕ(2a|u − c|)dμ

(9.4.62)

D

hold for any strictly increasing convex function ϕ on [0, ∞) with ϕ(0) = 0 and a function u in the domain D ⊂ Rn with ϕ(|u|) ∈ L1 (D; μ), and μ({x ∈ D : |u − c| > 0}) > 0 for any constant c. The choice ϕ(t) = ts , s ≥ 1, in (9.4.61) and (9.4.62) is very useful in estimating the Ls -norms. Also, we stress the importance of the arbitrary of c on the right-hand side of (9.4.62). In particular, c may or may not be zero, and we do not need to concern ourselves with the diﬀerence between uB and uB,μ in the integrand. Theorem 9.4.20. Let u ∈ L1 (Ω, ∧0 , μ) be a solution of the A-harmonic equation (1.2.2), where the measure μ is deﬁned by dμ = wdx and w ∈ Ar (Ω) for some r > 1. Then, there exists a constant C, independent of u, such that u1,Ω,w ≤ Cu,Ω,w

(9.4.63)

for any bounded L1 (μ)-averaging domain Ω ⊂ Rn . Proof. Choosing ϕ(t) = t and c = uB0 in (9.4.62), where B0 is a ball in Ω, we have u1,Ω,w = Ω |u|wdx = Ω |u|dμ (9.4.64) ≤ C1 Ω |u − uB0 |dμ. Thus, it follows that 1 μ(B0 )

C1 |u|dμ ≤ μ(B 0) Ω

|u − uB0 |dμ.

(9.4.65)

Ω

From (9.4.60), (9.4.64), (9.4.65), and (9.4.48), we ﬁnd that μ(B0 )−1 u1,Ω,w C1 ≤ μ(B |u − uB0 |dμ Ω 0) 1 |u − u |dμ ≤ C2 sup4B⊂Ω μ(B) B B = C2 sup4B⊂Ω (μ(B))−1 u − uB 1,B,w ≤ C3 u,Ω,w ,

(9.4.66)

362

9 Lipschitz and BM O norms

which is equivalent to u1,Ω,w ≤ C3 μ(B0 )u,Ω,w ≤ C4 u,Ω,w . Note that if w ∈ Ar (Ω) and α is a constant with 0 < α ≤ 1, then wα is also an Ar (Ω) weight. Hence, we may replace w by wα in Theorem 9.4.20, to obtain the following version of weighted norm inequality: u1,Ω,wα ≤ Cu,Ω,wα .

(9.4.67)

Now, from inequality (9.4.67) and Theorem 9.4.18, we have the following norm comparison theorem for harmonic functions in an L1 (μ)-averaging domain. Theorem 9.4.21. Let u ∈ L1 (Ω, ∧0 , μ) ∩ locLipk (Ω, ∧0 , wα ), 0 ≤ k ≤ 1, be a solution of the A-harmonic equation (1.2.2), where the measure μ is deﬁned by dμ = wdx and w ∈ Ar (Ω) for some r > 1. Then, there exist constants C1 and C2 , independent of u, such that u1,Ω,wα ≤ C1 u,Ω,wα ≤ C2 ulocLipk ,Ω,wα

(9.4.68)

for any bounded L1 (μ)-averaging domain Ω ⊂ Rn . Similarly, as in Theorem 9.4.17, we have the following weighted norm comparison inequalities for Green’s operator G and the projection operator H. Theorem 9.4.22. Let u ∈ Ls (Ω, ∧l , μ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded domain Ω, G be Green’s operator, and H be the projection operator, where the measure μ is deﬁned by dμ = wα dx and w ∈ Ar (Ω) for some r > 1 with w(x) ≥ ε > 0 for any x ∈ Ω. Then, there exists a constant C, independent of u, such that G(u)locLipk ,Ω,wα ≤ Cdus,Ω,wα (9.4.69) and H(u)locLipk ,Ω,wα ≤ Cdus,Ω,wα ,

(9.4.70)

where k and α are constants with 0 ≤ k ≤ 1 and 0 < α ≤ 1. Applying Theorems 9.4.18 and 9.4.22, we obtain the following estimates for the BM O norms for Green’s operator G and the projection operator H, respectively.

9.4 Lipschitz and BM O norms

363

Theorem 9.4.23. Let u ∈ Ls (Ω, ∧l , μ), l = 0, 1, 2, . . . , n − 1, 1 < s < ∞, be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in a bounded domain Ω, G be Green’s operator, and H be the projection operator, where the measure μ is deﬁned by dμ = wα dx and w ∈ Ar (Ω) for some r > 1 with w(x) ≥ ε > 0 for any x ∈ Ω. Then, there exists a constant C, independent of u, such that G(u),Ω,wα ≤ Cdus,Ω,wα (9.4.71) and H(u),Ω,wα ≤ Cdus,Ω,wα ,

(9.4.72)

where α is a constant with 0 < α ≤ 1. Now, we make the following observation. For any diﬀerential form u, we have uB s,B ≤ Cus,B . Also, if c is a closed form, then c = cB . Thus, it follows that G(u) − (G(u))B ,Ω

= supσB⊂Ω |B|−1 G(u) − (G(u))B − (G(u) − (G(u))B )B 1,B

= supσB⊂Ω |B|−1 G(u) − (G(u))B − (G(u))B + (G(u))B 1,B

≤ supσB⊂Ω |B|−1 G(u) − (G(u))B 1,B

≤ supσB⊂Ω |B|−1 G(u) − c − ((G(u))B − cB )1,B

≤ supσB⊂B |B|−1 G(u) − c − (G(u) − c)B 1,B = G(u) − c,Ω . Therefore, the following result holds. Theorem 9.4.24. Let u ∈ BMO(Ω, ∧l , μ), l = 0, 1, 2, . . . , n, 0 ≤ k ≤ 1, be a smooth diﬀerential form and G be Green’s operator, where the measure μ is deﬁned by dμ = wdx and w ∈ Ar (Ω) for some r > 1. Then, G(u) − (G(u))B ,Ω ≤ G(u) − c,Ω

(9.4.73)

holds for any bounded domain Ω and any closed form c. It should be noticed that in Theorem 9.4.24 the diﬀerential form u need not be a solution of any version of the A-harmonic equation. Further, similar to the case of Green’s operator, similar estimates hold for the homotopy operator and the projection operator, and also for the compositions of these operators.

364

9 Lipschitz and BM O norms

Theorem 9.4.25. Let u ∈ BMO(Ω, ∧l , μ), l = 0, 1, 2, . . . , n, 0 ≤ k ≤ 1, be a smooth diﬀerential form, T be the homotopy operator, and H be the projection operator, where the measure μ is deﬁned by dμ = wdx and w ∈ Ar (Ω) for some r > 1. Then, H(u) − (H(u))B ,Ω ≤ H(u) − c,Ω

(9.4.74)

T (u) − (T (u))B ,Ω ≤ T (u) − c,Ω

(9.4.75)

and hold for any bounded, convex domain Ω and any closed form c. We have presented the BM O norm estimates above. By the same method, we can develop a similar estimate for the Lipschitz norm as follows: G(u) − (G(u))B locLipk ,Ω

= supσB⊂Ω |B|−(n+k)/n G(u) − (G(u))B − (G(u) − (G(u))B )B 1,B

= supσB⊂Ω |B|−(n+k)/n G(u) − (G(u))B − (G(u))B + (G(u))B 1,B

≤ supσB⊂Ω |B|−(n+k)/n G(u) − (G(u))B 1,B

≤ supσB⊂Ω |B|−(n+k)/n G(u) − c − ((G(u))B − cB )1,B

≤ supσB⊂B |B|−(n+k)/n G(u) − c − (G(u) − c)B 1,B = G(u) − clocLipk ,Ω . Hence, we have established the following Lipschitz norm inequality for smooth diﬀerential forms. Theorem 9.4.26. Let u ∈ locLipk (Ω, ∧l , μ), l = 0, 1, 2, . . . , n, 0 ≤ k ≤ 1, be a smooth diﬀerential form and G be Green’s operator, where the measure μ is deﬁned by dμ = wdx and w ∈ Ar (Ω) for some r > 1. Then, G(u) − (G(u))B locLipk ,Ω ≤ G(u) − clocLipk ,Ω

(9.4.76)

holds for any bounded domain Ω and any closed form c. Now we state a result which is analogous to Theorem 9.4.25. Theorem 9.4.27. Let u ∈ locLipk (Ω, ∧l , μ), l = 1, 2, . . . , n, 0 ≤ k ≤ 1, be a smooth diﬀerential form, T be the homotopy operator, and H be the projection operator, where the measure μ is deﬁned by dμ = wdx and w(x) ∈ Ar (Ω) for

9.4 Lipschitz and BM O norms

365

some r > 1. Then, H(u) − (H(u))B locLipk ,Ω ≤ H(u) − clocLipk ,Ω

(9.4.77)

T (u) − (T (u))B locLipk ,Ω ≤ T (u) − clocLipk ,Ω

(9.4.78)

and hold for any bounded, convex domain Ω and any closed form c. If we choose μ to be the Lebesgue measure in the deﬁnition of the Ls (μ)averaging domain, we have

1 |B0 |

1/s

|u − uB0 |s dx

≤ C sup

4B⊂Ω

Ω

1 |B|

1/s

|u − uB |s dx

, (9.4.79)

B

which can be considered as the deﬁnition of the Ls -averaging domain. We also note that when ϕ(t) = ts and c = uB0 in (9.4.62), we obtain |u|s dx ≤ C |u − uB0 |s dx. (9.4.80) Ω

Ω

We are now ready to prove the following inequality in the Ls -averaging domain. Theorem 9.4.28. Let u ∈ BMO(Ω, ∧0 ) be a solution of the nonhomogeneous A-harmonic equation (1.2.10) in an Ls -averaging domain Ω, 1 ≤ s < ∞. Then, there exists a constant C, independent of u, such that us,Ω ≤ Cu,Ω .

(9.4.81)

Proof. From the weak reverse H¨older inequality, we have u − uB s,B ≤ C3 |B|(1−s)/s u − uB 1,σB

(9.4.82)

for any ball B and some σ > 1. Now, applying (9.4.79) and (9.4.80), we ﬁnd that

1/s us,Ω = Ω |u|s dx ≤ C1 |B0 |1/s

1 |B0 |

≤ C2 supσB⊂Ω

Ω

1 |B|

|u − uB0 |s dx

|u − uB | dx s

B

1/s 1/s

≤ C2 supσB⊂Ω |B|−1/s u − uB s,B .

(9.4.83)

366

9 Lipschitz and BM O norms

Finally, substituting (9.4.82) into (9.4.83), we obtain us,Ω ≤ C2 supσB⊂Ω |B|−1/s u − uB s,B ≤ C2 supσB⊂Ω |B|−1/s (C3 |B|(1−s)/s u − uB 1,σB ) (9.4.84)

≤ C4 supσB⊂Ω |B|−1 u − uB 1,σB ≤ C5 u,Ω .

9.4.8 Applications Let u be an (n−k)×(n−k) subdeterminant of Jacobian J(x, f ) of a mapping f : Ω → Rn , which is obtained by deleting the k rows and k columns, k = 0, 1, . . . , n − 1, say, u = J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k ) i1 fxi1j2 fxi1j3 · · · fxi1j fxj1 n−k i2 i2 i2 i2 fxj2 fxj3 · · · fxj fxj1 n−k = . .. .. .. , .. .. . . . . f in−k f in−k f in−k · · · f in−k xj1

xj2

xj3

xjn−k

where {i1 , i2 , . . . , in−k } ⊂ {1, 2, . . . , n} and {j1 , j2 , . . . , jn−k } ⊂ {1, 2, . . . , n}. Also, it is easy to see that J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k )dxj1 ∧dxj2 ∧ · · · ∧ dxjn−k is an (n − k)-form. Thus, all estimates for diﬀerential forms are applicable to the (n − k)-form J(xj1 , xj2 , . . . , xjn−k ; f i1 , f i2 , . . . , f in−k )dxj1 ∧ dxj2 ∧ · · · ∧ dxjn−k . For example, choosing u = J(x, f )dx and applying Theorems 9.4.14 and 9.4.18 to u, respectively, we have the following theorems. Theorem 9.4.29. Let J(x, f )dx ∈ locLipk (Ω, ∧n ), 0 ≤ k ≤ 1, where J(x, f ) is the Jacobian of the mapping f = (f 1 , . . . , f n ) : Ω → Rn and Ω is a bounded domain in Rn . Then, J(x, f )dx ∈ BMO(Ω, ∧n ) and J(x, f ),Ω ≤ CJ(x, f )locLipk ,Ω , where C is a constant.

(9.4.85)

9.4 Lipschitz and BM O norms

367

Theorem 9.4.30. Let J(x, f )dx ∈ locLipk (Ω, ∧n , μ), 0 ≤ k ≤ 1, where J(x, f ) is the Jacobian of the mapping f = (f 1 , . . . , f n ) : Ω → Rn , w ∈ Ar (Ω) is a weight for some r > 1, and Ω is a bounded domain in Rn . Then, J(x, f )dx ∈ BMO(Ω, ∧n , wα ) and J(x, f ),Ω,wα ≤ CJ(x, f )locLipk ,Ω,wα ,

(9.4.86)

where C and α are constants with 0 < α ≤ 1. Notes to Chapter 9. Considering the size of the monograph, we will not discuss the norm comparison theorems for the compositions of three operators here. For recent results on BM O spaces, Lipschitz classes and domains, and related domains, see [315–324, 137–143, 154, 167–169, 309]. We also encourage readers to see [325–333] for diﬀerent versions of the Poincar´e inequality.

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Index

Ar (Ω), 21

k(x, y) = k(x, y; Ω), 38

A∞ (Ω), 22

L(X, Y ), 5

Aloc ∞ (Ω), 22

L(X, R), 5

Aλ r (E), 25–26

L(Y, R), 5

Ar (λ, E), 23

Lp (log L)α (Ω), 48

Ar,λ (E), 27

Lϕ (Ω), 48 Lp (Ω, ∧l ), 2

A(x, du) = d v, 11 A(x, g + du) = h + BM O(Ω, ∧l ),

d v,

12

locLipk (Ω, ∧l ), 340

|Df (x)|, 9

IM, 225 IMs , 225 IMs , 225 IMs,α , 316

div(∇u|∇u|p−2 ) = 0, 8

W 1,p (Ω, ∧l ), 3

div A(x, ∇u) = 0, 8

1,p (Ω, ∧l ), 3 Wloc

340–341

BM O(Ω, μ), 340 Df (x), 9–10

du|du|p−2

=

d v,

12, 16

RHβ (Ω), 22

D (Ω, ∧l ), 2

uB,μ , 38

d(d v|d v|q−2 ) = 0, 12

W D(Ω), 21

d , 3

W RHβ (Ω), 21

d A(x, dω) = 0, 10

T , 30

d A(x, dω) = B(x, dω), 12

T , 5

d (du|du|p−2 ) = 0, 12

X, 5

F (ω),

∧, 2

6

G, 87

∧(V ), 6

GL(n), 9

∧l , 2

H, 87

∧l Ω, 86

H 1,p (Ω; μ), 119

∧l (Rn ), 2

H01,p (Ω; μ), 1,p (Ω; μ), Hloc

[f ]Lp (log L)α (E) , 48

J(x, f ), 9

119 120

f p,E,w , 2 f Lp (log L)α (E) , 48 385

386

Index

f ϕ , 48

Ls (μ)-averaging domains, 38–41

f p,E,w , 30

Lϕ -domains, 93

· W 1,p (Ω,∧l ) , 3

Exact forms, 7

· W 1,p (Ω,∧l ),w , 3

Exterior derivative, 3

· locLipk ,Ω , 340

Exterior diﬀerential, 3

· locLipk ,p,Ω,w , 352

Finite distortion, 9

· locLipk ,Ω,wα , 357

Green’s operator, 86

· ,Ω , 341

Hardy-Littlewood inequalities

· ,p,Ω,w , 353

in the unit disk, 29

· ,Ω,wα , 357

in John domains, 32

n l ⊕n l=0 ∧ (R ), 2

in Ls (μ)-averaging domains, 37

A-harmonic equation

with two-weight, 43

for functions, 8 for diﬀerential forms, 10

with · Ls (log L)α , 52 Harmonic l-ﬁelds, 87

A-harmonic tensors, 10–11

Hodge star operator, 3

Algebra

Hodge codiﬀerential operator, 3

exterior, 6

Homogeneous A-harmonic equation, 11

Grassmann, 6

Homotopy operator, 30

Beltrami system, 9 BMO spaces, 339 BMO norms, 339 Caccioppoli inequalities

H¨ older’s inequalities with Lp -norms, 7 with · Lp (log L)α -norm, 327 John-Nirenberg Lemma, 329

without weights, 115

K-quasiconformal mappings, 9–10

with Ar (Ω)-weights, 120–122

K-quasiregular mappings, 9

with Ar (λ, Ω)-weights, 123–125

Laplace Beltrami operator, 87

with Aλ r (Ω)-weights, 125–126

Lipschitz spaces, 340–341

with two-weights, 127, 129–133

Lipschitz norms, 340–341

with Orlicz norms, 133

Luxemburg functionals, 47

Carnot group, 56

Luxemburg norms, 48

Closed form, 11

Maximal operators

Coclosed form, 11

fractional, 316

Conformal distortion tensors, 10

Hardy-Litlewood, 225

Conformal structure, 10

sharp, 225

Conjugate A-harmonic equation, 11 Conjugate A-harmonic tensors, 11 Conjugate p-harmonic equation, 12

Nonhomogeneous A-harmonic equations for functions, 12 for diﬀerential forms, 12

Conjugate p-harmonic tensors, 12

Orientation preserving, 9

Covering Lemma, 32

Orientation reversing, 9

Diﬀerential k-forms, 3

Orlicz functions, 47

Distortion equation, 9

Orlicz norms, 47

Domains

Orlicz space, 47

John domains, 29

Orthogonal complement, 87

Ls -averaging domains, 37

Outer distortion functions, 9

Index

387

P -harmonic equation, for functions, 14

Solutions of the homogeneous A-harmonic

for diﬀerential forms, 14 P -harmonic functions, 14 P -harmonic tensor, 14 Poincar´ e inequalities for diﬀerential forms, 72–73 for Green’s operator, 86–88

equation, 11 of the nonhomogeneous A-harmonic equation, 12 Two-weights, 27–29 Weak reverse H¨ older inequalities for A-harmonic tensors, 78

for harmonic forms, 78–83

for Ar -weights, 24

for the homotopy operator, 275–281

for du, 200

for Jacobians, 107–108

for supersolutions, 188

for the projection operator, 111

in John domains, 195

for Sobolev functions, 75

in Ls (μ)-averaging domains, 195

in

Ls -averaging

domains, 83

with Ar (Ω)-weights, 189

in Ls (μ, 0)-averaging domains, 83

Aα r (Ω)-weights, 201–204

in Lϕ -domains, 93

with Aλ r (Ω)-weights, 205–208

with Ar (E)-weights, 77

with Ar (λ, Ω)-weights, 209

with Aλ r (E)-weights, 84–86

with two-weights, 214–219

with Orlicz norms, 99 with two-weight, 100–107

with Orlicz norms, 219–220 Weighted BMO norms, 359

Poincar´ e lemma, 6

Weighted Lp norms, 2

Poisson’s equation, 87

Weighted Lipschitz norms, 358

Projection operator, 87

Weights

Pullback, 5

Ar (Ω)-weights, 21

Q-harmonic equation, 14

Aλ r (E)-weights, 25–26

Q-harmonic tensor, 14

Ar (λ, E)-weights, 23–25

Quasihyperbolic distance, 38

Ar,λ (E)-weights, 27

Riemannian metric, 10

doubling weights, 21

Reverse H¨ older inequalities, 21

Muckenhoupt, 21

Sobolev space of l-forms, 3

Young functions, 48