B Opic and A Kufner Czechoslovak Academy of Sciences
Hardy-type inequalities
Longman NNW
Scientific &
- Technical
Copublished in the United States with John Wilev & Sons. Inc.. New York
Longman Scientific & Technical, Longman Group UK Limited, Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world. Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158
© Longman Group UK Limited 1990 All rights reserved: no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1990 AMS Subject Classification: 26D10, 46E35 ISSN 0269-3674
British Library Cataloguing in Publication Data Kufner, Alois, 1934Hardy-type inequalities 1. Mathematics. differential inequalities 1. Title II. Opic, B. 515.3'6
ISBN 0-582-05198-3 Library of Congress Cataloging-in-Publication Data Kufner, Alois. Hardy-type inequalities / A. Kufner and B. Opic. p. cm.-- (Pitman research notes in mathematics series, ISSN 0269-3674 ; 219) ISBN 0-470-21584-4 (Wiley) 1. Inequalities (Mathematics) I. Opic, B. II. Title. III. Series. QA295.K87 1990 512.9'--dc2O
89-14502
CIP
Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
Contents
Introduction Chapter 1.
1
The one-dimensional Hardy inequality
5
1. Formulation of the problem
5
2. Historical remarks
14
3. Proofs of Theorems 1.14 and 1.15
21
4. The method of differential equations
35
5. The limit values of the exponents
p
45
q
,
6. Functions vanishing at the right endpoint. Examples 7. Compactness of the operators
HL
HR
and
8. The Hardy inequality for functions from 9. The Hardy inequality for
0 < q <
ACLR(a,b)
65 73
92 129
1
10. Higher order derivatives
142
11. Some remarks
161
Chapter 2.
The N-dimensional Hardy inequality
170
12. Introduction
170
13. Some elementary methods
186
14. The approach via differential equations and formulas
204
15. The Hardy inequality and the class
A
226 r
235
16. Some special results Chapter 3.
Imbedding theorems for weighted Sobolev spaces
17. Some general necessary and sufficient conditions 18. Imbeddings for the case
1 < p <_ q <
243 243
249
19. Power type weights
269
20. Unbounded domains
287
21. The N-dimensional Hardy inequality
304
Appendix
22. Level intervals and level functions References
315
315 327
Preface
Inequalities involving integrals of a function and its derivatives appear frequently in various branches of mathematics and represent a useful tool e.g. in the theory and practice of differential equations, in the theory of approximation etc. During the last decades, this topic has been further extended, comprising additionally general measures and in particular measures generated by weight functions, i.e. functions measurable and almost everywhere positive. The authors decided to collect some of the numerous results obtained in this 'weighted' direction and to present here an - at least partial - survey. The one-dimensional case, treated in Chapter 1, is more or less closed, while the more-dimensional case, dealt with in Chapters 2 and 3, is as yet solved only partially. Nevertheless, the authors hope that the reader will find here some interesting and/or useful information. In any case, they would
welcome any remarks and comments concerning the content
as well as the exposition.
The text is organized in a usual manner. Each of the 22 sections has its own numeration (both for the formulas and for the subsections). The end of a proof is marked by the sign
D
.
We use the occasion for expressing our gratitude to all who supported us and helped us to bring the manuscript to its final form. Mainly, we want to mention Dr. Petr GURKA who not only contributed his own results but also improved the text by offering valuable comments; Ms. Hana VALACHOVA who read the manuscipt; Dr. Jifi JARNIK who substantially improved the authors' English;
Ms. Ruiena PACHTOVA who carefully typed the text.
Prague, November 1989
B. 0.
A. K.
List of symbols
(In the first column, the number indicates the page where the symbol occurs)
N
the set of natural numbers
R
the set of real numbers
R+
the set of positive numbers
Z
the set of integers
(a,b)
an open interval in R
[a,b1
a closed interval in R
(a,b, [a,b)
}
half-open intervals in R
RN
the N-dimensional Euclidean space
SI
an open set (domain) in RN
BSI
the boundary of
the closure of x = (x1,x,,...,xN) SI = f J BSI
Q
9
1
a point in
T
RN
186
x = (xi,xi)
186
xi = (xl" .. 'xi-11xi+1'
187
Pi(S2)
the projection of
187
Ci(Sc;xi)
a cut of
mN(E)
the N-dimensional Lebesgue measure of the set
'N-1(E) XE = XE(x)
an t a bn y b X 73
X
un - u
73
j
73
C,
a point in
...,xN) SI
RN-1
onto the hyperplane
the characteristic function of the set
the monotone convergence
a Banach space and its dual space the weak convergence (in
X )
the symbol of the continuous imbedding
y
E
the (N-1)-dimensional Lebesgue measure of the set
1 J}
xi = 0
SI
the symbol of the compact imbedding
E
E
Special sets
206
C0,1
269
CO,K
224
224
ClO'l im(x0)
COim(x0,°°) types of domains in
s pecial
288 289
RN
ZJ
"b0,1
308
308
0,1 {y E RN;
< r}
220
B(x,r)
t he ball
233
S(x,r)
t he sphere
231
S(0,1)
t he unit s phere
226
Q(y,r)
t he cube
170
supp u
t he suppor t of the function
219
aC+(x0)
219
aG-(x0)
ly - xl
{y ` RN; ly - xl = r}
{x
{y C- RN;
lyl =
}
RN; lxi - yil < r, i = 1,2,...,N} u
Special classes of functions and function spaces
5
AC(I) = AC(a,b)
5
ACL(1) = ACL(a,b)
5
ACR(1)
6
ACLR(I) = ACLR(a,b)
classes of absolutely continuous functions
14 2
AC R(a b)
AC(k-1)(a,b) m,n
on
1 - (a,
) %,- R
187
ACi(52)
187
ACi,L(Q)
188
ACi,R('C)
196
ACi,LR(Q)
89
classes of absolutely continuous functions on
CO(a,b)
170
C1(1)
170
CO(R)
special 170
classes of continuous functions
CM(Q) Cm
210
M(h)
218
C1 (Q)
46
Lp(a,b)
the Lebesgue space
46
lullp,(a,b)
its norm
89
Lloc(a,b)
45
Lp(a,b;w)
the weighted Lebesgue space
45
hull
its norm
p,(a,b),w
206
the Lebesgue space
206
its norm
227 227
228 228
8
Lp(S2;w) lullpPS2,w
LP,*
(S2;w)
hull*,Q,w
M+(I) = M+(a,b)
the weighted Lebesgue space its norm
the weak weighted Lebesgue space its quasinorm
Q
CRN
8
W(I) = W(a,b)
classes of weight functions 226
Ar = AR(RN )
226
A(S2)
238
A.(RN)
240
S = {vn,vl,...,v j
89
W1,P(a,b;v0,v1)
89 91
a collection of weights
weighted Sobolev spaces and the corresponding WL'P(a,b;v0,v1)
89 11u
norm
lll,p,(a,b),v0'"1
206
W1'P(2)
the Sobolev space
206
jlul,1
its norm
PSZ ,
240
W1'P(St;S)
240
240 241 241
lllullll,p,O,S
weighted Sobolev spaces and the corresponding
W1'P(S2;v0,v1)
norms (seminorms)
241
241
llulll,p,Q,"O,vl
241
llluIII 1,p,S2,v1
Operators
6
HL
6
HR
76
HL
76
HR
213
div g
205
Vu = grad u
211
IQul
233
IDulp
205
Au
Special constants and functions
p' = 1
if if
pf _ °°
if p=1
p
t
p
_ p
289
AL = AL(a,b,w,v,q,p)
66
AR = AR(a,b,w,v,q,p)
84
AL = AL(a,b,w,v,q,P) Jr't
p=w
= .A (a,b,w,v,q,p)
12
BL = BL(a,b,w,v,q,P)
65
BR = BR(a,b,w,,t,q,P)
109
< p < = or
a*
12
119
1
Q =
J3 (a,b,w,v,q,p)
12
FL(x) = FL(x;a,b,w,v,q,P)
65
FR(x) = FR(x;a,b,w,v,q,p)
13
k(p,q) _
1/q
(1 + P
186
Di = Di(S2)
186
Si = Si(Z)
1/p'
(1 + P
0
1
Introduction
These lecture notes are devoted to the inequality
111/q 1u(x)Iq w(x) dxJ
1j
N
J
i=1
1/p
p
(x)
'13u
< CI
v.(x) dx
axi
I
2 S2 is a domain in and to some of its modifications and consequences. Here N are positive real numand q p the N-dimensional Euclidean space R q > 0 ) and w,vl,...,vN are bers (in fact, we will consider p ? 1 ,
,
weight functions, i.e.
measurable and positive a.e. in
S2
.
We are concerned with the question what conditions on the data of our problem
-
on the domain
i.e.
w,v1,...,vN
weight functions
(0.1) for all functions
u
12
-
,
on the parameters
p
,
q
and on the
guarantee validity of the inequality K ,
from a certain class
K D CO(Q) C > 0
with a constant
independent of the function
estimates for the best possible constant
C
u
.
In some cases,
in (0.1) will be given.
The inequality (0.1) will be called here
Terminological remarks. (i)
the (N-dimensional) Hardy inequality, the reason being the following: In 1920, G. H. HARDY proved an inequality (see Section 1) which can be easily rewritten in the form 1l1/P
l11/P
u'(t)lp to dtJ
lu(t)1p to-p dt]
(0.2)
p + 11
0 J' J
0
wher e
u' = du/dt
.
We shall call (0.2) the (classical) Hardy inequality;
it holds, e.g., for all functions
p>
1
and
u E CO(0,-)
provided
e x p - 1.
The inequality (0.2) is a special case of the general (one-dimensional) Hardy inequality 1
bll1/q IIbl/p
[JIu(t)j
(0.3)
CIJu'(tp v(t) dt1
w(t) dtl J
lllla
1J
a
- - = a < b < -
where
and
w(t)
,
are weight functions.
v(t)
So, we obtain (0.2) from (0.3) taking p = q
>
1
a=0
,
b=0,
,
w(t) =
tE-p
v(t) = tE
,
.
On the other hand, the inequality (0.3) is again a special case of the inequality (0.1) for the case
N=
1
S2 = (a,b)
,
It can be said that the conditions of validity of the inequality (0.3) are investigated (almost) completely; we will deal with them in detail in Chapter 1.
In the literature, the following inequality has been intensively
(ii)
investigated:
CIfIVU(x)Ip dx]
_
S2
S2
where
l1 /p
((
ll1/q
{Jju(x)I q dx]
(0.4)
(Du
x = (x1,x2,...,xN)
Vu =
,
N
,
1
au
p
x
.
ax
and
8x
IDulp =
2
This inequality is known, e.g., as the SoboZev inequality and
i=1 axi
holds (e.g.) for 1
provided
u E C0(0)
,
1
and
< q < Np N-p
is a bounded domain with Lipschitzian boundary
52
'worse' domains
52
,
the admissible values of the parameter
(see, e.g., R. A. ADAMS [
1
32 ; for q
may change
1).
The inequality (0.4) is a special case of the inequality (0.1) if we take
w(x) = vi(x) =
1
,
i = 1,2,...,N
a special case of the inequality (0.4) (for equality
S2
2
JVu()I2 dx
Jdx < C2
(0.5)
S2
;
p = q = 2
)
is then the in-
which holds (e.g.) for all functions
u E
C0OO(0)
Friedrichs inequality) or for all functions is zero: J0 u(x) dx = 0
u
(then it is called the
whose mean value over
0
(then it is called the Poincare inequality). There-
fore, the inequality (0.1) could be called also the weighted Friedrichs-
ppincare inequality, and it appears under this name sometimes in literature. In connection with the inequality (0.4), also the name weighted SoboZev
inequality is used for the inequality (0.1).
However, we will show in Chapter 3 that the inequality (0.1) is closely connected with the properties of the so-called weighted Sobolev spaces, but nevertheless we will use the name Hardy inequality for the inequality (0.1) as well as for the inequality (0.3) and all inequalities appearing here can
be collected under the common name Hardy-type inequalities.
Chapter 1. The one-dimensional Hardy inequality 1. FORMULATION OF THE PROBLEM
1.1. 1
In the twenties, G. H. HARDY proved the following inequality: Let and denote
< p < -
t
f (x) dx
for
e< p -
1
(x) dx
for
e> p -
1
J
0
F(t) =
(1.1)
r J
t where
f
f
is a non-negative measurable function defined on
(1.2)
Ir Fp(t) to-P dt
C
J
0
(O,=)
.
Then
fp(t) tE dt
0
with a constant
C > 0
independent of
f
.
For a proof see, e.g., G. H. HARDY [1] or G. H. HARDY, J. E. LITTLEWOOD, G. POLYA [1]. The exact value of the constant
was given in 1926
C
by E. LANDAU [1]: it is P
1e-p+1I lp 1.2. Definition.
Let
-
I = (a,b)
a < b < +
,
and denote by
AC(I)
the set of all functions absolutely continuous on every compact subinterval [c,d] C I Further, denote by .
AC
(1)
L
and
AC R(I)
the sets of all functions
u E AC(I)
for which
5
lim
(1.4)
u(x) - 0
x+a+ and
u(x) = 0
lim
(1.5)
x+brespectively. (So, the indices tion
and
L
express the fact that the func-
R
vanishes on the Left and right end of the interval
u
I
,
respect-
ively.) Finally, denote by ACLR(I)
ACL(I) flACR(I)
the intersection
.
If it is necessary to point out the concrete form of the interval I - (a,b)
, we will use the notation AC(a,b)
ACL(a,b)
,
,
ACR(a,b)
,
ACLR(a,b)
Further, let us introduce the notation xr
(HLf)(x) =
f(t)
dt
f(t)
dt
J
a
b (HRf)(W )
= J
x
Using the operator
(1.7)
I
HL
(HLf)p(x)
we can rewrite (1.2) in the form
,
xe-p dx s C
0
for
1
fp(x) xe dx
0
e < p - 1
e > p -
J
,
and similarly with the help of the operator
HR
for
.
From the inequality (1.2) we obtain the Hardy inequality (0.2) as an easy corollary:
1.3. Lemma. (1.8)
Or 6
Let
1
c< p- 1
< p <
and
,
e x p -
u E ACL(O,m)
1
,
and suppose
E > p -
(1.9)
and
1
u
AC R(0,00)
.
Then
JIu(x)P X'-p dx
(1.10)
0
C I1u'(x)jp xE dx
<_
0
with the constant proof.
from (1.3).
C
Assume that (1.8) is fulfilled and denote
(i)
J = Ilu'(x)IP xE dx 0
If
J = cc
then the inequality (1.10) holds trivially. Therefore, let us
,
assume that the integral
J
is finite. Then for
x E (0,a)
we have, by
H81der's inequality, that
x
x
flu' (t) l
dt = JIu'(t) l
0
t-6/p
dt
0
lflui(t)Ip te
0
<
tE1
l0
J
(due to (1.8), we have
1
< J
x(P-1-E)/(P-1)(P-1)/P
1/P(--p-_1 P -
dt](P-1)/p
dt}1/P (f t-E/(P-1)
- E
J
c < p -
1
,
i.e.
- E/(p-1) > -
1
). Consequently,
x
Jlu,(t)l dt < -
for every
x > 0
.
0
Further x(
u(x) = J u'(t) dt + u(c)
for
c > 0
C
since
u c- AC(0,-)
.
Moreover,
u E ACL(0,-)
,
and therefore, we obtain for
e -` 0+ that
7
u(x) = J
u'(t) dt
0
Finally,
J
Jiu'ti dt = (HLlu'I)(x)
u'(t) dt
0
0
and (1.10) follows from (1.7) for (ii)
f = lu'I.
The case (1.9) can be handled analogously. 11
1.4. Definition.
denote by
W(a,b)
or
W(I)
I = (a,b)
For
the set of all weight functions on
I
,
i.e.
the set of all functions
measurable, positive and finite almost everywhere (a.e.) on
I
.
Further, denote by M+(I)
M+(a,b)
or
the set of all measurable functionsnon-negative a.e. on
1.5. Problem.
Let
1
< p,q < -
is there a (finite) constant
C > 0
v, w C- W(a,b)
Under what conditions
.
such that the inequality
111/q
b((
(1.11)
Let
.
I
l/p
((b
CIIIu'(x)lp v(x) dx
IJIu(x)Iq w(x) dxl
a
a
holds (i)
for every
u E ACL(a,b)
(ii)
for every
u E ACR(a,b) ?
1.6. Example and remark.
the inequality (1.11) for to
.
(i)
,
or
The inequality (1.10) is a special case of
p = q
,
a = 0
,
b = -
,
w(t) = tE P
,
v(t)
Consequently, Problem 1.5 is solved in this special case by Lemma
1.3. (ii)
8
Analogously as in this lemma, Problem 1.5 can be reduced to a
HL
problem concerning inequalities involving the operators
and
HR
from
(1.6). Let us now formulate this second problem.
Let
7. Problem.
1
< p,q <
are there (finite) constants
CL
,
Under what conditions
W(a,b) .
v, w
Let
such that
CR
the inequality
(i)
b
[j(Hf)(X) w(x) dxlll1/q < CL
(1.12)
rb
l1/p fp(x) v(x) dxl
a
JJJ
11
a
holds for every
f e M+(a,b)
the inequality
(ii)
(b 11
f(
1/p
< CR[J fp(x) v(x) dx
IJ(HRf)q(x) w(x) dxJ
(1.13)
br
1/q
I
a
a holds for every
f E M+(a,b)
?
1.8. Remark. Each of the problems mentioned, 1.5 as well as 1.7, represents
we consider the inequality
in fact a pair of problems: In Problem 1.5 (1.11) on two different classes of functions two
different
u
,
in Problem 1.7 we consider
Nonetheless, using elementary tools, we can
operators.
reduce Problem 1.5 (ii) to Problem 1.5 (i) and similarly
the investigation
of the inequality (1.13) can be reduced to the investigation of the inequality (1.12). Indeed, the substitution x = - y yields
b
,
bf
J[J ax
t = - s
lq
q
(t) dt] w(x) dx = f[J i(s) ds] w(y) dy
aa
and
b
B
r
( J
fp(x) v(x) dx =
fp(y) v(y) dy J
a Where
a
(a,6) = (-b,-a)
Y E (a,B)
.
Obviously
,
f(y) = f(-Y)
T E M+(a,6)
an analogue of (1.12) on
(a, B)
,
w(y) = w(-y)
,
v, w G W(a,B)
for
in Problem 1.5, since the substitution
f
,
v
, w
x = - y
.
,
v(y) = v(-y)
for
and (1.13) reduces to
A similar situation occurs transforms
u E ACL(a,b) 9
into
ACR(a,s)
u
u(y) = u(-y)
,
inequality on
ing
for
(a,s)
u
and reduces (1.11) to the correspond-
,
,
v
w
,
.
1.9. Convention. In accordance with the previous remark, we restrict ourselves in the sequel to the investigation of the inequality (1.11) only for u e ACL(a,b)
and to the investigation of the inequality (1.12) (i.e.
,
the operator
two inequalities are v
-
f E M+(a,b)
HL ) for
of
The following lemma states that these
.
under certain conditions on the weight function
-
equivalent, which means that Problems 1.5 and 1.7 are in some sense
equivalent, too.
1.10. Lemma. Let
< p,q < -
1
.
Let
v, w E W(a,b)
and assume
x
vl-p'(t)
J
(1.14)
dt <
a
for every
x E (a,b)
for every every
u E ACL(a,b)
f e M+(a,b)
best constant Proof.
with
CL
.
p' =
Then the inequality (1.11) holds
.
p P
1
if and only if the inequality (1.12) holds for
The best constant
C
in (1.11) coincides with the
in (1.12).
Assume that the inequality (1.12) holds and denote, for
(i)
uEACL(a,b)
, brr
J =
Jlu'(x)lp v(x) dx a
If
that
J = J
then the inequality (1.11) holds trivially. Therefore, assume
,
is finite. Then we have by Holder's inequality that for
x
x
Ju'(t)l dt = JIu'(t) lv11P(t) v-11P(t) dt < a a
xl1/P
[Jiu'tiP a x(
<
J 1/P
J
a
10
v(t) dt1
vJ ,(t)
1'P
dt
x i
v' -p (t) dt I [I
a
1/p I <
1/p' <
x E (a,b)
in view of (1.14). Since
u E AC(a,b)
,
we have
x u(x) =
u'(t) dt + u(c)
for every
c G (a,b)
J
c
u E ACL(a,b)
Z(oreover,
c -p a+
and, therefore, for
we obtain
x(
u(x) = J u' (t) dt a
Further,
xJiu't Iu(x)I <
dt = (HLlu'I)(x) )
I
a
and (1.11) follows from (1.12) for shown that the best constants C < CL
(1.15)
C
f = lu'I
.
in (1.11) and
Simultaneously, we have CL
in (1.12) satisfy
.
Assume that the inequality (1.11) holds for
(ii)
u E ACL(a,b)
.
Let
and denote
f E & (a,b)
b(
J=
J
fp(x) v(x) dx
a If
ttt
J = m , then the inequality (1.12) holds trivially. Therefore, assume J
is finite. Similarly as in part (i), Holder's inequality yields x I
IJ
a fo every
1 gyp'
(x
f(t) dt 5 J1/p
vl-p'(t) dt 1
a
x E (a,b)
.
Then the function
x(
u(x) =
J
f(t) dt = (H
LOW
a
obviously belongs to ACL(a,b) The inequality (1.11) applied to this function u yields immediately the inequality (1.12). Simultaneously, we have .
shown that the best constants
C
in (1.11) and
CL
in (1.12) satisfy 11
CL < C
,
which together with (1.15) completes the proof.
1.11. Remark.
(i)
11
Analogously, it can be shown that under the assumption
b(
(1.16)
J
vl p/(t) dt < -
x C (a,b)
for every
x
the inequality (1.11) holds for every inequality (1.13) holds for every (ii)
if and only if the
u c ACR(a,b)
f E
(and that
C = CR ).
In the proof of Lemma 1.10, the assumption (1.14) was essential.
Nevertheless, it can be shown that Problems 1.5 and 1.7 are equivalent also without the condition (1.14) and (1.16), respectively. (See Remark 3.7.)
Our aim is to establish necessary and sufficient conditions on
1.12. q
,
v
,
w
p
,
under which the Hardy inequality (1.11) holds. The corresponding
assertions will be formulated for the inequality (1.11), but of Lemma 1.10 and Remark 1.11
-
-
in view
we will proceed via Problem 1.7. More-
over, according to Convention 1.9 it suffices to deal with the inequality (1.12). The corresponding conditions concerning the inequality (1.13) the inequality (1.11) for
i.e.
u E ACR(a,b)
-
-
will be summarized in
Section 6.
First, we introduce some important auxiliary functions and constants.
1.13. Notation.
For
1
< p,q <
,
v, w C W(a,b) b
(1.17)
FL(x) = FL(x;a,b,w,v,q,p) =
f w(t)
x (1.18)
and (for (1.19)
12
BL = BL(a,b,w,v,q,p) =
p > q
)
AL = AL(a,b,w,v,q,p) =
sup FL(x) a<x
and 1/q
x E (a,b) x
dtl
1/pi
vl-p,
(t) lJ
a
denote
dtI
br(b J
a LIJ x
1-
xr
l1/q w(t) dtj
v
I
r
11/q'
(t)
p
1/r
1-
dtll
p (x) dx
v J
a
where _
1
1
(1.20)
Recall that for (1.21)
_
q
r
1
p 1
s' = s s
< s <
l, i.e.
S+ s- = 1
bll 1/
1.14. Theorem.
(1.11)
Let.
1
< p < q <
v, w E W(a,b)
,
q
1/ p
brr ll
< CLIJ' u'(x)Ip v(x) dx]
a
(1.22)
Then the inequality
(
[ju(x)I q w(x) dx]
holds for every
.
a u E ACL(a.b)
if and only if
BL = BL(a,b,w,v,q,p) < °°
Moreover, for the best possible constant
CL
in (1,11) the following
estimate is satisfied: (1.'3)
BL i CL < k(q,p)BL
where (1.24)
I /q
k(q,p) _ (1 +
P 1.15. Theorem.
Let
)
< q < p < -
b}} 1
(1 +
(1.11)q[Jiux w(x) dx]
1/q
(1.25)
,
)
1
p
v, w C W(a,b)
.
Then the inequality
b
}1/p
CIJ u'( x)Ip v(x) dx
a
holds for every
q
a
u E ACL(a,u)
if and only if
AL = AL(a,b,w,v,q,p) < -
Moreover, for the best possible constant
CL
in (1.11) the following
estimate is satisfied: q
(1.26)
q1/q
AL
c CL < ql/q(p,)1/q'AL
13
1.16. Remark. Theorems 1.14, 1.15 will be proved in Section 3. Let us note that they can be extended, in a modified form, also to the 'limit' values we will deal with these special results p and q namely or of 1 ;
,
in Section 5. Moreover, an analogue of the assertion of Theorem 1.15 holds also for the case 0 < q <
1
,
I< p<
(see Section 9). However, let us first give some historical remarks concernthe inequality (1.11), i.e.
ing
the inequality (1.12).
2. HISTORICAL REMARKS
Isolated partial results have been appearing in literature from the
2.1.
very beginning. Let us mention here the result of G. A. BLISS [1] from 1930,
who extended the classical Hardy inequality (1.2) to the case for the special value
c = 0
1
< p < q < -
He showed that the inequality (1.12) holds
.
x-1-q
for
(a,b) = (0,°°)
,
v(x) E
1
,
w(x) =
p
corresponding inequality (1.11) holds for
q x-1-q/p
(2.1)
f u(x)
1/q dx]
,
and consequently, the
u E ACL(0,°°)
too:
1 1/p
(r
=
,
CLIJIu'(x)lp dx
0
0
2.2. The approach via differential equations.
A systematic investigation of
the (generalized) Hardy inequality starting in the late fifties and early six-
ties was connected with the name of P. R. BEESACK. He dealt with the case p = q
and showed that the inequality (1.11), i.e.
in this case the inequality
br
(2.2)
Jlu(x)Ip w(x) dx
a
Ju'(x)(P v(x) dx a
holds if there exists a positive solution y'
14
on
(a,b)
)
y
of the differential equation
(with a positive derivative
P-1
(2.3)
dx Lv(x) d )
I + w(x) yp-1(x) = 0
p. R. BEESACK [1] investigated the equation (2.3) under certain additional
restrictive conditions on the solution (growth conditions for X -+a+) and on the weight functions (differentiability of is not a variational one, although
y
if
v ); his approach
the equation (2.3) is the Euler-Lagrange
equation for the functional bJ[(yP(x))P
v(x) - yp(x) w(x)] dx
J(y) = a
It should be mentioned that P. R. BEESACK dealt not only with the case p >
1
,
but also with
p < 0
and even with
0 < p <
1
;
in this last case,
the inequality sign in (2.2) must be reversed. BEESACK's approach was extended to a class of inequalities containing the Hardy inequality (2.2) as a special case; see, e.g., P. R. BEESACK [2] or D. T. SHUM [1].
Some of the restrictions on the solution
y
and on the weights
v , w
were removed by G. TOMASELLI [1] in 1969. He followed up the earlier papers of K,. TALENTI [1], [2], who considered a little more special weight functions.
Moreover, the paper of G. TOMASELLI [1] implicitly (see his Lemma 2)
contains the assertion that the solvability of the equation (2.3) is not only sufficient but in a certain sense also necessary for the validity of
the inequality (2.2). We will deal with this equivalence in Section 4 for the more general case
1
< p < q < -
.
2.3. Differential equations; the case
p
q-.
The approach mentioned in
Subsection 2.2 was extended to the case < p < q <
I
by P. GURKA [1]. He investigated the inequality (1.11), i.e.
(2.4)
b11 1/
[JIu(x)I q w(x) dx]
a
q
((br
ll
< CIJIu'(x)Ip v(x) dx]
a 15
on
(2.5) on
in connection with the differential equation
ACL(a,b)
A
(a,b)
dx
[vq/P (x) (a ) q P
with
y'
AC(a,b)
w(x) yq/P' (x) = 0
]+
and
y
,
positive. The number
y'
is closely connected with the constant
C
A > 0
in (2.4).
GURKA's results will be dealt with and slightly improved in Section 4.
2.4. The approach via formulas connecting the weight functions.
In the six-
ties, a number of authors dealt with the inequality (2.4) (almost at the same time but independently) in a way which can be described as follows: Given one of the weight functions
v , w
in (2.4),
find formulas expressing the other one so that (2.4) u E ACL(a,b)
holds, say, for every
.
First, the case p = q
was investigated. The result can be formulated in the following form: Assume
(i)
_
x
w(x) =
(2.6)
is given and define
v E W(a,b)
v1-p'
dt1-p
vl-p'
(x)
(t) ,
x E (a,b)
If
a
Then there exists a constant
b (2.7)
b(
I u(x)IP w(x) dx
-`-
C1J
P v(x) dx
u'(x)
a
a holds for every (ii)
such that the inequality
C1 > 0
Assume
u E ACL(a,b) w E W(a,b)
.
is given and define
b
lP
(
(2.8)
v(x) = w1-p(x)
l1
w(t) dtj
,
x E (a,b)
x
Then there exists a constant C1 > 0 Ib
b
Iu(x) IP w(x) dx = C1JIu'(x) IP v(x) dx
(2.9) a 16
such that the inequality
a
holds for every
2.5. Comments.
u f: ACL(a,b)
.
Obviously, we suppose that all terms appearing in the
(i)
foregoing formulas are meaningful. In particular, this means of (2.6) and (2.8)
-
that
x
b 1-pr
v
(2.10)
r
(t) dt < -
and
I
a (ii)
in view
-
w(t) dt < m
for
x E (a,b)
x
Formulas (2.6) and (2.8) are in a certain sense mutuaZZy inverse:
Multiplying, for instance, the right-hand side in (2.8) by a suitable constant A
,
denoting v(x) = A v(x)
and computing the function coincides with
w
w
from the formula (2.8), we obtain that
from (2.6). Therefore, we can omit the tildes. A
description of the formulas for
1
w full
will be given in Subsection
< p < q < -
2.6.
(iii)
stead of
If we compare the integrals appearing in (2.10) (with
w ) with the definition of the function
FL(x)
w
in-
from (1.17), we
set that the conditions (2.10) are quite natural. Therefore, we will not give the original (direct) proof of the assertions from 2.4(i) and 2.4(ii): it can be easily shown that the choice of
v
,
w
according to (2.6) im-
mediately yields BL(a,b,w,v,p,p) < so that our problem is solved by Theorem 1.14. (iv)
Nevertheless, the approach just described precedes the approach
via Theorem 1.14. Formulas of the type (2.6) have been derived by V. R.
PORTNOV [1] in 1964, by F. A. SYSOEVA [1] in 1965 and in the same year by
V. N. SEDOV [1] and A. KUFNER [1] who published this result later. (v)
In the formulas (2.6) and (2.8), the weight function
pressed in terms of
v
w
is ex-
and vice versa. A. KUFNER, H. TRIEBEL [1] gave the
following formulas in which both functions
v , w
are expressed simuZtane-
ously in terms of an auxiliary function; namely,
17
w(t) = e'X(t) A'(t)IeA(t) - e'X(a)] -p (2.11) e(1-p)A(t)
v(t) = where
A
[A,(t)]1-p ,
is a continuously differentiable function on
A'(t) > 0
and
lim
tib-
a(t) = m . Here
,1(t)
and
such that ,1(a) _ - W
(2.7) holds for every
v
,
u ; ACL(a,b)
2.6. Formulas; the case
p
<_
q
in the form (2.11), the inequality
with the constant
Let
.
w
1
C = [p/(p - 1)lp
< p < q < - , and either
,
x V'-P'(t)
(2.12)
dt <
J a
and (2.13)
V'-p'(t)
w(x) = pI v(x) If
dt
a
for
x E (a,b)
,
or
w E W(a,b) br
(2.14)
w(t) dt < J
x
and (2.15)
v(x) _ (P
1-P
b
w1-p(x)
w(t) dtl
l!
p-1+p/q
x
for
x E (a,b)
.
Then the inequality
b (2.16)
b
111/q
[JIu(x) lq w(x) dx]
< k(q,p)
a holds for every
u E ACL(a,b)
k(q,p) _
(1 + p )
with ,
(1 + 9
18
11/p
{JIu'(x)IP v(x) dx]
a
1/q (2.17)
is
t- a+
an admissible value. If we choose
v E W(a,b)
A(a) = lim
(a,b)
1/p'
J
,
2.7. Comments.
(i)
The foregoing result is due to P. GURKA [1], who in pl/q(p,)l/p'
fact obtained the constant
instead of (2.17). His proof is v
based on Theorem 1.14; it can be easily shown that for
, w
chosen
according to Subsection 2.6, BL(a,b,w,v,q,p) = 1 -
see (1.18). (ii)
The formulas (2.11) mentioned in Subsection 2.5 (v) were ex-
tended by P. GURKA C1] also to the case t
w(t) = e1
(2.18)
while the function (iii)
v
[e,1(t)
V(t)
- e
1
< p < q < -
:
he obtained that
A(a) -1-q/p
is the same as in (2.11).
Formulas (2.13) and (2.15) cover also the result of G. A. BLISS
[1] mentioned in Subsection 2.1.
2.8. The approach via Theorem 1.14.
Let us consider the Hardy in-
(i)
equality in the form (2.7), i.e. for p = q
Necessary and sufficient conditions for its validity (for the case of gene-
ra weights FL(x)
v , w ), expressed in terms of the boundedness of the function
from (1.17), appeared probably for the first time in G. TOMASELLI [1]
(1969). Apart from the condition
BL < m he derived also another necessary and sufficient condition of the form
x BL
sup
a<x
tr
vl-pI(Y)
[1 w(t)
(J
p
x
1
dtJ
dyl
<
JJJ
a
a
a
He started from the differential equation of the type (2.3) and substantially used the auxiliary constant
t
x
11
(2.19)
K =
1
inf
sup
f
a<x
(J w(t)
vlp(y) dy]dt
[f(t) + J
where the infimum is taken over all positive measurable functions
f
on
19
(a,b)
We will use an analogous constant for the case
.
p
<_
q
(see Section
4 ).
G. TOMASELLI extended the ideas of G. TALENTI [1], [2] (1966, 1967), (a,b) = (0,-)
who considered the case
and more special pairs of weights
connected by the formula
v , w
w(x) = x-P v(x)
x E (O,=)
,
.
In 1972, B. MUCKENHOUPT [1] published a direct proof of Theorem
(ii)
1.14 for the case
p = q
.
Moreover, he considered (for
(a,b)
the more general inequality
fP(x) dv(x)
1(HLf)P(x) dµ(x) < C
(2.20)
J
0 where
µ
,
0
are Borel measures, and showed that this inequality holds if
v
and only if 1-p'
1/P (2.21)
where
B
v*
L
[([x=))]
=
0 dt
x>O
*1
1/pr <
dt [J[_]
denotes the absolutely continuous part of
v
For the case
(iii)
< p < q < m
1
,
the first proof of Theorem 1.14 was published in 1978 by J. S. BRADLEY [1]. He considered the interval
(0,"')
and, similarly as other authors mentioned
above, investigated in fact Problem 1.7 (i). One year later, the same result was published independently by V. G. MAZ'JA [1] and V. M. KOKILASHVILI [1]. The latter author considered also the corresponding analogue of the inequality (2.20), i.e.
(2.22)
11/q
{(Hf)(X)
ll1/p
Cl fP(x) dv(x)J dµ(x)J
0
0
where the necessary and sufficient condition reads as follows:
20
I/q
r((dt*11-pIdt11/p'
sup p [u(Ex,-))
BL
L0Il
J
< m
.
J
J
This result can be found also in V. G. MAZ'JA [1].
q < p
2.9. The case
This case has been treated in literature only in
.
the way indicated by Theorem 1.15, which was proved by V. G. MAZ'JA [1] in 1979 (again for the case of Problem 1.7 (i)).
the inequality (2.22) for
The case of Borel measures, i.e.
q < p,
[1], and the corresponding necessary
was also investigated by V. G. MAZ'JA
and sufficient condition for the validity of the inequality (2.22) reads x(dv* 1-p'
r/q
J[u([x,-))I 0
AL = sup X >O
where
r = pq/(p - q)
2.10. The case the case
1
< p,q <
°°
llr/q'
dtl
0
dv*l1-p/ 1/r dx l dx J1
<
.
< p < W
1
[Jfdt )
.
,
0 < q <
1
Up to now we have dealt only with
.
Recently, G. SINNAMON [1] investigated Problem 1.7
alto for the case mentioned above
see Section 9.
-
3. PROOFS OF THEOREMS 1.14 AND 1.15 Let us start with an auxiliary assertion.
The Minkowski
A modification of the Minkowski integral inequality.
3.1.
integral inequality
(3.1) ll
U
a
c
K(x,y) dyJ
Kr(x,Y) dx] l/r c
a
holds for every non-negative measurable function r ?
1
(for
r =
1
,
dy
lf
r dxJ l/r
K
on
(a,b) X (c,d)
and
we have the equality sign since (3.1) is in fact a
consequence of the Fubini theorem)
-
see, e.g., G. H. HARDY, J. E. LITTLE-
WOOD, G. POLYA [1] (Theorem 202) or N. DUNFORD, J. T. SCHWARTZ [1] (Chap. VI, 21
Section 11). We will use (3.1) in the following special form
If
If Y(y)
(1 41(x)
a
where
0,
1/r
r
x (3.2)
b
dxJ
dYJ
a
a
4' E M+(a,b)
1/r
b
j Y(y)
<
If
'P(x) dxJ
dy
y
.
In what follows, we will use the notation from Section 1, in particular the constants and functions introduced in Subsection 1.13.
Let
3.2. Lemma.
ber
< p < q < m
1
BL = BL(a,b,w,v,q,p)
holds for every
and
v, w E W(a,b)
.
Assume that the num-
from (1.18) is finite. Then the inequality (1.11)
u E ACL(a,b)
and the best possible constant
satisfies the following estimate: CL = k(q,p)BL
(3.3)
with
k(q,p)
Proof.
from (1.24).
The assumption
BL
implies that the integral
t
vl-p'
(y) dy
a
is finite for every
t E (a,b)
.
t (I
(3.4)
h(t) =
II
v
1-p
Consequently, the function 1/(p's)
'
(y)
dyI
a
where
s
is a fixed number from
0 < h(t) < Let
f E M+(a,b)
.
(1,00)
for every
00
satisfies
t E (a,b)
.
Then Holder's inequality yields
x
x f(t) dt =
j
f(t) v1/p(t) h(t) h-1(t) v-1/p(t) dt J
a
a
x r
111/p
fp(t) v(t) hp(t) dtj J
a Further, we have 22
h-p,(t) vl p'(t) dt If
a
CL
in (1.11)
x dyl-1/s
J h-P (t) v1-p' (t) dt =
f 1
a
vl-p'(y)
vl -p'
If
a
(t) dt
a
x
( s
s
(s-1)/s
vl-p
,
= s
s
1
h(s-1)p (x)
(y)
l
dy] a
and consequently x
b
1P/q
q
r
f(t) dt)
(I(
w(x) dx]
<
j
a
a
<
(s
11q/p
s
-
1)
lJ a
fp(t) v(t) hp(t) dt]
h(s-1)q(x) w(x) dxl11P/q
a
Estimating the right-hand side with help of the inequality (3.2) (for = q/p ), we obtain
(fx
b
(3.5)
[J
f(t) dt)
q
r =
p/q
w(x) dx I
a
a
P/p, b
b
I
J fp(t) v(t) hp(t)
(s S 1)
J
a
p/q
w(x) dx
dt I
t
From the definition of the number
h(s-1)q(x) =
h(s-1)q(x)
BL
-
see (1.18) and (1.17)
-
we have
(s-1)/s
[hs(x)](s-1)q/s < BL
w(y)
L
dyI
,
and hence b J
b
b
h
(s-1)q
(s-1)q/s (x) w(x) dx < BL
t
B(s-1)q/s L
1+1/s
w(x) dx =
[J w(y) dy]
J
x
t
s
_ 1
r
b
I
1/s
IJ w(y) dy l jJ
t
This estimate together with (1.17) and (1.18) implies
23
b
p/q
h(s-1)q(x)
b
11
<
w(x) dx
(
L
<
= s
P/q
(s-1)p/s
w(Y) dyI
t
t
BL
f
lJ
j
t
p/(qs) '
sp/q B(s-1)P/s
l-p'
[BL{J v
1/p/1P/s (Y) dy1
=
p/4
P P Bh (t)
JJJ
a
and from (3.5) we have x(
(br
1/q
q
II I f(t) dt)
(3.6)
a
a
s
/ <
w(t) dt]
is
b(
l1/p'
sl/q
BLIJ fp(t)
1J
v(t)
hp(t) h-P(t) dt
a
b
= g(s) BLl1
1/p
fP(t)
v(t) dt
,
I
a
where g(s) = sl/q(s s Here
s
was arbitrary,
s >
inf g(s) = g(l + s>1
1
,
and since
= k(q,p)
p
we finally conclude from (3.6) b
(I J f(t) dt) q w(x) dxJ l/q
k(q,p) BLIJ fp(t) v(t) dt 1/p l
111
a
a
a
Thus, we have shown that the inequality (1.12) holds for every
f E M+(a,b)
and the assertions of Lemma 3.2 follow from Lemma 1.10.
3.3. Remark.
In Lemma 3.2 we in fact proved the
'if '
part of Theorem
1.14. Let us point out that we approached Problem 1.5 via Problem 1.7; in
Lemma 3.2 we proved that if
24
BL < -
,
them Problem 1.7 (i) has a solution.
Let
3.4. Lemma. ity
1
< p,q < -
v, w E W(a,b) . Assume that the inequal-
and
(1.11) holds with the (finite) constant
CL
for every
u E ACL(a,b)
Then BL < CL
(3.7)
Proof.
Assume in addition that
(i)
x vl_P
(3.8)
r
(t) dt < m
J
for every
x E (a,b)
a
Then it follows from Lemma 1.10 that the inequality (1.12), i.e.
(3.9)
b111/4
[J(Hf)(x) w(x) dx]
b(
< CLIJ fp(x) v(x) dx)
a
a
holds for every
f E M+(a,b)
b
.
E E (a,b)
Let b
j(HLf)q(x) w(x) dx Z J(
a
be fixed. Then
x lll
Ifa
4
w(x) dx
f(t) dt ))))))
b
If w(x) dxI If f (t) dt] a
E
This estimate together with (3.9) implies b
1/q
(3.10)
If w(x) dx]
f(x) dx]
-< CLIJ fP(x) v(x) dx]
a
f E M+(a,b)
1/p
(b
ll
II
E
for every
(
a
Let us take
.
r
f (x) _
vl-P
I
(x)
x E
for
for x e E ,b)
0
This function obviously belongs to
f(x) dx
J
J
a
a
M+(a,b)
,
we have
vl Pr(x) dx
and moreover
25
(b
0 <
IJ
1/P
fp(x) v(x) dx
1/P
v1-P'(x)
=
dx
IJ
J
a
]
a
The last integral is finite due to (3.8) and positive since
v ` W(a,b)
Consequently, from (3.10) we have vl1/p,
ll1/q
(b(
w(x) dx]
(3.11)
(x) dx]
(I
< CL
a (ii)
Let
v
be a general function from
(3.12)
vn(x) = v(x) + n(1 + x2/(p'-l))
Obviously
vn E W(a,b)
too, and for
,
x
x E (a,b)
b dt J
a
we have
vn-p'(t) dt =
n
J
n E N define
For
.
x E (a,b)
,
b v1 p'(t) dt <
(3.13)
W(a,b)
a
a
-1
w(t) +
t2/(P'-1))]p'-1
n(1 +
1
p,-1
Pr-1
n
dt +
t2
l
np'-1
dt+J
2
dt
-1
PI-1
= 4n
Using the inequality (1.11) and the estimate (3.14)
v(x) < vn(x)
x E (a,b)
bll 1/
we obtain
(3.15)
for
{Ju(x)j q
q
w(x) dx]
n c- N
ll 1/ p
b (
< CLIIIu'(x)Ip vn(x) dx]
a for every
and
a
u E ACL(a,b)
.
Since (3.13) is nothing else than the condition (3.8) for the function vn
,
we obtain from part (i) of our proof that in view of (3.15) b
(3.16) 26
1/q (
li w(x) dxl
((
l1
a
vl-p
n
(x) dxl J
1/p'
< CL
holds for every
E (a,b)
Further, for a.e. vn-p (x)
0
-
cf.
x E (a,b)
(3.11).
we have
vI-P (x) n+I vn-p'(x) = vl-p'(x)
(3.17)
lim
<=
n-
and the monotone convergence theorem together with (3.16) yields that b ff
ll1/p
IJ w(x) dx] =
[J
v
1-
(x) dx]
C -
ll
L
a E (a,b)
for every
3.5. Remark. if '
,
which immediately implies the estimate (3.7). U
If we use Lemma 3.4 for p < q , we have in fact proved the 'only
part of Theorem 1.14, and consequently, in view of Remark 3.3, Theorem
1.14 is proved completely. Let us mention that this theorem could be formulated also in the following form: Problem 1.5 (i) has a solution if and only if the number
3.6. Comments.
BL
from (1.18) is finite.
Lemmas 3.2 and 3.4 provide a complete proof of Theorem 1.14.
Let us mention that the proof given here is 3.2
-
-
especially as concerns Lemma
a modification of the former proofs given by B. MUCKENHOUPT [1]
(the case
p = q
)
and J. S. BRADLEY [1], V. C. MAZ'JA [1]. An important
role is played by the constant
k(q,p)
from (1.24). Instead of the in-
equality (3.3), the authors establish the following estimates (3.18)
(J.
CL < p1/q(p')I/p' BL
S. BRADLEY [17, V.
S. KOKILASHVILI [1] and - by another method -
P. GURKA [1]) or (3.19)
CL < q1/q(q')1/P'B
(V. C. MAZ'JA [1]). All constants mentioned above are closely connected via the function (3.20)
g(s) = sI/q(s s I)
1/p
27
introduced in the proof of Lemma 3.2. For the constant from (3.18) we have pl q(p,)1/Pr
= g(p)
while q1/q(q')l/Pr
= g(q)
;
both constants are greater than the constant
from (3.3) defined as
k(q,p)
g(l + P = inf g(s) s>1 provided
< p < q < -
1
Consequently, the constant
.
CL
up to now best estimate of
;
leads to the
k(q,p)
this estimate is due to B. OPIC and is
published here for the first time. For
p = q
,
all three estimates
(3.3), (3.18) and (3.19)
-
coin-
-
cide and, moreover, give the best possible estimate for the constant
CL
in (1.12). This follows from the fact that the right-hand side in any of these estimates is (for
and for the special weights
p = q
w(t) = tc-p ) equal to the best possible value
C
v(t) = te
from (1.3).
The proof of Lemma 3.4 implies that the assertion of Lemma
3.7. Remark.
1.10 holds without the assumption (1.14). Indeed, according to Remarks 3.5
and 3.3 we have to show that
the following implication holds:
Problem 1.7 (i) is solvable (with a finite constant ,, I
CL )
f
BL < m Let us prove it. Suppose
v E W(a,b)
is general (not necessarily satis-
fying (1.14)) and Problem 1.7 (i) is solvable with
CL < m
.
This means
that the inequality (1.12) [ i.e. (3.9) 1 holds. Using the functions
v
n
from (3.12), we obtain in view of (3.14) that the inequality
`1/p
[j(Hf)(X) w(x) dx]
<
a
a holds for every vn
28
CL[J fp(x) vn(x) dx
f E M+(a,b)
(with the same constant
fulfil the condition (1.14)
-
CL ). The functions
see (3.13). Therefore, according to
Lemma 1.10, we have (3.15) and due to the proof of Lemma 3.4, the estimate (3.16), too. Consequently
which implies that
BL <_ CL
BL < m
Now, we are ready to give in two lemmas the proof of Theorem 1.15. The first concerns the 'if' part.
Let
3.8. Lemma.
ber
< q < p < w
1
AL = AL(a,b,w,v,q,p)
holds for every
v, w E W(a,b) . Assume that the num-
and
from (1.19) is finite. Then the inequality (1.11)
u E ACL(a,b)
and the best possible constant
CL
in (1.11)
satisfies the following estimate:
(3.21)
Proof.
CL < ql q(p,)l/q AL
AL < -
The assumption
implies that the integral
t
vl-p'(Y)
dy
J a
t E (a,b)
is finite for every to show
In accordance with Lemma 1.10, it suffices
.
that the inequality (1.12) holds for every
f E M+(a,b)
the estimate (3.21) holds for the best possible constant For such
f
CL
and that
in (1.12).
, we have x
b
b
q
J(HLf)q(x) w(x) dx a
J
U
a
b
x(Y
= q J [J a
a
I
f(t) dt]
w(x) dx =
f(y) dy 1
a
q-1
f(y) dy w(x) dx I
J
a
and the Fubini theorem yields b (3.22)
J(Hf)(x) w(x) dx = q
y
( ( J IJ
f
q-1
w(x) dx] dy
f(y)
f(t) dt 1
a
a
y
Rewriting the right-hand side in an appropriate form and using Holder's inequality for the product of three functions (with exponents and
p/(p - q)
p
,
p/(q - 1) ) we obtain
29
b
(3.23)
1(HLf)q(x)
(1-q)/p
w(x) dx = q(p - 1)
a b
b
dtlq-1 v(1-P')(P-q)/P(Y)
dx]iY
f
f(y) vl/p(y)
vl P'(t)
If w(x) IJJJ
a
y
a Yf
(p - 1) (q-l)/p [HLf(y)
l vl-p'(t) dt)q-1 v(1 P')(q-1)/P() dY
a 1/p
b 1)(1-q)/p
q(p -
elf fP(Y) v(Y) dy] a
b
(q-1)/p
{J(Hf)P(y)
w(Y) dy]
a
where the new weight function
w
is given by
y
dtJ-p
ll
w(Y) = (p - 1) If vl-p'(t)
However, for this function
w
BL(a,b,w,v.P,P) =
vl -P, (y)
and for
we have
v
1
and consequently, according to Theorem 1.14 (with b
q = p ),
11/p
b
l1/P
w(y) dYJ
If
(a,b)
y
a
k(p,p) if f P(x)
v(x) dx]
a k(p,p) =
Using (3.23), the last estimate and the fact that
pl/p(p,)'/P,
we arrive at the inequality b
[J(Hf)(X) w(x) a
b
dxll/q l < ql/q(P,)l/q
dxll/p
ALIJ
fp(x) v(x)
a
which implies (3.21). 11
3.9. Lemma. Let
30
1
< q < p < -
and
v, w t W(a,b).
Assume that the
inequality (1.11) holds with the (finite) constant u e ACL(a,b)
.
for every
CL
Then 1/qr
ql/q(r
(3.24) 1
where
Proof.
__
_
1
q
r
CL
L
1
p
Due to Lemma 3.4 (where
we have
BL <_ CL
p
,
q
are arbitrary numbers from
(1,-)
< W ) and consequently
(
b
vl-p(t)
dt <
w(t) dt <
(3.25)
I
J
x
for every
x c (a,b)
numbers such that
.
an
Let us choose two sequences a
and for
bn t b
,
n C-' N
{an}
{b n}
,
of real let us
x C- (a,b)
,
introduce auxiliary functions
(3.26)
fn (x) = (b(
x
w(t) dt
=
vl-p,(t)
(f
lr/(pq')
lJ
Ifx Obviously,
l_ p
'
(x) X(a
V
dt j
n bn
)(x)
an > 0
fn (x)
a.e. in
(an,bn)
,
and hence
b (3.27)
f fp(x) v(x) dx > 0 n
.
a
If we define
bn b (3.28)
r/q'
x vl-p'(t) 1r/q ((
vl_p'(x)
dtj
II w(t) dt
An = If
11/r dx]
an
an x
we obtain in view of (3.25) b b
(3.29)
r/q fn(
x
l
f w(t) dtI an
vl-p1-p ,(t) dtir/q'
v
(
An
J
an
l
'(x)
dx =
j
an
31
.
bn
b
r
w(t)
f
ll
dtjr/q
dt]r/P'
[Jvl-P'(t) 1
an
an
Further, we can easily verify that bn
b r
fn (x) v(x) dx = Jfn(x) v(x) dx = An an
(3.30) a
Now, we will derive lower estimates of the left-hand side in (1.12) for
fn E M+(a,b)
lie can deal with (1.12) instead of (1.11) in view of
.
Lemma 1.10 and the second condition in (3.25).] Since b
If (HLfn)q(x)
ll1/q
w(x) dx]
=
a
bx( ( (
[q
J
lJ
a
) a
a
1l/q
q-1
ly(
r
fn(t) dt)
w(x)
dy
,
I
the Fubini theorem yields b
1 /q
[J(Hf)(x)
dx`
w(x) a If b
y
[bf
= q 1/q
fn(t) dtJq-1 fn(Y)
w(x) dx] dy]l/q =
lJ
a
y
a
bn y ql/q
[J fJ fn(t) an an
=
Further, for
q-1
dtl
b
1/q
fn(Y) If w(x) dxJ dY] y
y E (an,b) , we obtain from (3.26) that
y J
fn(t) dt =
an
y (b =
J
an at
32
lr/(P4) ((t
J w(x) dxJ
J
v
an
1- P '
(x) dxJ
r/(Pq')
1_
v
P
(t) dt
b
If w(x)
Z
dxJ
r/ (Pq) yt, (x) vl-p
dxl
fan Ifan
y
RE (fb w(x) dx lr/ (Pq) J
J
y
If
r/ (Pq' ) v 1_ P
(t) dt
=
J
}r/ (qP' )
v1-P (x) dxJ
an
y
b/q1/qr
and consequently,
(3.31)
[I(H Lfn)() w(x) dxJ qa
rrbn L
an
br/q
((y
Ij w(x) dxJ
f
v
1- P '
(x) dxJ
r/q' v
1- P '
11/q (y) dy)
an
y
qq (gel l r l
1/q r A
n
This inequality together with the formula (3.30) used in the inequality (1.12) yields r
1/q
1/q qk
Ar/q < C Ar/P
r
q
Ln
n
This implies (3.32)
since
1/q'
1/q q
0 < A
(
n
An ` CL
r
in view of (3.30), (3.29) and (3.27). Letting n
<
in
(3.32), we obtain (3.24). 11
3.10. Remark.
Lemmas 3.8 and 3.9 give a complete proof of Theorem 1.15.
The procedure used here is a small modification of the original proof of V. G. MAZ'JA [1].
In the conclusion of this Section we give a proof of an auxiliary assertion, which will be useful in the sequel.
3.11. Lemma.
Let
1
< q < p < m
and
v, w E W(a,b)
.
Assume that
33
and
1 w(x) dx < -
vl-P'(x) dx <
1
a
a
Denote 1
1
__
Let
{a
bn t b
}
{b
,
n
_
1
q
r
n
p
be two sequences of real numbers such that
}
and for
n E N
and
r /4 /q
r/q'
x
vl-P,(t)
lJ
An =
LJ
lJ
w(t) dt
dt
b
r/(Pq)
1/r
vl-P (x) dxI
l
lr/(Pq')
vl-P,(t)
(Ix
fn(x) = If w(t) dtJ
y a
1J
an
an x
n
define
x C- (a,b)
fbn (b
a
dt]
IJ
lan
JJJ
x
v1-p'(x)
X(a,b )(x) n
n
and x
gn(x) = Anr/P J fn(y) dy
(3.33)
a
Then
gn E ACL(a,b)
,
b
J(gF)P(x) v(x) dx = 1
(3.34)
a
and b
1/q
1/q
(3.35) (I
gn(x) w(x) dxI
,
1/q
- q
{
r
An
a
for every Proof.
n E N
We have used the functions
of Lemma 3.9 (3.36)
.
-
n
and the numbers
A.
in the proof
see the formulas (3.26) and (3.28). Therefore, denoting
fn(x) = Anr/P fn(x)
we obtain from (3.30) that 34
f
b
fn (x) v(x) dx =
(3.37)
1
.
a
According to (3.31), we have b
(3.38)
,
l/
[J(H?)(x)
w(x)
qq(2) l/pA n 1/ r
dxl4
a x
Moreover, in view of (3.33) and (3.36),
gn(x) = J fn(y) dy
,
and since
a b
bn b
fn(y) dy = Anr/p
J
fn(y) dy = Anr/p f fn (y) dy J
a
b
b
Anr/P (J w(t) dt
.i
lr/(Pq)
-qL A
we/conclude that
dt]r/(Pq')
vl-p'(x)
ll
dx) =
bn
r/(pq)
r/pf
r/(P'q)
vl-p'(t)
w(t) dtJ an
<
dtl
1(
n
r
vl-p'(t)
an an
b _
x
lJnlJ
an
Since
an
a
lJ
an
gn E- ACL(a,b)
gn = fn
,
we have (3.34) from (3.37); further, since
(3.35) follows from (3.38) and the lemma is proved.
g n = HLfn,
O
4. THE METHOD OF DIFFERENTIAL EQUATIONS Let us just formulate the main result.
4.1. Theorem. (4.1)
Let
1
< p < q < -
,
v, w e W(a,b)
.
Moreover, assume that
v E AC(a,b)
and x (4.2)
J vl-P'(t) dt < -
for
x E (a,b)
a
35
bll 1/
Then the Hardy inequality (1.11), i.e.
[JjU(X)I q
(4.3)
q
a
a
there is a number
A dx
has a solution
x
> 0
such that the differential equation q/p']
rvq/p (x) (a )
+ w(x) yq/p' (x) = 0
satisfying the conditions
y
y(x) > 0
y' E
(4.5)
if and only if
CL
with a (finite) constant
u E ACL(a,b)
holds for every
(4.4)
ll 1/ P
((b(r
CLIIIu'(x)Ip v(x) dx]
w(x) dx]
y'(x) > 0
,
for
x C- (a,b)
.
The assertion of Theorem 4.1 is a consequence of several lemmas, which deal with the inequality (1.12) for functions
f E M+(a,b)
;
the approach
via the inequality (1.12) is correct due to Lemma 1.10.
Let the assumptions of Theorem 4.1 be satisfied and assume that
4.2. Lemma.
A > 0 such that the differential equation (4.4) has a
there is a number solution
satisfying (4.5). Then the inequality
y brr
(br
ll 1/q
ll 1/p
CIJ fp(x) v(x) dxJ
IJ(HLf)q(x) w(x) dxJ
(4.6)
a
a
holds for every
f E M+(a,b)
with the constant
C = al/q
(4.7)
Proof.
Clearly, we can suppose that
f E M+(a,b)
satisfies
b
fp(x) v(x) dx < -
.
J
a Let
y
be the solution of (4.4) satisfying (4.5). For (dx)
(4.8)
fi(x) _ - A
(4.9)
Y(t) = fp(t)
Then (4.4) yields that 36
x, t E (a,b) denote
q/pr
dx [vq/p(x)
[Y'(t)] -p/P'
.
O(x) = w(x) yq/p'(x)
and Holder's inequality
together with (4.5) implies x
(HLf)q(x) w(x) =
f(t) dt q w(x) _
[r
I
a
x (t)]-1/P'
(1 f(t) [y'
[y'
dt lq w(x)
(t)]1/p'
<-
a
w(x)
'Y (t) dt] q/P
I
lJ
a
y' (t) dt]
[
la
q/p'
x
x
yq/p'(x)
< w(x)
lJ
ID(x)
Y(t) dtJq/p =
lJ
a
bll 1r
Consequently, denoting
[J(Hf)(X)
r = q/p
Y(t)
dtlq/p
a
xr
we have
b <
w(x) dx]
[r
o(x)
(J
bb (b a
a
W(t) dt)
l 1r dx]
a
and the Minkowski inequality (3.2) yields
< j Y(t)
a
l dxl/r
r
dxJl/r
(4.10)
[J(Hf)(X) w(x)
a
O(x)
dt
t
Since according to (4.8) jb
l1 /r O(x) dx]
<
XP/q v(t) Ey'(t)lp/P,
t the inequality (4.10) together with (4.9) implies b (f
1/q
<
x1/q
lJ(HLf)q(x) w(x) dxl
[l w(t)
a
v(t) ry'(t)]P/P'dt]ll1/p
=
a (b
ll
( (
=
b
a1/q
1/P
fp(t) v(t) dt a
This is the inequality (4.6) with the constant
C
from (4.7).
U 37
Let the assumptions of Theorem 4.1 be satisfied. Let us denote
4.3. Lemma.
t
x K =
(4.11)
q'
sup a<x
inf f
w(t) [f(t) +
f
f(x)
111
a
a
where the infimum is taken over the set of all
positive on
(a,b)
tion (4.4) has a solution
(and, consequently,
which are
A > 0
such that the differential equa-
satisfying (4.5), then
y
K<W).
If K <
(ii)
then the differential equation (4.4) has a solution
satisfying (4.5) for every
Proof.
fE M+(a,b)
K < A
(4.12)
y
V'-P' (s) ds]q/p'+l dt
.
If there exists a number
(i)
I
JJ
Let
(i)
.
be the solution of (4.4), (4.5) and put
y
z = ry
A > K
v1-p'
y It can be easily verified that
on
(a,b)
is a positive solution of the equation
v1-pr
It =
(4.13)
z
zq/p'+l +
Qw
Since
.
x(
z(x)
z'(t) dt
J
a
we have according to (4.13) x(
xJ
z(x) > 9 I w(t) zq/P'+1(t) dt +
a
vl-p'(t) dt
a
and,consequently, denoting x ,
f(x) = z(x) -
vl-p (t) dt
J a
we have
f(x) > 0
on
(a,b)
and
t
x A
38
q
f (x) )I w(t) a
V'-P' (s) ds] q/p'+1 dt
[f (t) + J
a
In view of (4.11), we immediately obtain (4.12). (ii)
function
Let us fix
A > K
.
such that
f
t
x
(4.14)
there exists a positive
According to (4.11)
f (x)
vl-P, (s) ds] q/P'+1 dt
w(t) [f (t) +
>Q 1
a Further, on
(a,b)
a
let us define a sequence of functions
zn(x)
,
n15 N,
by the formulas x vl-p
z0(x) = f(x) +
(t) dt
J a
(4.15)
x z
pq
n+1 (x)
f
x(
w(t) zn/P'+l(t) dt + J vl-p'(t) dt
a
zn(x) > 0
Obviously,
for
a
and in view of (4.14)
x E (a,b)
x zq/Pr+l(t)
(4.16)
J
dt <
w(t)
a
Moreover, x
z0(x) - z1(x) = f(x) -
P! q
w(t) zq/P +1(t) dt > 0
f a
according to (4.14), and consequently x
zn(x)
-
zn+1(x)
Aq f w(t) [zn/p'+l(t)
-
zn/P'+l(t)] dt > 0
a
Thus we have shown that the sequence
{z
which together with the positivity of non-negative function z(x)
on
(a,b)
z
n
n
(x)} (x)
is decreasing on
(a,b)
yields the existence of a
,
z(x) = lim zn(x) nom
Taking into account (4.16) and applying the monotone convergence theorem, we obtain from (4.15) that 39
x z(x) _
J
x w(t) zq/p'+l(t) dt +
I
9
vl-p/(t) dt
1
a
a
This formula implies that the function
is positive, belongs to
z
AC(a,b)
and satisfies the differential equation (4.13).
The proof is now complete since it can be shown that the function x vl-pr(t)
y(x) = exp
z-1(t)
dt
If c
with
c
a fixed number from
(a,b)
is the solution of (4.4) satisfying
(4.5). 11
According to Lemma 4.3, we have shown that under the assump-
4.4. Remark.
tions of Theorem 4.1, the number there is a
from (4.11) is finite if and only if
such that the problem (4.4), (4.5) is solvable. Con-
(0,°°)
A
K
sequently, using in addition Lemma 4.2, Theorem 4.1 will be proved if we show that the validity of the Hardy inequality implies the finiteness of the number
K
This will follow from the next assertion (and Lemma 3.4).
.
Let the assumptions of Theorem 4.1 be satisfied. Let
4.5. Lemma.
defined by (4.11), Let
be the number from (1.18) and let
BL
best possible constant in (4.3)
CL
be
K
be the
Then
CL < K1/q < k(q,p)BL
(4.17)
where )1/q
(4.18)
(l +
q 11/p,
k(q,p) _ (1 + p
Proof.
The first inequality in (4.17) will be proved by contradiction.
(i)
To this end, let us assume that (4.19)
Since
K1/q < CL
,
and choose
such that
< CL
K1/q < A0
K < Ao
,
the problem (4.4), (4.5) is solvable for
Lemma 4.3. Formula (4.7) in Lemma 4.2 implies that CL < al/q = A0 40
A0
,
A = aq
due to
which contradicts (4.19). (ii)
BL = w
If
ally. Therefore, assume
fact that
then the second inequality in (4.17) holds trivi-
,
w E W(a,b)
BL < -
Then, due to the definition of
.
BL
and the
we have
,
b 0 <
J
w(y) dy < -
for
t E (a,b)
t
Consequently, the function b
,/q
tf
(r
f(t) = sBL,IJ w(y) dy, -p-
vl-p,(y) dy
t is continuous on
(a,b)
J
a for every
s E (1,-)
b
dyJP,/q
.
Moreover, for such
s
we
have
t (r v1-p'(y) dyl
sBp,
> BPI -
L
L
( f lJ
w(y)
J
Il
t
a
which implies f(t) > 0
for
t E (a,b)
The( definition of the number
K
-
x
see (4.11)
yields
-
b
q / p '+1
r
w(t) < K - 1?! q
[sB'[I w(y) dyJ-p /q
a
sup
b
t t
a < x
sBL,IJ w(y) dyJ
-P/q
v'-P'(y)
a
x
h(x)
sup
= q (sBL
sBp,
-
(
lJ x
w(y)
P, /q [J vl-p' dy]
(y) dy]
a
q/p'+l
(sBP L q
x
b
a<x
,
dy
- J
q/p'+1 (4.20)
dt J
(s - 1)Bp
sup
h(x)
a<x
where
41
b
x
h(x) =
J
(-P '/q)(q/P'+1)
(( dy) w(t) lJ w(y)
a
b
b p
If
w(y) dy)
r
l
pf [1 -
(J x
t
(4.21)
p '/q
b
dt
<
w(y) dy
w(y) dy J-Pr/q
a
.
P p
If
x
If we denote 1/q
for
s E (1,m)
,
s1/P
g(s) = (Ss 1)
(4.22)
we have from (4.20), (4.21)
,
K1/q < g(s) BL
.
Since
k(q,p) =
inf g(s) 1<S<-
the second inequality in (4.17) follows immediately. 11
4.6. Remarks. (i)
CL < K1/q < k(q,P)BL
BL
(4.23)
Combining the inequalities (4.17) and (3.7), we have
and consequently
Theorem 4.1 is proved completely: the validity of the CL < - ) implies
Hardy inequality (with the best constant
BL < m
due to
the first inequality in (4.23), while the last inequality in (4.23) then implies
K < -
(ii)
(see Remark 4.4).
From the inequality (4.23) we again have the estimate (1.23),
but now under the restrictive assumptions (4.1), (4.2). In what follows we will show that the less natural assumption (4.1) can be removed. For this purpose, let us make some slight transformations of the data and inequalities considered.
Let us assume that
(iii)
1
< p < q < m
,
v, w
W(a,b)
and
x
v1-P'(t) dt < m
(4.24)
for every
x e (a,b)
J
a
This enables us to introduce new variables
42
z
and
s
(instead of
x
and
t
) by the formulas
(4.25)
x = O(z)
t = 0(s)
,
where
x
t
vl-pr(y) dy
-1(x) =
z =
vl-p
0-1(t) = s =
(y) dy
J
J
r
a
a
Obviously, lim 0-1(x) = 0 xaa+
If we denote br
(4.26)
0-1(x)
L = lim xib-
= J
vl-pr(y) dy
a
new functions
f E M+(a,b)
and introduce for
f(s) =
f(O(s))vpr-1(0(s))
w(z) =
(z))
f
by
, w
,
s E (0,L)
,
,
z E (0,L)
,
(4.27)
we easily obtain b
x
r
(f
L l4 w(x) dx = f(t) dt]
IJ
z [j
f(s) dsj
w(z) dz
J
J
a
a
0
0
and
b
L
r
r
fp(z) dz
fp(x) v(x) dx = J
J
a
0
Therefore, the inequality (4.6) can be transformed into the inequality (Lf
(4.28)
IJ
q N
zr
(J
f(s) ds)
l1/q w(z) dz1 f C
1/p
(Lr
II
fp(z) dz
0 0
which is in fact the inequality from Problem the interval (4.29)
I
0
(0,L)
w(z)
and
1.7
-
see (1.12)
-
on
and with the new weight functions v(z) -
1
.
Obviously, these new weight functions satisfy all assumptions of Theorem 4.1, and,therefore, the following theorem holds: 43
Let
4.7. Theorem.
< p
1
q < - ,
<_
v, w E W(a,b)
(4.24) be satisfied. Let the number
L
and let the condition w
and the weight function
be
defined by (4.26) and (4.27), respectively. Then the Hardy inequality
b
b l/q
dxj1/P
< C
lllu'(x)IP
v(x)
lu(x)Iq w(x) dx]
l
a
a
if and only if there is a number
u E ACL(a,b)
holds for every
A > 0
such
that the differential equation
dd[1ci P
has a solution
satisfying the conditions
y
y' E AC(0,L)
4.8. Remark.
r
J+ w(z) yq/P (z) = 0
y(z) > 0
,
y'(z) >0 for
,
Note that in Theorem 4.7
(i)
z c (0,L)
.
the condition (4.1) was not
mentioned. This is a consequence of the fact that we reduced our problem to an analogous problem for weight functions
v , w
from (4.29) where the
conditions analogous to (4.1) and (4.2) are fulfilled automatically. (ii)
For the same reason, a formula analogous to (4.23) holds for the
numbers l1/q (z l1/P' IJ w(s) dsJ dsJ = II
1/q
(Lr
BL =
sup
0
sup
I/ P
z
P
If
w(s) dsj
0
0 z
z zr
K = q
inf
sup
f
0
I
f(z)
w(s) [f(s) + s]q/p,+1 ds
0
where the infimum is taken over the set of all positive on
(0,L)
,
CL = sup
II
0
J
0
1/q
q
(z
f(s) ds
w(z) dz I
where the supremum is taken over the set of all L
f fP(z) dz = 1 0
44
.
which are
and L
f
f E M+(0,L)
This formula reads
f E M+(0,L)
such that
(4.30)
BL < CL <- K1/q = k(q,p)BL
Using the substitutions (4.25), we can easily show that
(4.3)
CL
-
CL = CL
and
is the best possible constant for which the Hardy inequality
CL
where
K=K
BL = BL ,
(4.31)
and consequently also the inequality (4.6)
holds. Besides,
-
is the best possible constant for which the inequality (4.28) holds. Due to (4.30) and (4.31), we have shown that the estimate (4.23)
(iii)
holds for every pair of weight functions
for
1
v, w E W(a,b)
(and, of course,
< p < q < m ).
5. THE LIMIT VALUES OF THE EXPONENTS
p
,
q
Up to now, we have dealt with
5.1. Convention.
p, q E (1,-)
.
will consider also the case when at least one of the exponents equal
to
1
Now, we p
,
q
is
Therefore, let us introduce the following 'arithmetic'
on
convention:
and, for
00 =
1
x > 0
,
=
0
,
5.2. Some function spaces.
- (-m) _
For
p
>_
1
and
w E W(a,b)
,
let us denote by
Lp(a,b;w)
(5.1)
the weighted Lebesgue space defined as the set of all measurable functions u
on
(a,b)
for which the norm b
(5.2)
Ilullp,
(a,b),w
1/p
[Juxi p w(x)dx]
for
1
ess sup Iu(x)I a<x
for
p = -
is finite. 45
w(x) =
For
LP(a,b)
,
1
we obtain the classical (non-weighted) Lebesgue space
whose norm will be denoted simply by
(5.3)
IIUIIp,(a,b)
Since
belongs to
u
if and only if the product
LP(a,b;w) LP(a,b)
belongs to the Lebesgue space
and
=
IIu'wl/pIIp,(a,b)'
is a Banach space with respect to
LP(a,b;w)
it follows immediately that
IluIIp,(a,b),w
the norm (5.2). (Note that we have used, for
p = - , Convention 5.1.) Now
the Hardy inequality (1.11) as well as the inequality (1.12) can be rewritten in the following form: IIu
q, (a,b),w
u
C
P, (a,bi,v
or
l/q
(5.4)
Iluw
1/n
<
IIq,(a,b)
C
IIp,(a,b)
Ilu v
and
IIHI.f Iq,
(a,b),w
s CLIIf p,(a,b),v
or
(5.5)
IIHLf'wI/q 11q,(a,b) S CLIIfv"/PIIp,(a,b)
,
respectively.
Similarly, the number (5.6)
where
BL =
p' =
BL
from (1.18) can be expressed as follows:
sup a<x
p
p - 1
These relations have been derived for under Convention 5.1 the numbers
BL
p, q - (1,m)
from (5.6) and
AL
.
Nonetheless, from (1.19) as
well as the inequalities (5.4), (5.5) are meaningful even for
p, q E[1'-1
and the assertions of Theorems 1.14 and 1.15 can be extended to the cases 1
<- p = q S -
and
1
= q < p
(1.23) and (1.26), i.e.
BL < CL S k(q,P)BL
46
,
respectively. Moreover, the estimates
q1/q
(L)1/q
C
Al
r
reduce, for
p = 1
or
q = °°
,
q1/q(p,)l/q'A
S
L
L
to the simple identities
BL = CL (cf. Lemma 5.4) and ql/q AL = CL
(cf. Lemma 5.6). [Note that for p = 1 or from (1.24) equals (p'q/r)1/q, =
1
1
the number
k(p,q)
according to Convention 5.1, and similarly
W) 1/q/ =
,
q =
for
1
q =
1
or
p = m .]
Now, we shall show that our considerations are not purely formal. However, we start with an auxiliary lemma.
Let
5.3. Lemma.
w E W(a,b)
.
Then
w(x) < ess sup w(t) a
(5.7)
Proof.
for a.e.
x E (a,b)
Suppose that (5.7) is not true, and denote
I = (a,b)
,
S(x) = ess sup w(t) a
A = {x F I; S(x) < w(x)}
,
An = {x E I; S(x) + n < w(x)} Then the Lebesgue measure exists a,
S E I
n0 E N
such that
a <
,
JAI
I
of
An0 I
such that for
A > 0
nEN
,
A = U An
is positive, .
and there
n= Further, we can choose numbers
An0 =
An0 n [a,s7
we have
An0I > 0
and consequently
0< Let
x
JAn
[
0
-a<
be the characteristic function of the set
g(x) = 1 X(t) dt The function
g
,
is continuous,
An
and define
xER. g(a) = 0
,
g(g) = IAn,I
.
Therefore, 47
there exists a number
g(z) =
such that
z E (a, 8)
n Al .
Putting
0 2
B = (a,z) P,
C = (z,8) f
,
An0
An0
we have IBI = ICI
Since
g
Z IAnOI > 0
=
is nondecreasing on
of points
xk E (a,z)
[a,z]
k = 1,2,...
,
g(xk) = (1 - 2-k)IBI Set
.
,
,
there exists an increasing sequence such that
.
and define
x0 = a
Bk = (xk-l,xk) (l B
kE N
,
Then obviously g(xk-1) = 2-kIBI > 0
IBkl = g(xk)
Bk C An
Since
,
we have
S(x) + n < w(x)
for
x E Bk
0
which together with the fact that the function
S
is nondecreasing on
I
yields
n
S(xk-1) +
< w(x)
,
x `c Bk
,
k E N
0
Consequently, <
S(xk_1) +
n0
ess sup w(x) = S(xk)
yEBk
and, therefore, (5.8)
For
S(x1) + k x e C
and
no
k GIN
1
< S(xk)
for
k (T N
we have
S(xk) + no < S(x) + no < w(x)
n
n
which together with (5.8) implies S(x1) + n
< w(x) 0
Since this inequality holds for every 48
k E N , we have
w(x) = -
for all
x E C ICI
.
>0
However, this contradicts the assumption
w E W(a,b)
,
since
.
5.4. Lemma.
Let
v, w C W(a,b)
p =
Let
.
1
q E [1,']
,
p E (1,m]
q = W ,
or
be defined by (5.6) and Zet
BL
CL
.
Let
be the best pos-
sible constant in (5.5). Then (5.9)
BL = CL
.
Proof. Obviously, it suffices to prove the following two implications: (5.10)
BL <
( 5 . 1 1 )
CL <
CL < BL
j B
C
[If (5.10) and (5.11) hold, then the numbers
BL
and
CL
are either
simultaneously finite (and equal) or simultaneously equal to (i)
The case
p =
1
1
,
< q <
In this case (5.12)
a<x
Assume that
5.13)
,
(
(J
]
x
ess sup v-1(t)J a
Using the inequality (3.2) for
.
q
f (t) dt] w(x) dxJ
1/q
J
a
a
w(t) dt
[[J
r = q
,
0 = w
we obtain
x(
br
(
BL < m
l
1
BL = B(a,b,w,v,q,l)= sup
T = f E M+(a,b)
1/q
b
bj
1/q
w(x) dx
dt I
Lemma 5.3 yields b
f
(b( 1
f(t) IJ w(x) dx]
1/q
dt =
t
a bf
(br
f(x) v(x)
IJ
11/q w(t) dt]
v-1(x) dx
J
a
x
b
b
f a
f(x) v(x)
r l IJ w(t) dtJ
x
1/q Less sup v-1 (0] dx < a
b
f(x) v(x) dx
BL J
a
and from (5.13) we have
b
b
1/q
BL J
If(HLf)q(x) w(x) dx
f(x) v(x) dx
I
a
a
However, in view of (5.5) the last inequality immediately implies that C
L
< B
L
'
The implication (5.10) is proved. Now, assume that
CL < -
Then the inequality (5.5) holds for every
and analogously as in the proof of Lemma 3.4 we obtain
f E M+(a,b)
ll1/q
b
ll
11
f(x) dx] < CL
If w(x) dx]
(5.14)
.
f(x) v(x) dx
If
If
E
with
E E (a,b)
S(x) = ess sup v-1(t) a
(5.15)
Then
(cf. the formula (3.10) for
S(x) > 0
for some
for every
x0 E (a,b)
implies that for
p = 1 ). Denote
.
x E (a,b)
,
since the assumption
leads to a contradiction with E (a,b)
,
S(x0) = 0
v (F W(a,b)
.
This
n E N such that
there exists
0 < SO - 1n (i-1)
(5.16)
Then
Assume in addition that S(x) < m
for every
0 < S(E) - 1 < m n
(5.17)
then
> 0
0< Define 50
.
IM
.
and if we denote
Mn = {x E (a,E);
IMnI
x E (a,b)
v-1(x)
> S(E) - n
l
,
Therefore, there exists a subset
I
<-
.
Mn C Mn
such that
fn (x) = x
(x)
x E (a,b)
,
Then (5.14) and (5.17) yield b
1/q (b
f(x) dx [In a
IJ w(x) dxJ (5.18)
J
a 1
n1
v(x) dx
CL
f(x) v(x) dx =
CL I
CL IMnI
[S(E) -
I
.
Mn Since, moreover, E
fn (x) dx =
J
dx = IMnI
I
,
Mn
a
we immediately conclude from (5.18) that
l1/q
b
CL [S(E) -
dx]
w(x) I I
I1 n
E
n - -
Letting
we obtain 1/q
(b(
l
S(E) s CL
II w(x) dxJ E
which in view of the definition of BL(a,b,w,v,q,l) < CL
(5.19)
(i-2)
Let
BL
-
see (5.12)
yields
.
be a general function from
v
-
W(a,b)
and for
n tN
define vn(x) = v(x) + n , Obviously,
w ,
,
too, and
ess sup vnl(t) < m O
(5.20)
Since
vn E W(a,b)
x E (a,b)
for every
vn(x) ? v(x)
on
(a,b)
with
vn
instead of
vn
(i.e.
,
x E (a,b)
the inequality (5.5) holds for the pair v ) with the same constant
CL
.
More-
over, (5.20) is nothing else than the additional assumption (5.16) for vn , and consequently
we can use all arguments from (i-1) arriving at the in-
equality 51
BL(a,b,w,vn,q,l) ` CL
n - -
as an analogue of (5.19). Letting arbitrary (ii)
v E W(a,b)
we again obtain (5.19) for an
The implication (5.11) is proved.
.
p =
The case
,
1
q = -
,
.
In this case (5.21)
BL = BL(a,b,w,v,°°,1) =
,v-1II
sup
(x,b).
a<x
a<x
ess sup v-1(t) = ess sup v-1(t) a
f ` M+(a,b)
and since for
x
b
w1/glq,(a,b) = IIHLfIm,(a,b) = ess sup J f(t) dt = a < x
HLf
a
f(x) dx JJJ
a
the inequality (5.5) has the form
(5.22)
b(
b(
J
f(x) dx < CL J f(x) v(x) dx
a
a
Assume that
BL < =
b
.
Lemma 5.3 yields
b
r
f (x) dx =
( J
f (x) V (X)
v-1
(x) dx <
J
a
a b(
b(
<
J
f(x) v(x)
a
[ess sup v-(t)] dx a
which in view of (5.22) leads to the inequality
BL
J
f(x) v(x) dx
a CL < BL
and the implica-
tion (5.10) is proved.
The proof of the implication (5.11) is completely analogous to that of the case (i). (iii)
The case
q = m
,
1
In this case (5.23)
52
BL - BL(a,b,w,v,°°,p) _
< p <
b (r a<sx
v
IJ
1-
'
p (t) dtj
1/p' =
II
l
p,
(t) dt v ,-P, (t)
ll
a
a and the inequality (5.5) has the form
((
(5.24)
f(x) dx <_ CLIJ fp(x) v(x) dx)
J
J
a
a BL < W . Then Holder's inequality yields
Assume that
b
b
(
(
f(x) dx =
J
a
f(x) vl/p(x) v-1/p(x) dx
J
a
(b(
y
f
(x) v(x) dxl
1/p
b IJ
v
1-p'
1/p'
(x) dxJ
)))
a
a b
11/p
= BLIJ fp(x) v(x) dx a
and in view of (5.24) we have Assume that
CL <
supp f C (a,E)
that
J
(5.25)
f(x) dx
.
-
°°
Let
.
CL -< BL
.
E ` (a,b)
and let
f E M+(a,b)
be such
Then the inequality (5.24) has the form
CLIJ
l1/p fp(x) v(x) dx]
a
a Putting
vl-p (x)
for
x E (a,)
for
x L [E,b)
,
we obtain
f(x) dx =
(5.26)
f vl-p (x) dx
J
a
a
(iii-1)
;
vl-p (x) dx
fp(x) v(x) dx =
.
a
Assume in addition that
53
x
for every
vl p/(t) dt < m
x E (a,b)
1
a
Using (5.26) in (5.25) we obtain l1/p' dtJ
vl-p (t)
a
< CL
JJ
which immediately yields the inequality
by using the functions
vn(x)
CL
, we obtain the inequality
v E W(a,b)
For general
(iii-2)
BL
BL
CL
from (3.12) and proceeding analogously as
in the case (i). (iv)
The case
In this case
x
BL = BL(a,b,w,v,°°,°°) =
(5.27)
sup a<x
Ii ( dt] = b - a a
and the inequality (5.5) has the form b (5.28)
f(x) dx
J
CL ess sup f(x) a<x
a
Assume that
BL < -
Then for
.
f E M+(a,b)
,
b
1 f (x) dx < (b - a) a and
CL < BL
ess sup f(x) = BL ess sup f(x) a<x
in view of (5.28).
Assume that
CL < m
.
Using (5.28) for
(b - a) < CL ess sup 1
f(x) =
1
,
we obtain
= CL
a<x
The investigation of the Hardy inequality is reasonable as BL = b - a far as the best possible constant CL is finite. The number 5.5. Remark.
54
from the case (a,b)
p = q = m
-
see (5.27)
-
is unbounded, and consequently also
is infinite if the interval in this case. This is
CL = -
w
caused by the fact that the weight functions
and
p = q
formulas (5.5) and (5.6): due to Convention 5.1, for
wl/q
= w0 =
,
1
vl/P
=
v- 1/p = v0
=
do not appear in
v
we have
1
Consequently, it seems not to be reasonable to investigate the Hardy inequality for
p = q = -
on unbounded intervals. This apparent discre-
pancy can be avoided if instead of the inequality (1.12) we consider the following inequality: br
(5.29)
[(Hf)(X)
ll 1/ q w(x) dx]
b
l I/ P
CL[J fp(x) v(x) dx]
a
a
with w(x) = wq(x)
,
v(x) = vp(x)
This approach looks very formal; in fact we slightly modify the definition of the weighted Lebesgue space:
f E LP(a,b;w) <
c-Lp(a,b)
Nevertheless, this approach has the following consequences: The number
BL
associated with the inequality (5.29) has the form BL = BL(a,b,w,v,q,P) =
sup
q,(x,b)
a<x
sup a<x
llwll(x,b) q,
11v
v-1/p
Iwl/qji
p',(a,x)
BL
p',(a,x)
'
and this expression contains the weight functions Let us note that the finiteness of
'l
w ,
v
even if
p = q
again fully characterizes the
validity of the inequality (5.29). This can be shown by the methods used in the foregoing subsections. See also J.
S. BRADLEY [11, V. C. MAZ'JA [1].
Now we shall consider the limit cases corresponding to the Hardy inequality for p > q
.
Let us recall that in this case an important role was
played by the number (5.30)
AL = AL(a,b,w,v,q,p) =
55
J
[
1
_
vl-p'(t) dtll/q'Ir vl-p' (x) dx
dtI1/q
w(t)
If
[J
where 1
__
1
(5.31) r
p
q
Now we extend the formula (5.30) also to the limit values or
p = m
q E [1,°)
,
Let
5.6. Lemma. v, w c- W(a,b)
.
q =
Let
q =
1
,
p E(1,m)
using again Convention 5.1.
,
1
or
p E (1,-)
,
p = -
q E [1,-)
,
be defined by (5.30) and Let
AL
CL
.
Let
be the best
possible constant in (5.5). Then ql/q AL = CL
(5.32)
.
Analogously as in Lemma 5.4, it suffices to prove the following two
Proof.
implications: q1/qAL
(5.33)
AL < .
CL <
(5.34)
CL < -
q1/qAL <
CL
(cf. (5.10), (5.11)).
The case
(i)
In this case
q =
1
q' = -
,
p E (1,W)
,
r = p'
and
b r
AL = AL(a,b,w,v,1,p)
(5.35)
.
p,
(b I
J
a
Assume that (with
p
,
AL < -
.
11/p'
w(t) dtvl p (x) dxl
[
x
Using the Fubini theorem and Holder's inequality
p' ), we obtain for
f E M+(a,b)
b J
If
a
a
f(y) dy I w(x) dx =
f(y)
p
[J w(x) dx] a
dy
y
1/p
fp(y) v(y) dyJ
56
w(x) dx I
a
Ibf
a
lj
J
y
1/pi
r
vl-p' (y)
dyJ
b lll/P
= ALIJ f
P(y) v(y) dy]
a which is the inequality
(a, b) As
< ALl1fv1/Pllp,
(a,b)
is the best possible constant in (5.5), we have
CL
C
L
L
which is (5.33) since
q1/q =
Now assume that
CL < -
1
.
Analogously as in the proof of Lemma 3.4
.
we obtain b
l/p,
vl-p
(f
r
w(x) dx
l
(x) dx
IJ
J
a
E
for every
E (a,b)
.
Therefore, it is meaningful to define 1/(P-1)
(
fn
(5.36)
< CL J
,
v1-p (x) X
(x) = II w(t) dt]
lJ x for
n E N , where
an 1 a
bn t b
,
{an}
,
(in fact,
(an,bn)
(x)
are two sequences of real numbers such that
{b n}
is the function from (3.26) for
fn
q =
1
).
Denoting b
A
n
=
n
(3.30)
-
l
IJ w(t)
1
an
we obtain
b
(f
1/ P
,
,
v1 p,(x) dx]
dtJp
x
analogously as in the proof of Lemma 3.9, cf. (3.27), (3.29), that
-
bn (5.37)
0
fn (x) v(x) dx = An
<
<
an
In view of (5.36), the Fubini theorem yields b
br
w(x) dx = J
a
a
xJ(HLfn)(x)
If
fn(t) dt
w(x) dx = I
a
57
b
b fn(t)
l w(x) dxJ dt =
f
I
J
a
bn
bl ,
If
w(x) dxjp
vl-p/ (t) dt = Any
J
an t
t
This inequality together with (5.37), used in the inequality (5.5), yields Ap,/p Ap/ < C n L n and consequently A
<_
n
Letting
n - -
(ii)
C
L
,
q1/q = 1
we obtain (5.34) (again
The case
< q < p = -
1
In this case,
p' = 1
.
r = q
,
and b(
AL(a,b,w,v,q,)
).
x(
b(
dt]]1/
l
=
[J
al x
[w(t) dt]
l [J
(5.38) dx]l/q
(( = [J(x - a)q-1
IJ
w(t) dt`
b1 / a
x
The inequality (5.5) has the form
(5.39)
q
[J(Hf)(x) w(x) dxJ
s CL ess sup f(x) a<x
a
Assume that
AL <
Using (3.22) and (5.38), we obtain for
that
f t; M+(a,b) (b(
lJ
a
1/q
(HLf)q(x) w(x)
= q 1/q
rb( (y(
IJ
q_1
f(t)
b(
f(y)
dt}
IJ
l
l/q
t
w(x) dx] dy]
J
a
a
y
b
< q1/q ess sup f(x) a<x
58
IJ
ql/q AL ess sup f(x) a<x
(b
l IJ w(x) dxl l
JJJ
yq1/ q If
dt]
dy]
and a comparison with (5.39) yields CL < ql/q AL
Assume that
proof, and for
CL < m
Let
.
x e (a,b)
an
,
bn
be the numbers from part (i) of the
define
fn(x) = X(a ,b ) (x) n n
nEN.
,
Using again the formula (3.22) and the inequality (5.39) for the function f
,
n
we obtain br
11/q
1J(HLfn)q(x) w(x) dx a b
= q
_1
y
b
1/q
q
1/q
If f dt I
X(an,bn)(Y)
an
a
w(x) dx]
dy]
lJ
y
x CL ess sup fn(x) = CL a<x
n -+ -
the monotone convergence theorem yields w(t) dt l
dtIq-1
q1/q
dy]l/q
=
ql/q
Cj
a
lj y
[I
AL < CL
J
a
and (5.34) is proved. 11
5.7. The equivalence of the inequalities (5.4) and (5.5) in the limit cases. In Sections 1 and 3 we have shown that Problem 1.5 is equivalent to Problem 1.7 provided
'1
< p,q < -
(see Lemma 1.10, Remark 3.7). In this section,
we have extended Problem 1.7 (i) to the limit values of
p
,
q
,
rewriting
the inequality (1.12) in the form (5.5): (5.5)
q, (a, b)
CLIIfv1/P p, (a, b)
Analogously we can extend Problem 1.5, i.e. (5.40)
for
IIuw1/gllq,(a,b)
u e ACL(a,b)
-
the Hardy inequality (1.11)
CLIIu'v1/P
IP,(a,b)
(see (5.4)). 59
In this section we have dealt up to now with the inequality (5.5), with the (extended) Problem 1.7. Therefore we should investigate
i.e.
whether its equivalence with Problem 1.5 occurs also in the limit cases. An important role in the equivalence proofs was played by the condition
x (5.41)
vl-p'(t) dt <
J
a -
cf. (1.14). This condition can be rewritten in the form
llv-lipll
(5.42)
< m p',(a,x)
which is meaningful for
[1,-]
p
.
In particular, for
p =
1
it assumes
the form ess sup v- l(t) a
(5.43)
<
Since the proof of the equivalence in the limit cases uses almost literally the arguments of the proofs of Lemma 1.10 and Lemma 3.4, we will give it here only for one case:
p= (i)
1
Assume
5.4, the number
q E [1,m)
,
.
that the inequality (5.5) holds. Then, according to Lemma BL
(5.43) is fulfilled.
from (5.12) is finite, and consequently For
u E ACL(a,b)
the condition
denote
bJu'xJ J =
v(x) dx a
If
that
J = J
then the inequality (5.40) holds trivially. Therefore, assume
,
is finite. Then
J Iu'(t)Idt = a
JIu'(t)I v(t) v-1(t) dt a
and the inequality (5.40) follows from (5.5) with arguments as in the part (ii)
60
sup v- l(t) a
<
by the same
(i) of the proof of Lemma 1.10.
Assume that the inequality (5.40) holds for
u E ACL(a,b)
and
define
v(x) = v(x) + 1n n vn(x) '_ v(x)
Since
nEN.
,
and (5.40) holds, we obtain
luwl/q
(5.44)
x E (a,b)
,
s C Lllu vn 1,(a,b)
q,(a,b)
for every
u E ACL(a,b)
(5.45)
.
Let us show that then the inequality
S CLl
(a,b) f E M+(a,b)
holds for every
.
fvnlll,(a,b)
Denote
b(
in = J
f(x) vn(x) dx
a If
Jn = w
that
J
n
than the inequality (5.45) holds trivially.
,
< -
.
Therefore assume
Similarly as in the point (i) above, we obtain
x
1(t) <
f(t) dt = J
J
a
a
n
n
<
and consequently, the function
x u(x) =
J
f(t) dt
a
belongs to
ACL(a,b)
.
For this function we obtain the inequality (5.45)
from (5.44).
Denoting by
BL(n)
the number
BL(a,b,w,vnoq,l)
and by
CL(n)
the
best possible constant in (5.45), we have from Lemma 5.4 that BL(n) = CL(n) S CL Since
BL(n) -+ BL = BL(a,b,w,v,q,1)
= CL < m
.
for
n -+
In view of Lemma 5.4, the finiteness of
,
we have that BL
BL <
implies that the
inequality (5.5) holds, too.
The other limit cases can be investigated in an analogous manner.
5.8. Summary.
In the foregoing subsections, we have shown in fact that the 61
assertions of Theorems 1.14 and 1.15 can be extended to the limit values of p
and
q
provided we consider the Hardy inequality (1.11) in the form and
BL
(5.4) [(5.40)] and the numbers
AL
in the forms (5.6) and (5.30).
For the convenience of the reader, we will list the forms of the Hardy inequality as well as the formulas for the best possible constants the
for the individual cases.
CL <
(i.e. (i)
(bll1/q
p =
1
<1,")
q !E
,
w(x) dx]
b(
< CL
u'(x)I v(x) dx 1
a
a
`1/q
(br
BL =
CL
1IJ w(t) dt]
sup
a<x
(ii) p= 1
l
x
ess sup v-1(t)] a
q=
,
(b
ess sup Iu(x)I
_<
CLllll IIu'(x)I v(x)
a<x
dx]
a
CL = BL = ess sup v-1(x) a<x
(iii) p e (1,°°)
,
<
q=1/p ((b
ess sup lu(x)I a<x
<
CL{J1u'(x)Ip v(x) dx] l
a
1/p r
b v1-p
i
< 0
(t) dt
CL = BL =
I a
(iv)
p=q=m ess sup lu(x)I a<x
_<_
C 1
CL = BL = b - a < m (v)
62
q =
1
,
and
sufficient conditions of the validity of the inequality
and
necessary
CL
P E (1,m)
ess sup Iu'(x)I a<x
< W
.
ll
((b(
1/p
w(x) dx < CLIJIu'(x)Ip v(x) dx] a
bbr CL = AL = [j[
,
(J w(t) dt) p
1
vl-p'(x)
p'
dxl
x
<
111
a
(vi)
q E El,-)
,
p=-
b
ll 1/q <- CL ess sup lu'(x)I
lu(x)lq w(x) dxJ
a < x
a b(
CL
ql/qAL = ql/q
1/ q
[fx
w(t) dt] dx]
LJ(x - a)q-1
a
<
a
Theorems 1.14 and 1.15 together with the results listed in Subsection 5.8 enable us to formulate the following two more general theorem:
5.9. Theorem.
(5.46)
Let
1
< p < q < m ,
v, w E W(a,b)
FL(x) = FL(x;a)b,w,v,q,P) =
.
Define
Ilwl/gllq,(x,b)*lly-1/Pllp',(a,x)
and (5.47)
BL = BL(a,b,w,v,q,P) =
sup FL(x) a<x
.
Then the Hardy inequality (5.48)
CLllu'vl/Pllp,(a,b)
lluwl/gllq,(a,b)
holds for every
u E ACL(a,b) [and, consequently, the inequality 1/q
(5.49)
1/p
<
Iq,(a,b)
holds for every (5.50)
`
f E M+(a,b)]
CLllf°
llP,(a,b)
if and only if
BL = BL(a,b,w,v,q,P) < m
Moreover, the best possible constant
CL
in (5.48) [in
(5.49)] satis-
fies the estimate
63
BL < CL < k(q,p)BL
(5.51)
where 1/q
k(q,p)
(5.52)
(5.53)
1 < q < p < m
Let
5.10. Theorem.
1/p
,
(1 + q
(1 + P )
v, w E W(a,b)
,
.
Define
AL = AL(a,b,w,v,q,p) =
lJ
l
a
dtl
w(t)
l/q
(x
vl-p' (t) dt 1/q /J
j
x
1/r
v1-p' (x) dx
a
where 1
_
r
1
1
q
p
Then the inequality (5.48) holds for every the inequality (5.49) holds for every (5.54)
AL = AL(a,b,w,v,q,p) < W
u E ACL(a,b)
f e M+(a,b)
Land, consequently,
] if and only if
.
Moreover, the best possible constant
CL
in (5.48)
L in
(5.49)] satis-
fies the estimate (5.55)
ql/q
l/q A
5.11. Formulas connecting
ql/q(p,)1/q'
C
L
r
v
A-L
L
and
w
.
formulas connecting the weight function -
In Subsection 2.6 we introduced w ,
v from (1.11) for
1< p < q<
cf. (2.13), (2.15). The main tool for the proof was Theorem 1.14. Theorem 5.9, the extended version of Theorem 1.14, allows us to extend
at least one of the formulas just mentioned to the case Assuming
w e W(a,b)
and
b(
(5.56)
J
w(t) dt < m
,
x
and putting for
64
x E (a,b)
1
< p < q <
1-P
v(x) _ q
(5.57)
_
p-1+p/q
(br
w1 1(x)
If w(t) dt I
(we use Convention 5.1! ), we can show that the inequality (5.48) holds
for every
u E ACL(a,b)
with the constant
CL = k(q,p)
6. FUNCTIONS VANISHING AT THE RIGHT ENDPOINT. 6.1.
from (5.52).
EXAMPLES
Up to now, we have investigated the Hardy inequality (1.11) for
functions
u E ACL(a,b)
,
i.e.
the Problem 1.5 (i), which is the same as
to consider the inequality (1.12) from Problem 1.7. In Remark 1.8 we have shown that the Hardy inequality (1.11) for functions
u E ACR(a,b)
,
i.e.
the inequality (1.13) from Problem 1.7,can be reduced to the former case.
Therefore, let us now shortly summarize the results concerning Problem 1.5 (ii). The following two theorems are analogues of Theorem 1.14 and Theorem 1.15. extended to the limit values of
p
,
q
in the sense of
Section 5. Note that we use Convention 5.1 and the spaces introduced in Subsection 5.2.
6.2. Theorem.
Let
1 < p < q < -
v, w E W(a,b)
,
1/q
FR(x) = FR(x,a,b,w,v,q,P) =
Define lq,(a,x)*lly-1/pIIpr,(x,b)
lw
FR(x)
sup
BR = BR(a,b,w,v,q,P) =
.
a<x
Then the Hardy inequality (6.3)
lluwl/q
q,(a,b)
holds for every (6.4)
with
CRurv
u E ACR(a,b)
1/p
p,(a,b)
[and, consequently, the inegauZity
CRllfvl/Pllp,(a,b)
HR
from (1.6) holds for every BR = BR(a.b,w,v,q,P) < ,
f E M+(a,b)
7 if and only if
.
65
Moreover, the best possible constant
CR
in (6.3) [and in (6.4)] satisfies
the estimate BR ` C R < k(q,p)BR
(6.5)
where
k(q,p) _ ( 1 + P ) 1/q ( 1 + q l p
(6.6)
Let
6.3. Theorem.
1
< q < p
<_ W
v, w ` W(a,b)
,
. Define
AR = AR(a,b,w,v,q,p) _
(6.7)
[bj
dtll/q
x w(t) b
[
vl-p'(t) dt]l/q'Jr vl-p'(x) dx
lJ
a
a
x
J
where 1
1
__
(6.8) r
_
1
p
q
Then the Hardy inequality (6.3) holds for every
u C ACR(a,b)
sequently, the inequality (6.4) holds for every
f E M+(a,b) ] if and only
[and con-
if AR = AR(a,b,w,v,q,p) < W Moreover, the best possible constant
CR
in (6.3) [and in (6.4)] satisfies
the estimate q1/q (p'g) 1
(6.9)
q
r
< C R
<
q1/q(p,)1/q'
R
Similarly, we can extend to
the approach via the for-
u c ACR(a,b)
mulas connecting the weight functions
w
and
v
(see Subsections 2.6 and
5.11):
6.4. Theorem.
(i)
Let one of the following two assertions be fulfilled:
1< p
q < m, v E W(a,b)
b
(6.10)
J
x
66
vl-p'(t) dt <
,
and
b
(6.11)
w(x)
vl-p/ (x)
=
P for
x E (a,b)
vl-p' (t) dt,
-1-q/p'
If
x
;
(ii) 1< p
q<m
w E W(a,b)
,
x(
(6.10*)
w(t) dt < -
J
a
and 1-p
(6.11*)
v(x) = q
xp-1+ /4
1
w -p(x)
If
w(t) dt
p I
a
for
x E (a,b)
.
Then the inequality (6.3) holds for every constant
CR = k(q,p)
u e ACR(a,b)
with the
from (6.6).
As concerns the approach via the differential equations, the following analogue of Theorem 4.1 holds:
Let
6.5. Theorem. (6.12)
1
< p < q < w
v, w E W(a,b)
;
. Moreover, assume that
v E AC(a,b)
and b(
(6.13)
J
vl-p'(t) dt <
°°
for
x e (a,b)
x
Then the Hardy inequality (6.3) holds for every (finite) constant
CR
u E ACR(a,b)
if and only if there is a number
with a
A > 0
such that
the differential equation (6.14)
A
[vq/p(x)(-
d-
has a solution (6.15)
y
(i)
d
r
- w(x)
yq/p
= 0
satisfying the conditions
y' E AC(a,b)
6.6. Remarks.
)q/ P
,
y(x) >0 ,
y'(x) <0 for
x E (a,b)
.
The restrictive condition (6.12) can be removed in the 67
same way as in Section 4 (ii)
cf. Remark 4.6 (ii) and Theorem 4.7.
-
If we introduce the number b
L
b
q/P'+1
i
KR = q inf
supI,
w(t) [f(t) +
`
f(x)
dt
J
x
t
'
from (4.11), we can again derive the estimates
K
which is an analogue of
vl-p(s) dsI
BR < CR :5 KR/q <= k(q,P)BR
For illustration, we will now give several examples in which we will consider particular weight functions
w , v
,
conditions of the validity of
the Hardy inequality with these weights and, sometimes, also estimates for the best possible constant. Although it would be useful to know the exact values of the constants
BL
BR
,
,
AL
AR
,
(which would make it possible
to obtain good estimates for the best possible constants in the Hardy in-
equality - see, e.g., (6.5) and (6.9)), in most cases we will give only necessary and sufficient conditions for their finiteness.
6.7. Example.
Let us consider
p, q E: [l,-)
u(x)Iq xdxl [Ti 0
Then we have
l1/p
(
ll1/q
(6.16)
C
(u'(x)Idxl 0
111111
(a,b) = (0,m)
JJJ
w(x) = xa
,
p = q
If we consider the case
and the inequality
,
,
v(x) = x8
a, aE R
,
then the numbers
BL
from
BR
and
(1.18) (or (5.47)) and (6.2), respectively, are given by the formulas B L
= (
(6.17)
1/q
q
(p
- 1) 1/P'(p - 1 -
S)-(1/q + 1/P')
for
S
<
p-
1
for
S >
p -
1
(q)l/q
(p - 1)1/P' (6 + 1 - p)-(1/q + 1/P')
BR =
(recall Convention 5.1).
If we consider the case
p > q
both infinite,and consequently
,
68
if
1
= q < P <
°°
.
AL
and
AR
are
we can assert that the inequality (6.16)
does not hold (with a finite constant ACR(O,W)
then the numbers
C ) on the class
ACL(0,-)
or
Therefore, it is natural to consider (6.16)
only for <- p =< q < .
I
;
then the inequality (6.16) holds (i)
for
u e ACL(O,m)
B < p - 1
(6.18)
(ii) for
The constant
a= S p- p -
C
1
,
if and only if
u E ACR(0,=)
1, a= B p- p -
B> p
(6.19)
,
if and only if
1
can be chosen in the form
C = k(q,p)B k(q,p)
where
is defined in (5.52) and
For
6.8. Example.
p, q E [1,-)
B
0 < b < -
,
BL
or
and
a,
is
from (6.17).
BR
6 E R
consider the
inequality b
(6.20)
Jlu(x)lq xo' dxllJq _ 0
xs dxll/p
C[[IU'(XP 0
JJ1
)
This inequality holds (i)
u E ACL(O,b)
for
if and only if one of the following two con-
ditions is fulfilled:
(6.21)
1=p`q<
p-
-
(6.22) (ii)
1fq
8
,
u e ACR(0,b)
,
1
R
p
P
if and only if one of the following two con-
ditions is fulfilled:
(6.23)
1
< p < q < °° and
S
>p-
1
,
a
>_
s p
or
Bfp-
1
,
a> - 1,
P,
-
1
69
(6.24)
1q
B> p - 1,
and
a > B g p
or
B
p -
<=
1
a > -
,
-
1
p
1
.
[Let us emphasize the influence of the finiteness of the right endpoint
in comparison with Example 6.7, the set of admissible values of
a
,
B
b
:
is
substantially richer.
For
6.9. Example.
p, q E [1,r)
0 < a < -
,
and
B E R consider the
a,
inequality W (6.25)
((
{ju(x)I
qx a
l1/p
1/q
< CIJIu'(x)IP x
dx
B
dx]
I
a
a This inequality holds (i)
if and only if one of the following two condi-
u E ACL(a,-)
for
tions is fulfilled:
(6.26)
q < m and
1
or
(6.27)
p -
1
1 =q< p <m and B< p -
1
or
(ii)
for
u E ACR(a,m)
g
B < p - 1, a < B B
>=
6 ' p -
1
p
a < -
,
1
-
-
p
1
,
a
,
1
;
if and only if one of the following two con-
ditions is fulfilled:
< p < q <, B> p - 1, a< B
(6.28)
1
(6.29)
1= q < p < m
6.10. Example.
For
,
B> p - 1,
p, q E [1,-)
,
a<
- 1,
P-
g p
P -
RP,
1
a, B E R consider the inequality
1r
1
(6.30)
1/p
ll1/q
Iln xla dxJ
11r
<
C[Jlu'(x)IP xP-1 Iln xl6 dx
0 0
This inequality holds (i)
70
for
u E ACL(0,1)
if and only if
(6.31)
for
(ii)
(6.32)
B> p_ 1,
epSq<
1
u E ACR(0,1)
p_q<
1
6 1- g. - 1;
a
if and only if
B< p- 1, a= 6
' ,
P
P
P-
p
-
1
It can be easily shown that similarly as in Example 6.7, in the case the corresponding numbers
AL
and
are infinite. The constant
AR
(6.30) can be chosen as in this example, i.e. or
C = k(q,p)B
with
p > q in
C
B = BL
from (6.17).
B = BR
For the special case
p = q E (1,m)
and
a = B - p
,
the inequality
(6.30) was investigated by J. KADLEC, A. KUFNER [1], in the early sixties.
6.11. Remark.
(i)
It can be easily shown that the inequality
q
}}
(
P
(rrr
(fJu(x)q x (ln x)a dxJl
(6.33)
CIJIu'(W )IP xp-1 (In x)B
dxl/ 1
1
1
can be transformed into the inequality (6.30) for the function = u(1/x)
ACR(1,-)
class
( ACL(1,-)
)
dity of (6.30) on the class
are the same as the conditions for the valiACL(0,1)
( AC (011) ). R
Here it is important that we consider the interval
(ii)
the interval
(1,=)
.
w t W(O,=)
(0,1)
or
If we investigate the corresponding Hardy inequality
for instance on the interval function
u(x) _
Consequently, the conditions for the validity of (6.33) on the
.
and ask whether there is a weight
(0,m)
[or a weight function
v c-: W(0,-)]
such that the
1
1/P
inequality l1/q
(6.34)
(mr
{fIu(x)I q w (x) dxl
CJIu'(x)IP xP
Iln x1 6 dxJ
l0
JJ
0
[or the inequality 1/q
(f
(6.35)
l
1/p
11
Iu(x) q x
In xla dx
CIJ Iu'(x)IP v(x) dx] J
0
holds for every
0
u E ACL(0,m)
or for every
u E ACR(O,=)
provided 71
< p < q < -
then the answer is negative: it can be easily shown that the corresponding numbers BL BR are infinite for any choice of w or 1
,
'
v
,
respectively. On the other hand, the answer is affirmative if we con-
sider the inequality (6.34) or (6.35) on the narrower class of functions ACLR(O,m) = ACL(0,=)(` ACR(O,-) w(x) = x Iln
We can choose
- P -
a = B
xp-1
v(x) =
or
xlB
Iln
with
B > p -
1
,
see Section 8, Example 8.6 (v).
-
1
xla
.
p
p, qE [1,")
For
6.12. Example.
and
a,
B E R consider the inequality
1/q (6.36)
[Jlu(x) lq eax dxj
Bx
Jlu'(x)lp e
< CI
l
dxj
This inequality holds (i)
(6.37) (ii)
(6.38)
u E ACL(-°°,m)
for
1
p
`-
if and only if
B< 0
<= q<
u E ACRif and only if
for
1 <_p<
0
The constant
a= B gp
,
C
a = B 9
,
.
p
in (6.36) can be chosen in the form
=
1)1/P'(-8)-(1/q+1/p' )
(p ()1/q q
B
L
B
l/q R = (q)
(p -
p, q
1,-)
C = k(q,p)B
where
,
for a < 0
B =
6.13. Example.
For
(6.39))l Jux aq11/q dx [ this inequality appears
-
for
1)1/p/ 6-(1/q + 1/p' )
and
C
l
a,
B >
0
.
B E R consider the inequality
Jlu'(x)I peBx 2
p = q = 2
for
,
dx
a = B > 0
) 1/P
-
in F. TREVES [1]
(see L. HORMANDER [11, p. 182). The investigation of the validity of this
72
inequality on the class ACLor ACR(-°°,m)
leads to the calculation
of the integrals
x
x eat2
eat2
dt
J
,
dt
,
_m BL
which appear in the definition of the numbers a > 0
a < 0
(6.40)
or
8 5 0
dt ,
f
x
x
bers are infinite if
e-Bt2/(P-1)
(e-St2/(P-1) dt ,
,
BR , AL
,
AR. These num-
Consequently, the condition
.
g > 0
,
is necessary for the validity of (6.39) on the classes mentioned. The condition
mentioned above seems to contradict our necessary condition
a = 8
(6.40). But,in fact, TREVES investigated the inequality (6.39) on the more
special class C0defined in Subsection 7.11. We will resume the study of this inequality in Section 8.
7. COMPACTNESS OF THE OPERATORS
HL
AND
HR
For two Banach spaces
7.1. Notation and some auxiliary results.
X
,
Y
we denote by [X,Y]
(7.1)
K[X,Y]
or
the set of all linear mappings from
into
X
which are continuous or
Y
compact, respectively. If
X(_ Y
(7.2)
,
then the symbols
X(,, Y
and
X c (, Y
denote that the identity mapping [X,Y]
and
K[X,Y]
,
I
lu = u
,
X
u E X
,
belongs to
respectively. We will say that the imbedding
continuous (compact) or that the space bedded into
for
I
is
is continuously (compactly) im-
X
.
The symbol
(7.3)
un - u
will denote the weak convergence of
u
n
to
u
(in
X
The symbol 73
X = Y
(7.4)
and
T = [X,Y]
Finally, if
and
X
will denote that the spaces
Y
X
are isometrically isomorphic.
,
Y
T
,
are the dual spaces to
X
,
Y
then T*
will denote the adjoint operator to
acting from
Y*
X
into
We will use the following assertions whose proofs can be found, e.g., in N. DUNFORD, J. T. SCHWARTZ [1]. (i)
If
T E [X,Y]
(iii)
T E [X,Y]
Let Let
Tu
then
and
T* E [Y*,X*]
= II T*
T II
(ii)
,
.
T E K[X,Y]
Then ,
if and only if
T E K[X,Y]
{un} C X
,
un - u
.
T* C- K[Y*,X*].
Then
n -+ Tu
We will work mainly with the weighted Lebesgue spaces introduced in Subsection 5.2. For
1
and
< p < -
LP(a,b;v)
v E W(a,b)
,
the mapping
defined by
(P
@(u) = uv1/P
(7.5)
is obviously an isometric isomorphism of
LP(a,b;v)
into
r
simultaneously an isometric isomorphism of
LP(a,b)
,
r
LP (a,b)
into
LP (a,b;v
and 1_
r
P ).
This fact together with Riesz' representation theorem leads to the following assertion: [LP(a,b;v)]*
(iv)
Let
g E LP
< p < W ,
1
G E
.
Then there exists an element
such that
(a,b;v1-P
)
b
G(u) = Jg() u(x) dx for every u e LP(a,b;v) a Moreover, G II
=
,
Ilg
.
p,,(a,b),v1-P Consequently, (7.6) 74
[LP(a,h;v)]*
--
Lp'(a,b;v 1-p?
Further, the following two assertions will be used: (v)
(R. A. ADAMS [1] , Theorem 2.21) is precompact in
S C LP(a,b)
there exists a number such that for every
LP(a,b)
1 <
p <
.
A bounded set
if and only if for every
and a closed interval
6 > 0
h E R with
and every
u E S
Let
c > 0
G = [c,d] C (a,b) Ihl <
6
we have
b
1+ h) - u(x)dx
(7.7)
Ep
a and u(x)Ip dx < cp
(7.8)
u(x) = 0
(We define
for
.
x tt (a,b)
.)
(N. DUNFORD, J. T. SCHWARTZ [1], Theorem IV.8.7)
(vi)
{un} C LP(a,b)
The sequence
converges weakly to
Let
u E LP(a,b)
1 < p < - .
if and only
if the following two conditions are fulfilled: (a)
sup
< W
un
(S)
I
;
p,(a,b)
n
un(t) dt -
for every measurable subset M C (a,b)
u(t) dt J
M
M
bll
7.2. Introduction. ACL(a,b)
or on
Let us note, that the Hardy inequality (1.11) (on
ACR(a,b)
[JITf(x)I q
(7.9)
)
is equivalent to the inequality
1/q
w(x) dxj
CrJIf(x)lp v(x) dxj xj
a with
Tf = HLf
or
Tf = HRf
l1/P
b(( <_
a ( HR
and
HL
are defined by the formulas
(1.6)). The inequality (7.9) means in fact that (7.10)
T E [LP(a,b;v), Lq(a,b;w)]
.
According to our foregoing results, the (necessary and sufficient) condition for (7.10), i.e.
for the continuity of the operator
HL
or
HR , reads as
follows:
75
BL <
or
BR < W
AL <
or
AR < m
(7.11)
if if
1
< p <= q 5 °°
1
s q < p S °°
,
Therefore, it is natural to ask what are the additional conditions guaranteeing the compactness of the operator
HL
or
HR .
The corresponding results will be formulated in Theorems 7.3, 7.4 and 7.6. These results are due to V. D. STEPANOV C1],
12] who in fact con-
sidered a little more general operator. The proofs given here are slight
modifications of his proofs.
We will need some information about the operator
adjoint to
T*
T
from (7.10). According to the assertions (i) and (iv) from Subsection 7.1 (cf. (7.6)), it is Lp'(a,b;v1-p')l
[Lq'(a,b;w1-q'),
T* E
(7.12)
Moreover, it can be easily shown that b(
(H*f) (x) =
f (t)
dt = (HRf) (x)
f(t)
dt = (HLf)(x)
J
x (7.13) x(
(H*Rf)(x) =
J
a
Let
7.3. Theorem.
1
< p ` q <
°°
,
v, w E W(a,b)
.
Let
FL(x)
and
BL
be defined by the formulas (1.17) and (1.18). Then the operator Lp(a,b;v) -- Lq(a,b;w)
HL
(7.14)
is compact if and only if BL = BL(a,b,w,v,9,P) <
(i)
(ii)
lira
x+a+
FL(x) = Jim FL(x) = 0 xabHL
be compact. Then
HL
is continuous and consequently
Proof.
(I)
BL <
(see Subsection 7.2). Now, we will show that
Let
lira
x+a+ us choose a fixed
76
C (a,b)
and define
FL(x) = 0
.
Let
vl-p'
for a < x < C x
(x)
f (x) = 0
, .
Obviously, E
P,(a,b),v =
1/ p
vl-P'(t) dt lJ
.
I
a we have
Denoting z = fE/IIfEIIp,(a,b),v (7.15)
zE - 0
'
for
LP(a,b;v)
in
C --
a+
.
Indeed, according to the assertion 7.1 (vi),it suffices to verify the condition (s), since (a) is fulfilled
M be a measurable subset of
zE(x) dx = J M
IJ
-
(a,b)
Therefore, let IIzdp,(a,b),v = 1 and denote ME = M n (a,E) . Obviously, .
fE(x) dxl l/ IIfEIIp, (a,b),v
M E
-
(( vl-p
1/p
'
i
(
(x) dxl / (( ll v1 -P (x) dx l
lJ
(x) dxl
,
11
ME
a
The last integral tends to
111/p'
r
v1-P
a
C -4 a+ ,
for
0
since
BL <
Thus, the con-
dition (a) is verified and (7.15) holds.
The assertions 7.1 (iii) and (7.15) imply that (7.16)
IIHLzE
q,(a,b),w
-. 0
- a+
for
.
Since
x b
Jr fE(t) dt
q
a HLzEI q,(a,b),w q =
J
a b
E
q
fE(t) dtJ J
l
w(x) dx '_
IIfIIp,(a,b),v
w(x) dx
b
1
a
q
IIfcp,(a,b),v
=
vl-p' (x)
w(x) dx
1
9 - 4/P
dx )
a
77
lim FL(T) = 0 &+a+
the equality
follows from (7.16).
lim FL(x) = 0 x+b-
We will show that
as well. Due to the assertion
7.1 (ii), the operator HL = HR C- [Lq (a,b;w
Lp,(a,b;vl p,)
1-q')
I
,
(see (7.12), (7.13)), is also compact. Choosing again
for a < x for E < x < b
10
t
IS
w(x)
i; E (a,b)
and
,
&
,
we have
(b ligE11
=
q,,(a,b),wl-q
I
subset
I
J
Denoting z = g /IIgE11
1/q'
w(x) dx
l-q,
,
we have for an arbitrary measurable
q ,(a,b),w
M C (a,b)
zE(x) dx
If
M
gE(x) dxJ/ 11g,11 q ,(a,b),w
M 1
w(x)
1
I
w(x) dxJ
If
I
M (I (E,b)
E
The last integral tends to Moreover,
liz
,
=
1
w(x) dxJ
E
for
0
_
bl1/q
1/q'
dxJ/[b
(
1-q'
i; -+ b-
due to the condition
BL < w
and in view of the assertion 7.1 (vi),
I
E
(7.17)
z
--' 0
for
in
-i b-
The assertions 7.1 (iii) and (7.17) imply that
(7.18) Finally,
IIHL
lim
p ,(a,b),v FL(T) = 0
1-P, --+ 0
-- b-
for
follows from the estimate
l *bb
Jr ge(t) dt
b
pI
p, 1-p'v
H* L
(x) dx ? lp,,(a,b),vl-p'
J
a
78
II g ',(a,b),wl-q,
'
((b
>a
dtj p v1-p' (x)
ge (t)
J[ 1
dx
4 p,
Ig
l-P
II
q',(a,b),wq r
=
IUJ
vl-p (x) dx
t p'
(br
I
-p'/q'
w(x) dx]
Fp,() L
J
a (II)
Let the conditions (i), (ii) of the theorem be fulfilled. Let LP(a,b;v)
be the unit ball in
B
H
,
L
B
its image in
Lq(a,b;w)
and
,
denote by S = (HLB)w1/q
(7.19)
the set of all functions of the form
with
the assertion 7.1 (v) we have to show that for
u C S
(7.8) (with
instead of
q
p
)
c
implies the existence of numbers 0 < FL(x) <
.
According to
the conditions (7.7),
are fulfilled.
For this purpose, choose c> 0
(7.20)
f C B
l/q
E
.
The condition (ii) of our theorem
,
d E (a,b)
c < d
,
,
such that
x c-: (a,c) U (d,b)
for every
.
4 k(q,p) 3 If
u E S
,
then
u =
u(x)lq dx
(7.21) J
(a, c) U (d,b)
b
x
c
I
a
d
I1
w(x) dx = 11 + 12
a
using the inequality (1.12) for the interval
and (7.20): c
I1 = (7.22)
I
J
a
We estimate the integral (a,c)
[JJf(t) dtj
w(x) dx +
dt I
J
xll
( J
a
x
IJ If(t) I dtj
w(x) dx
a
[k(q,p) BL(a,c;w,v,q,P)-1A
P,(a,c),v
]q <
79
(recall that
eq
sup FL(x)1q a<x
[k(q,p)
IfIIp,(a,b),v
3.4q
since
1
:-5
u E S
f E B )
and consequently
Using Holder's inequality, the inequality (1.12) for the interval (d,b) (with the corresponding estimate for the best constant) and (7.20), we have for the integral
12 b (
12 =
J
d b
x
[Jit ) Idt
q
w(x) dx =
I
a x
d
(
(
dtlw(x)
[JIf(tI dt + llf(t)I J
J
d
dx
J
d
a
b
d
[Jti dt
((
J
x
jw(x) dx + J [f I f (t) I dtJ
a
d
w(x) dx
d
(7.23)
2q-1
lJ
w(x)
dx
vl-pr(t) dtl
((
Ijfj1qp,(a,b),v +
J
lJ
a
d
+ [k(q,p) BL(d,b,w,v,q,P) lfllp,(a,b), 2q-1 fFq(d) L+ kq(q,P)
[2 k(q,p)
sup
d<x
The estimate (7.8) (with
q
sup FL(x)]g} d<x
[
Eq
F (x)] q
3.2q
L
instead of
p
) now follows from (7.21), (7.22)
and (7.23).
Further, choose some points
c' E (a,c)
,
b(
Ju(x + h) -
(x)dx =
a
(
+ J
J
u(x) = 0
for
d'
u(x + h) - u(x)lq dx <
80
=
.
1 + J2 + J3
c'
x ip (a,b) ). We have
c'
Then
r(7.24) J
a
(recall that we define
+
d' E (d,b)
c' u(x + h)
q
u(x)lq dx
dx + J
a If
h E R is such that
lhl < H
,
where
H = min (c-c', c'-a, d'-d, b-d') then
c' J
,
c'+h lu(x + h)lq dx =
J
c(
u(Y)lq dy < Jlu(x)lq dx
a+h
a
I1
a
c
c
u(x)lq dx < Jlu(x)Iq dx < I1 J
a
a
which together with (7.22) implies
E
(7.25)
J1 <
3
Analogously we have bJiu
( x+ h) - u(x)lq dx <
J2 = d'
(7.26)
b
2q-1
i
b
JIu(x + h)lq dx + J
l d
JJu(x + h)lq dx
due to (7.23), since
lu(x)lq dx1 <- 2q I2 3
d
,
12
< H/4
lhl
1 2
d'
d'
Finally, let
J lu(x)lq dx
and denote
c = c' - H/4
d' + H/4
.
Then
dJiux '
+ h) - u(x)lq dx =
J3 = c'
d(' (7.27)
=
J
(HLf)(x + h) w1/q(x + h) - (HLf)(x) w1/q(x)
q
dx
<-
c' d'
< 2q-1 {
(HLf)(x + h)
[Wl/q(x
q
+ h) - w1/q(x)]
dx +
) c'
81
[(HLf)(x + h) - (HLf)(x)] wl/q(x)
q
dx } < 2q-1 (131 + 132)
Holder's inequality yields d' x+h Iwl/q(x + h) -
131 =
d'+h
y (
dt)q Iwl/q(y) - wl/q(y - h)Iq dy
Jjf(t) I
1
wl/q(x) q dx =
dtjq
f(t)I
I J c' a
1
l
c'+h
a
(7.28)
d'+h
vl-p' (t) dt)q/p,Iwl/q(y) - wl/q(y-h) Igdy <
(I lJ
J Iifllqp, (a,b),v c'+h
a
d
q/p '
< 1JvP' (t) dt
Iwl/q(y)
-
wl/q(y - h) Iq dy
.
1
j
a
c
vl-p
E L1(a,d)
Since
number
HO > 0
and
wl/g E Lg(c - H/4, d + H/4)
such that for
h E R ,
IhI
< H0
,
there exists a
we have
r
lwl/q(y) -
w1/g(y - h)Ig dy < eg / 13.28
c
a
which together with (7.28) implies (7.29)
eg
J31 <
3.2q
provided
IhI
< min (H/4, H0)
For
IhI
< H/4
we have
d' x+h 132 =
J
c'
(7.30)
I
J
x dt -
f(t)
f(t) a
a
d' x+h q
111 If(t) c' 82
vl-p'(t) dt
x
dtI
w(x) dx 5
dt
d'
x+h Iq/p'
r J
_<
J
c'
x
(dr'
x+h
l IJ w(x) dx]
c
sup c'<x
1
w E L (c',d')
Since
w(x) dx <
vl-p,(t) dt
Ifq,(a,b),v
and
h E R ,
such that for
I
x
'
E LI(c,d)
< H1
Ihl
,
q/Pl
d' w(x)
q/ [3.2q
vl p (t) dt
I
there exists a number
H1 > 0
we have
x+h sup c'<x
q/p'
v1-p,(t) dt
(J r
dxJl]
IIc'
T
x which together with (7.30) implies Eq
(7.31)
32
3.2q
provided
Ihi
< min (H/4, H1)
For
Ihl
< 6 = min (H/4, HO, H1)
(7.32)
,
(7.27), (7.29) and (7.31) yield
e
J3 <
3
Now, the estimate (7.7) (with
instead of
q
p
)
follows from (7.24)
in view of (7.25), (7.26) and (7.32). 11
Let us formulate the
7.4. Theorem.
Let
1
'right endpoint' analogue of Theorem 7.3.
< p _< q < -
,
v, w E W(a,b)
.
Let
FR(x)
and
BR
be defined by the formulas (6.1) and (6.2). Then the operator HR : Lp(a,b;v) -- Lq(a,b;w)
is compact if and only if (i)
BR = BR(a,b,w,v,q,p) <
(ii)
lim
FR(x) = lim
For the case
FR(x) = 0
x+b-
x4a+ p > q
,
the situation is quite different:
83
Let
7.5. Theorem.
1
< q < p < W
v, w E W(a,b)
,
Let
.
AL be the number
defined by the formula (1.19). Then the operator HL
LP(a,b;v) -+ Lq(a,b;w)
:
is compact if and only if AL = AL(a,b,w,v,q,p) < is continuous if and
HL
A comparison with Theorem 1.15 shows that
only if it is compact. Therefore we can reformulate Theorem 7.5 as well as from the formula (6.7)
'right endpoint' with AR
its counterpart for the as follows:
Let
7.6. Theorem.
1
v, w E W(a,b)
< q < p <
Then the following
.
conditions are equivalent: (i)
Hi E [LP(a,b;v), Lq(a,b;w)]
(ii)
H. E K[LP(a,b;v), Lq(a,b;w)]
(iii)
A.(a,b,w,v,q,p) <
i = L or i = R
where
.
The proof of Theorem 7.5 is based on a lemma in which the following notation is used with
1
=
r
1
q
-
1
p r
l-awq AR(a,b,vl-p
(7.33)
b =
and for (7.34)
r
= AL(a,b,w,v,q,P) =
1/p,
x
Ja L
lJ
UJ
w(t)
dtj
1/r
r
1/p
b
vl-p,(t) dtj
,P',qI) _
J
w(x) dx j
4 E (a,b)
(P L(4) = IPL(E;a,b,w,v,q,p) =
dill/q
dxIl/r
[xf
v'-p'(t) dtJl/q,]r vl-p,(x) = {
J
a
(7.35)
84
4
x
a
(Pi(C)
L [f w(t) x
a
= (DL(&;a,b,w,v,q,p) _
b
x(
J
[ J
dtl
Let
1
l1/r
w(t) dtl dt)
IJ
w(x) dx? J
x
= AL(a,b,w,v,q,p) <
1
,
=
r
_
1
1
v, w E W(a,b)
.
Let
AL =
p
q
Then
AL = AL(a,b,w,v,q,P)
(i)
(br
a
C
7.7. Lemma.
l1/p'
-'-P' (t)
<
1/r (ii)
FL(T) <
)
ID L(E)
-<- AL
<_ AL(E)
-< - AL
1/r (iii)
(g)
FL(E)
(iv)
lim
(D LM = 0
->a+ lim
(v)
(DL (E) = 0
E+bProof.
(i)
AL < -
The condition
is equivalent to
HL E [Lp(a,b;v), Lg(a,b;w)]
According to Subsection 7.2, this implies q'(a,b;w1-q'),
HL = HR E [L
Lp'(a,b;vl-p')]
However, this is equivalent (due to Theorem 6.3) to AR(a,b,vl-p',wl
gr,p',gr) <
and the assertion (i) follows from (7.33). (ii)
Let
E E (a,b)
.
Then
b
dtJr/q' 'Ip,
w(t) dtl/q
r
1/r
=
(x)
[ J a
rJ
dx]l/r
v1-P'(t)
]
[J
a
b
1/q
(J w(t) dt) J
IJv
l-P
r
1/r
1/p'
(t) dtj
=
(p') r
FL(T)
la
which is the estimate (ii). 85
The proof is an analogue of that from part (ii).
(iii)
(iv) and (v) are consequences of the absolute continuity of the integral (see (7.34) and (7.35); moreover,
c' (a) _
and
There is a close connection between the numbers
7.8. Remark.
AL
from
from (7.33), namely,
Ai
(1.19) and
0L(b) - AL < -
1/r
,Q
AL(a,b,w,v,q,p) =
(7.36)
AL(a,b,w,v,q,p)
)
This can be easily shown if we suppose that vl-p
E L1(a,b)
w,
.
x,
Indeed, integration by parts then yields (bf
br
AL =
l JI w(t) dtl
I
r/q
Ifa
1
a
lr/q-1
IJ
rf
If
(q ) since
functions
v , w
(x) dx
w(t) dt]
x {r
,
w(x) (Q, + 1)-1IJ vl-p (t) dt
r/q'+l
dx =
a
(Ai)r
r/q - 1 - r/p If
dtvl-p
(t)
x
r
q
r/q,
vl-p
,
r/q' + 1 = r/p'
.
are general functions from W(a,b)
v n, wn E W(a,b)
,
we introduce auxiliary
by the formulas (p x2/'
+
v(x) n = v(x) +
1
1
,
n wn(x) = min (w(x) , n/x2, n) vn-p
Obviously
wn,
E L1(a,b)
the identity (7.36) holds for
cf., e.g., (3.13)
-
wn , vn
and
AL < = (ii)
86
(i)
w
,
v
.
The formula
convergence theorem.
(7.36) then follows by the monotone
7.9. Proof of Theorem 7.5.
instead of
and consequently,
-
HL
If
is compact, then it is continuous
according to Theorem 1.15. Let
AL(a,b,w,v,q,p) < -
.
Similarly as in the proof of Theorem
7.3, we have to show that for
L
conditions (7.7) and (7.8) (with For this purpose, choose
instead of
q
c > 0
7.7 imply the existence of numbers (7.37)
0L(c) < K
OL(d) '
.
cf. (7.19)
-
u E S = (H B)w1/q p
)
the
-
are fulfilled.
The assertions (iv), (v) of Lemma
c, d E (a,b)
c < d
,
,
such that
5 K
where 1/r
K = 4.31/q(r)
max (k(q,p), k(p',q'))
q
k(s,t) =
s1/s.(t,)1 s
,
for
s,
t E (1,W)
and (7.37) we have for the integral
Using the above introduced notation from (7.21) the estimate
I1
I
` [k(q,p) A (a,c,w,v,q,p)]q < [k(q,p) L
1
note that with help
(c)]q <
e
q
L
3.4q
of the function
k
,
the upper estimate of
CL
in
(1.26) can be rewritten into the form
CL ` k(q,p) AL I . If we denote by HR
,
HL
and
HR = HL
the analogues of the operators
but now acting on the interval
HL
and
only, then for their norms we
(d,b)
have the estimate
IHLd=I H*1 L
k(p',q') A*(d,b,w,v,q,p) < k(p',q') 0*(d)
Using this estimate, the first inequality in the assertion (iii) of Lemma 7.7 and (7.37), for the integral
from (7.23) we have the estimate
1 2
I2 < 2q-1 {FL (d)
2q-1
1/r +
1FL(d) + Lk(p',q') 0*(d)]q} <
0*L(d q
[2 k(p',q') (-) q
3.2q
The rest of the proof is the same as in Theorem 7.3. 11
7.10. Examples.
The examples from Section 6, in which the Hardy inequality
was considered, give at the same time necessary and sufficient conditions for particular pairs of weights HL
and
HR
v
,
as operators acting from
w
,
which guarantee the continuity of
Lp(a,b;v)
into
Lq(a,b;w)
.
Using 87
the foregoing results, we can give conditions under which these operators are compact. For simplicity, we will deal only with the operator (i)
Let
(cf. Example 6.7)
w(x) = xa
v(x) = xs
,
a
,
,
< p < q <
1
HL
(a,b)
,
being described by (6.18). The continuous
8
operator
Lp(O,-;x ) -a Lq(0,0;xa)
HL
FL(x) = const = BL
can never be compact since
and thus conditions (ii)
of Theorem 7.3 cannot be satisfied. (ii) (cf. Example 6.8)
w(x) = xa
v(x) = x6
,
HL
:
p, q
Let
-- (1,co)
a = 0
,
b <
,
Then the operator
.
L?(O,b;x6) __ Lq(O,b;xa)
is compact if and only if B < p -
1
a>
,
B
g p
-
-
p
1
,
(compare with the continuity conditions (6.21), (6.22)). (iii) (cf. Example 6.9)
w(x) = xa
v(x) = xB
,
HL
:
Let
p,
q E (1,-)
,
a > 0
,
b
Then the operator
.
Lp(a,-;x6) --> Lq(a,-;xa)
is compact if and only if B < p -
1
a<
,
B
5 p
-
p
-1
or
B? p[note that
a<- 1
B S - P - 1 = P (6 - p + 1) -
1
,
and consequently, the last
two conditions can be written in a unique form:
a < min (- 1
BER, (iv)
Let
p, q E (1,w)
w(x) = X-1 Iln xla
w(x) = e
ax ,
,
.
,
P(B - p + 1) - 1)
In the case of (a,b) _ (0,1) xlB or (a,b)
v(x) = xp-1 Iln
v(x) = e
Bx
and and
(cf. Examples 6.10 and 6.12), the situation is
similar to that of case (i) above and the operator
88
.
HL
cannot be compact.
7.11. Weighted Sobolev spaces.
For
1
s p <
°°
and
Vol v1 E W(a,b)
,
let
us define the weighted Sobolev space W1'p(a,b;v0,v1)
(7.38)
as the set of all functions u' G LP(a,b;vI)
.
such that
u r= AC(a,b)
u C LP(a,b;v0)
On this space, we define the norm by the formula l11
jlu
(7.39)
and
/p
u/lp,(a,b),v1J
1,p,(a,b),v0,v1
If, in addition, v-1/P E LP/ (a,b)
(7.40)
loc
1
which means that then
WI'P(a,b;v0,v1)
[c,d] CC(a,b)
for every interval
v1I/P E Lp'(c,d)
is a Banach space. Further, denote by
set of all infinitely differentiable functions
in the neighbourhood of the endpoints
a
on
(a,b)
the
which vanish
Then the inclusion
b
,
u
CC(a,b)
CO(a,b) C W"P(a,b;v0,v1) holds if and only if Vol vl
(7.41)
(a,b)
L1loc
Consequently, under the assumptions (7.40), (7.41),
the space
W0,P(a,b;v0lv
defined as the closure of the set
CO(a,b)
with respect to the norm (7.39)
is also a Banach space (normed again by (7.39)). (For the proofs of these assertions, see A. KUFNER, B. OPIC [2], Lemma 2.3 and Lemma 4.4.) Therefore, when
dealing with weighted Sobolev spaces, we will always assume
that the weight functions If
w, v
l
< C u'
llu q,(a,b),w
ACL(a,b)
V
E W(a,b)
VI
satisfy the conditions (7.40), (7.41).
or
p,(a,b)vI ACR(a,b)
then obviously
Cllulll,p,(a,b),v0,v1
lu11q,(a,b),w
for every
,
are such that the Hardy inequality
C W(a,b)
is satisfied on
v0
.
This means that
0
89
(7.42)
WI'P(a,b;v0,v1) Lj Lq(a,b;w)
In the sequel, we will establish conditions on the weights which guarantee not only the continuity, but also the compactness of the imbedding (7.42).
Vol v1, w E W(a,b)
Let
7.12. Lemma. H
:
LP(a,b;v1) -+ Lq(a,b;w) HL
be one of the operators
If
(i)
Let
.
,
HR
.
< p,q a - , p # - , and
1
H
is continuous, then
l. Lq(a,b;w) (ii)
If
and
< p,q <
1
Wp,p(a,b;v0,v1) Proof.
H
is compact, then
L'(a,b;w)
Obviously it suffices to give the proof for the case
follows from the continuity (compactness) of the operator numbers
BL(a,b,w,vl,q,p)
or
AL(a,b,w,vl,q,p)
H - HL
HL
that the
are finite, and conse-
quently r
vi-P
for every
E L1(a,x)
x C (a,b)
.
This implies, together with the density of the set 'WQ'P(a,b;v0,v1)
,
that
u
C0(a,b)
from the latter space fulfils
x r
u(x) _
`
u'(t) dt
for a.e.
x E (a,b)
a
(cf. the proof of Lemma 1.10). Thus, the identity operator I
:
W0'P(a,b;v0,v1) ---+ Lq(a,b;w)
can be expressed as the composition (7.43)
I = HL ° D
where the operator
90
.
in
It
D
W 1'p
:
(a,b;vopv l)
Lp(a,b;vl)
is defined by Du = u'
and is obviously continuous.
Now, the assertions of our lemma follow from the formula (7.43). D
As a consequence of Lemma 7.12 and Theorems 5.9, 5.10, 6.2, 6.3, 7.3, 7.4 and 7.6 we immediately obtain
Let
7.13. Theorem.
Let
vol v1, w E W(a,b)
i = L or i = R (i)
If
m
p
,
,
Cj Lq(a,b;w)
(7.44)
(ii)
If
1
lim
< p
v0
,
v1
satisfy (7.40), (7.41).
.
< p < q < m
1
,
<<-
Bi(a,b,w,vl,q,p) <
.
Bi(a,b,w,vl,q,p) < m
q <
Fi(x;a,b,w,vl,q,p) = lim
x±a+
then
and
Fi(x;a,b,w,vl,q,p) = 0
x+b-
then (7.45)
W0l'p(a,b;v0,v1)
y ( Lq(a,bw)
(iii)
If
1
= q < p < m
(iv)
If
1
< q < p < m
7.14. Remarks. (i)
,
,
Ai(a,b,w,vl,q,p) < m
,
then (7.44) holds.
Ai(a,b,w,vl,q,p) < m
,
then (7.45) holds.
Let us define the weighted Sobolev space
WL' (a,b;v0,v1)
as the closure of the set of all and
u' E Lp(a,b;v1)
.
If we take
u E ACL(a,b) i = L
such that
u E Lp(a,b;v0)
in the assumptions of Theorem
7.13, then the assertions (7.44) and (7.45) hold with
W1 1)(a,b;v0,v1)
91
instead of
W0''p(a,b;v0,v1)
The similar consideration concerning the
.
right endpoint is left to the reader.
The results mentioned in Theorem 7.13, part (ii), generalize
(ii)
certain results from the paper B. OPIC, A. KUFNER C3]. On the other hand, the results contained in Theorem 7.13 can be derived also under some modificated assumptions
-
see Theorem 8.23.
All conditions guaranteeing the continuity or compactness of
(iii)
the imbedding W10,P(a,b;v0,v1)
C_ Lq(a,b;w)
and appearing in Theorem 7.13 (and also in Theorem 8.23) are only sufficient. One of the reasons is that the weight function
v0
does not play
any role in these conditions. In Section 11, we will mention some recent results concerning necessary and sufficient conditions for the validity of (7.44) and (7.45) (see Subsection 11.3).
Examples to Theorem 7.13 can be easily constructed using
(iv)
Examples 7.10. Note that in function
W1'p(a,b;v0,v1)
from Examples 7.10, and
v
,
vl
v0 C W(a,b)
plays the role of the is arbitrary except for
the conditions (7.40), (7.41).
8. THE HARDY INEQUALITY FOR FUNCTIONS FROM
8.1. Two examples.
(8.1)
ACLR(a,b)
If we consider the Hardy inequalities
{jIux!q x
Iln
xIa dill/4
dxJl/P
C IIIu'(x)lp xp-1 Iln xls
J
J
0
0
or aax2
(
(8.2)
I
Jux) lq
ll1/q
dx]
Jlu'(x)l'
< C
for
l1/P a 2 e px /q dxI
l
1 a p 5 q < - ,a,P E R , then we have shown in Subsections 6.11, 6.13
that they hold neither on the class
ACL(a,b)
nor on the class
ACR(a,b).
On the other hand, we have mentioned in these examples that both inequalities hold on the class ACLR(a,b)= ACL(a,b) C) ACR(a,b) 92
provided
6>p-1,
(8.3)
a=8gp
-1
P
in the case of the inequality (8.1), or
a>0
(8.4)
in the case of the inequality (8.2)
[cf. Example 8.16, formula (8.97)].
The invalidity of these inequalities on the classes ACR(a,b)
ACL(a,b)
or
is a consequence of the fact that in our particular cases
BL=- and
(8.5)
BR = - .
This indicates that the obvious condition for the validity of the general Hardy inequality (8.6)
luwl/glq,(a,b)
=
IIu'vl/PIIp,(a,b)
C
on
ACLR(a,b) = ACL(a,b)O ACR(a,b) namely the condition min (B L(a,b,w,v,q,p), BR(a,b,w,v,q,p)) <
(8.7)
is only sufficient.
Therefore it is meaningful to look for necessary and sufficient conditions for the validity of the Hardy inequality (8.6) on the class ACLR(a,b)
,
on the class of functions vanishing at both endpoints
i.e.
of the interval
(a,b)
For the case
1
.
< p = q < m
,
such a condition was derived by P. GURKA
[ 2]. He has shown that the corresponding necessary and sufficient conditior reads (8.8)
B = B(a,b,w,v,q,P) <
where r
(8.9)
B = sup
(
L LIIw1IgI
min
Iv-1/p II P,,(a,c)'
q, (c,d)
the supremum being taken over all pairs
I
-1/PI Iv
11
P c
,
d
such that
,(d,b)}1
a < c < d < b
.
Simultaneously, P. GURKA obtained the following estimate for the best possible constant (8.10)
C
in (8.6):
B < C 5 4B
.
93
In the following theorem , 1
< p < q < -
his result is extended to the case
and the estimate (8.10) is improved.
Let
8.2. Theorem.
finite constant every function
1
< p < q < °°
W(a,b)
v, w
,
.
Then there exists a
such that the Hardy inequality (8.6) holds for
C > 0
if and only if the number
u E ACLR(a,b)
B
from (8.9) is
finite.
Moreover, the best possible constant
in (8.6) satisfies the esti-
C
mate (8.11)
for
2-1/p B < C < h(q) B
q E [1,m)
with g(s) inf 1<s<2
(8.12)
h(q) =
(8.13)
g(s) = s - 1
s
(s
q - 1) 11q
and (8.14)
2-1/P B < C < 4B
for q = m 8.3. Remark.
For
the estimate (8.11) can be replaced by a
q E
little rougher estimate (8.15)
2-1/p B <_ C 5 2(2q - 1)1/q B
such that the function increases on
(s012J
;
g(s) = thus
8.4. Proof of Theorem 8.2. fix numbers
c, d E (a,b)
1
(sq - 1)1/q
decreases on
Let
c < d
,
and
C < m
1
and define for
< p < q <
94
v
n
E W(a,b)
and
,
°°
.
.
Let us
n E N functions
(Pr_1))1
vn(x) = v(x) + n (1 + x2 Obviously
(1,s0)
s
h(q) = g(s0) < g(2) = 2(2q - 1)1/q
(i) ,
s
s0 = s0(q),
s0 E (1,2) ,
which follows from the fact that there is a number
x E (a,b)
and
b
(8.16)
0 <
I
vn P(t) dt <
a
(cf.
(3.13)). Let us define two sets N1
{ni
l
v(t) dtr
J v(t) dt
;
by
b
c =
N2
,
J
J
a b
c
N2 = fn E N ;
J
vn-p,(t) dt
f vn-p'(t) dta JJJ
d
At least one of these sets contains infinitely many elements; without loss
of generality we can suppose that it is the set N. Further, let
zn F[d,b)
be such that br
cr
vn-P'(t) dt
(8.17)
n GIN 1
,
Zn
a
and for
V'n-p,(t) dt
=
n E N1
and
x E (a,b)
define
x
(8.18)
un(x) = J gn(t) dt a
1-p'
r
gn = [X(a,c) - X(Zn,b)] vn
un E ACLR(a,b) and c r
(8.21)
vn-P
un(x) =
(t) dt
for every
x E [c,d]
a
which implies 1/9
Icj
d
r
(8.22) [ L
d
19
Illun(t)I q w(t) dt] J
J
c
l4
vn-P (t) dtJ
w(t) dt a
c
95
In view of (8.20), the Hardy inequality (8.6) holds for
u
n
and in view
,
of the estimate for a.e.
v(x) < vn(x)
x E (a,b)
we have b (8.23)
b
1/q
ll/p
IJIun(x)q w(x) dxJ
C
(( IJlgn(x)p
vn(x) dxJ xJ
If
a
a
b11
The formulas (8.17) and (8.19) imply
[Jg(x) lp
1 /P
vn(x) dxJ
=
a b
(8.24)
1/p p
[fiX(a,.)(x)
- X
(zn,b)
()IP vl-p'(x) dx n
cJ
a
b
c r
( =
IJ
vn-P (x) dx +
vn P
= 21/P
vn-P (x) dxJ
I
c
l 1 /P
r
=
I
Zn
a
which together with (8.23) and (8.22) yields
d(
lc(
If
w(t) dtj
C
(t) dtJ
vn-P
21/P C
a
c
a
In view of (8.16), we obtain the estimate d
c
IJ w(t)
vn-p
(t) dtJ
If
If
n -+ m ,
n E N1
,
-1/P
w1/q q,(c,d)'I1°
lc(
d(
IJ c
w(t) dt
J
If
a
Thus, in view of (8.9) we have
?6
<
21/p
C
a
c
and letting
111/p'
dtj1/4
we arrive at the estimate
1lp',(a,c)
=
v(t) dt
l/pr < J
21/P
(x) dxJ
a
vn-P
If
1 /P
r
C
(x) dxJ
B < 21/p C
(8.25)
,
which is in fact the first estimate in (8.11). Let
(ii)
(8.18) with
C <
and
n E N1
1
q = -
< p < - ,
.
For the function
un
from
we obtain in view of (8.21)
c I1unw1/gllq,(c,d)
Jr vn-p'(t) dt =
(8.26)
IunllL,(c,d) =
'
a
which is an analogue of (8.22). If we rewrite (8.23) in a form meaningfull q = -
also for
,
viz. b [jg(x)IP vn(x) dx f
unw1/gllq,(a,b) < C
1/P ,
a
we obtain in view of (8.26) and (8.24) the estimate c
1/p
f vn-p/(t) dt < 21/p C
v1_P (t) dt
7
a
and we again
(iii)
as in part (i)
-
C<m,
Let
p=
arrive at the inequality (8.25).
-
1< q <
1
Assume in addition that
(iii-1)
ess sup v-1(t)
<
tEM for every measurable subset c < d
,
denote
I1 = (a,c)
M C (a,b) ,
.
Let us fix numbers and define
= (d,b)
1
c, d E (a,b)
2
(8.27)
Si = ess sup v-1(t) = Iv-1/pI
t EIi
pI
,11
(recall Convention 5.1). Since
v E W(a,b)
loss of generality we can assume that
, we have
S1 < S2
Si > 0
.
Without
and, in view of our addi-
tional assumption,we have (8.28)
0 < S1 < S2 < -
There exists
n E N
.
such that
97
(8.29)
0 < S1 - n .
If we denote Min - {x E Ii
(8.30)
for
Mini > 0
then
i = 1,2
Si - n s v-1 (x)}
;
.
Therefore, there exists a subset
Min C Min
such that 0 <
(8.31)
by the formula
un
Define
IM2nI <
IM1nI
x
J CXMln (t)
u n(x) =
- XM2n (t)] dt
a Then
un E ACLR(a,b)
.
x E [c,d]
For
we have
c
un(x) = J XMI(t) dt
IM1nl
n a
and consequently (8.32)
I
un
IwIIq,(c,d) 1/q
wl/q'
.
Ilq,(c,d) =
IMinI
Further, using (8.28), (8.29), (8.30) and (8.31), we obtain b
unv1/plp,(a,b)
= JIXM In
(x)I v(x) dx =
(x) - XM 2n
a n)-
(8.33)
v(x) dx +
=
J
v(x) dx < IM1nI(S1
J
Mln
+ IM2nI(S2
n
M2n
1 -1
< 2IMIn
(S1 - n
The validity of the Hardy inequality (8.6) for the function estimates (8.32), (8.33) yield 1Iw1/g11q,(c,d) S 2C(SI -
Letting 98
n - -
,
n)-1
.
we obtain in view of (8.28), (8.29) that
un
and the
Ilwl
(8.34)
-1/p
{IIv-1/pIIp',(a,c),
II
(iii-2)
l
_
IIPI,(d,b)}
liv
gilq,(c,d)-min
wl/q.IIq,(c,d)S1 < 2C
Let
v
be a general function from
W(a,b)
and for
n C IN
define vn(x) = v(x) + n ,
Obviously
vn E W(a,b)
x C (a,b)
as well, and
ess sup v-(t) 5 n < tEM n for every measurable subset
M C (a,b)
.
Thus, the function
vn
fulfils
the additional assumption from part (iii-1). Moreover, the Hardy inequality (8.6) holds with the same constant w
of
,
v
vn(x) s v(x)
since
.
also for the pair
C
.
Letting
n -+ - , we finally obtain the inequality
fact the first estimate in (8.11) for (iv)
c < d
.
C < W ,
p = q = m
Taking a function
obtain from (8.35)
Let
.
p = 1
vn
instead of
B < 2C
which is in
.
Let us fix numbers
c, d E (a,b)
u E ACLR(a,b) , which is constant on
(8.6) that for
instead
Consequently, we can proceed literally
as in part (iii-1) arriving at the estimate (8.34) with v
w , vn
(c,d)
,
we
x E (c,d) Ilurvl/P
11wl/gllq,(c,d)
Iu(x)I < Iiuwl/gllq,(a,b) < C
p,(a,b)
C ess sup Iu'(x)I a<x
= min (c - a, b - d) = Iv-1/P
= min {iiv-1/pIIp,,(a,c).
choose a sequence
Ip',(d,b)}
{Ek}R+ such that
ak = c - Ek
bk = d + Ek
Ek t
E
'
and denote for
k E N
,
x uk(x) = JLX(ak,c)(t) - X(d,bk)(t)] dt a
99
uk e ACLR(a,b)
and
uk(x) = c - ak = Ek
for
Then obviously
Ilwi/q
q (c,d)
k - -
and letting
instead of
uk
Using (8.35) for
IIw1/q 11
x E (c,d)
u
,
kGN .
,
we obtain
< C ess sup lu'(x)I = C a<x
' Ek
we arrive at the estimate
,
q, (c,d) '
`C
,
B < C
which together with (8.36) and (8.9) implies that (v)
.
Thus, we have proved the implication
1< p < q <-, C< >
2-l/p B= C
It remains to prove the following two implications:
q<W,
(8.37)
1
(8.38)
1
<= p
<=
The function
°° > C < h(q) B
B<
B < m > C < 4B
,
belongs
u = 0
to
,
.
and the Hardy inequality
ACLR(a,b)
(8.6) holds for it with an arbitrary constant
C
.
Consequently, to verify
the implications (8.37), (8.38), it suffices to show that (8.6) holds for every function 4B
,
u E ACLR(a,b)
u 1 0 , with the constants
,
h(q)B
,
and
respectively. (vi)
(8.39)
B <
Let
u E ACLR(a,b) n s
,
<
u
0
sup a<x
1i p < q <
,
.
.
Fix
s E(1,-)
Then there is a number
Iu(x)I
<
n e Z
and take
such that
Sn+1
Denote
Zn = {k E Z; k < n} For
k E Zn
let
xk
(yk)
.
be the smallest (greatest ) number from
such that Iu(xk)I = sk
100
(
Iu(yk)I = sk ).
(a,b)
Obviously, <
xk
xk-1
<
yk
'
yk-1
and
(a,b) = U
kEZn
Exk-l' xkI U Cxn,ynI U U Eyk'yk-17
.
kEZn
Consequently, bJu(x)
x(k
(8.40)I" w(x) dx =
u(x)Iq w(x) dx +
J
kEZ n
a
xk-1 yk-1
y(n
Iu(x)Iq w(x) dx +
+ J
()lq w(x) dx < Z
J
n
xn
Yk
yn
s(n+l)q
w(x) dx +
xn
skq w(x) dx =
(
kEZ
J
J
n [xk-1'xk]U Eyk'yk-11
yn (.+1)q f
w(x) dx. +
xn
+
klxk,ykI w(x) dx -
x
skq w(x) dxl =
L[Xk_JYk_lJ
Yk f s(k+l)q
kEZn
J
Yk
w(x) dx -
J
skq w(x) dx =
kEZn x k
xk Yk
(sq - 1)
skq w(x) dx
J
k E Zn
xk
Further, we have xk
Xk
r
Iu W I dx f xk-1
I
j
u'(x) dx
Iu(xk) - u(xk-1)I >
xk-1 u(xk)I
- lu(xk-1)l =
Sk-1 (s
- 1) 101
and consequently, Xk s
sk
u'(x)l dx
1
s
J
xk-1
Holder's inequality yields
(8.41)
sk < s s
1
IIu'vl/PIIp, (x ' k-1 ,xk)
Iv-1/PIIp?
Ilu'vl/PIP,(Yk,Yk-1)
Jv-1/P
)
k
(x kk-1
and analogously (8.42)
s
sk
s
1
Ip,,(Yk,Yk-1)
If
I!v-1/p
(8.43)
l-1/PII
< 11
v
p',(xk-l,xk)
P',(Yk,Yk-1)
,
then from (8.41) we have Yk
w(x) dx
skq J
xk
Yk (
<
s
Jw(x)dx.
s-
v-1/pllq
lwl/glq,(xk,Yk)
1 f
min { v
.
p,(xk-l,xk)
q
s
(8.44)
Ilu'v1/pllq
p',(xk-l'xk)
xk
1/p q
-1/piq Ip,,(yk'yk-1)J
ll
p',(xk-l'xk)'
v
<
Ilu'vl/pllq
p'(xk-l,xk)
If
(8.45)
v-1/pi
p',(xk-l,xk)
>
v-1/PII
P',(Yk'yk-1)
then we proceed analogously using (8.42) and obtain
102
,
yk r
(8.46)
skq
w(x) dx < (s
f
s
1
lu'vl/pljq
Bq
p,(yk'yk-1)
xk
The formulas (8.44) and (8.46) yield yk
skq I w(x) dx < xk
s
<
q
u,vl/p q p'(xk-1'xk) +
1)
(s
for every
k E 7n
u'vl/pllq p'(yk'yk-1
Using this estimate in (8.40), we obtain
.
xk
bJ
lq/p v(x) dxJ +
r
u(x)w(x) dx
u'
(
(
J
(x)
nL
a
xk-1
yk-1
+II with the function
g
l q/P lu'(x)lp v(x) dxJ ]
yk from (8.13). As
g :
1
, we have
p
b
xrk
ju(x)jw(x)
r
g(s) BL
dx
u'(x)Iv(x) dx + k E
n xk-1
a
yk-1
llq/p
bl +
lu'(x)lp v(x) dx]1
<
J
yk
g( )
dx]
[JIu'(x
)lp v(x) a
Taking here the (1/q)-th power, we obtain the Hardy inequality (8.6) with the constant
g(s)B
where
s E (1,W)
is arbitrary. Consequently, for the
best possible constant we obtain the estimate C = inf g(s) B = inf g(s) B = h(q) B s>1 1<s<2
Thus, we have proved the implication (8.37). 103
(vii)
Let
u e AC1 R(a,b) (8.47)
u10
,
Iu
, <
s
such that (8.39) holds. Then
(8.42) we obtain
2
sn+1
u'vl/p'P,(x
< ss-
(8.48)
s`
n+l
s`
)
v-1/p Pi
'(Ynlyn-1)
-1/p
{Iv-1/P
min
s -
n-1" n
k = n ), then (8.48) implies
If (8.43) holds (with
-
lv-1/Pllp',(x
,x n )
p,(yn'vn-1)
s
sn+l
n-1
1/p,
,
u v
-
s
P" (xn-1'hn)'
1
u,vl/pll
<
Ilwl/gtq,(c,d) =
1
since
s2
s -
P'(xn-1'xn)
(recall that
s E (I,-) and take
Fix
n+1
and analogously as in (8.41),
(8.49)
l s p< m
q=m and n e L
B<W
P''(Yn,Yn-1)}
,v B iup,(a,b) 1/p 1.
1
If (8.45) holds, then we
q = m ).
obtain analogously from (8.49) that sn+l <
s2
B
Ilu'vl/ply
p,(a,b)
s - 1
In view of (8.47), we have shown that 1/q lluw
l/po
'q,(a,b)
_
B u'v
s2
s -
p,(a,b)
1
2
which is the Hardy inequality (8.6) with the constant
s- l
B
,
s
>
arbitrary. Consequently, for the best possible constant we obtain the estimate 2
C < inf
ss- 1 B = 4B
.
s>1 Thus we have proved the implication (8.38).
8.5. Remarks. (8.50) 104
(i)
11
Note that in the proof of the implication
C<>
2-1/p B< C
1
we have not used the assumption
p
q
-<
.
Therefore, (8.50) holds for
1
While the foregoing proof for the case
1
< p < q < -
the proof for
slight modification of the proof given by P. GURKA [2] , the case
p =
was a
is completely different and the proof for the case
1
q =
is new and published here for the first time. Both proofs are due to B. OPIC.
p = q = m
If
(iii)
,
then the number
from (8.9) is given by the
B
formula
B=
sup a
min (c - a, b - d) B = -
(cf. Subsection 8.4 (iv), formula (8.36)). Consequently,
if
(a,b)
is unbounded. Therefore, if we wished to investigate the Hardy inequality on
ACLR(a,b)
for
p = q = -
and unbounded intervals
(a,b)
,
we should
proceed in accordance with Remark 5.5.
Now, we will return to the examples considered in Section 6.
8.6. Examples.
We will give the necessary and sufficient conditions for the validity of the corresponding Hardy inequality, but this time on the class
ACLR(a,b).
Naturally, we will use Theorem 8.2.
(i) Let
1
,
a,
S E F.
m
.
Then the Hardy inequality
W [I1u((rr
(8.51)
x
C IJIU,p x S
dxJ
0
holds for every
(8.52)
Szp
dxJ
0
with a finite constant
u E ACLR(0,m)
1, a=
g p
C
if and only if
-- 1 p
Thus, the condition (8.52) combines the conditions (6.18), (6.19) from Example 6.7.
(ii)
Let
1< p < q <
,
0
a,
B E R .
Then the Hardy
inequality
105
b(8.53)
11/q
br
(Jju(x)IxdxJ
[Ju'(x)JP
C
0
0
u E ACLR(O,b)
holds for every
a2B (8.54)
8 dxJ
P
p
with a finite constant
C
if and only if
B x p- 1, for B=p - 1 .
for
1
a>-1
Cf. Example 6.8.
(iii)
0< a<,
15p=q<
Let
a,
B E R . Then the Hardy
inequality [jjU(X)Iq xa dx]
(8.55)
m
1/q
1/p
C[JIu'(x)IP x6 dx]
<_
a
J
a
holds for every
-p-1
a< B p (8.56)
with a finite constant
u E ACLR(a,-)
a <1
for
x p- 1,
for
B= p - 1.
C
if and only if
Cf. Example 6.9. (iv)
Let
1
< p 5 q < m
,
1
(8.57)
1
[11ux )Iq x
B E R . Then the Hardy inequality
l
q
dxl
Iln xI
1/1 p
a,
1
C[Iu'lP I(W
0
Iln
xI6
l
dxJ
0
holds for every
(8.58)
_
xp
u E ACLR(0,1)
Bxp-1
with a finite constant
a= B p- p -
C
if and only if
1
Cf. Example 6.10.
(v)
(8.59)
Let
1
[1U(X)I
1
qx in xI
a
106
a E R . Then the Hardy inequality
ll1/q
dxj
<
Iflu'(x)Ip xP
1
Iln xI
B
1
1/P
dxJ
C
0
0
holds for every
a,
u E ACLR(0,W)
with a finite constant
C
if and only if
(8.60)
(vi) (8.61)
8 > p -
Let
a= 8 P- P - 1.
,
1
Sc
q<-
[fIu(x)1q eax
,
a,
ll1/q
6 E R . Then the Hardy inequality 1/p
(
5 C[ JIu'(x)lp e8x dxJ
dxJ
u E ACLRwith a finite constant
holds for every
(8.62)
1
a
,
6x0
C
if and only if
8p
Cf. Example 6.12.
Let us return to the inequality (8.59). Since
8.7. Remark.
j
Jlu(x)lq 1 lln x
=
xla
dx =
0 W
1
Iu(x)lq X lln
xla
xla dx + Jlu(x)Iq I Iln dx = J1 + J2
0
1
and we are able to estimate u E ACR(1,-)
u E ACL(0,1)
for
J1
as well as
J2
for
(cf. Example 6.10 and Remark 6.11, formulas (6.30), (6.33)),
we obtain the following estimate for
J
:
1
14/P
J < Cq
+
f [11u' (x )Ip xp-1 lln xl8 dxJ [
0
14/P +
[Ju'(x)lp xP-1 Iln xl8 dx]
] <
1
< Cgl(lu'(x)lP
xP-1
lln
xl8
dx
l0
taking into account that
p < q
,
obtained the inequality (8.59) for
.
Consequently, we have
u E ACLR(0,m)
provided the corres-
i.e.
q/p >
1
ponding Hardy inequalities hold respectively on the subintervals
(0,1)
,
107
and for the classes
(1,m)
ACL(0,1)
,
ACR(1,-)
. According to Example
6.10 and Remark 6.11, the conditions which ensure the validity of these Hardy inequalities are given by (6.31) and coincide with the conditions (8.60). However, the approach used in this remark guarantees only the
sufficiency of the conditions (8.60), while their necessity follows from Theorem 8.2.
Therefore, a natural question arises whether this coincidence is of splitting the interval
accidental or if the 'trick' subintervals ACLR(a,b) ACR(c,d)
(a,c)
,
into two
(a,b)
and investigating the Hardy inequality on
(c,b)
via the investigation of the Hardy inequality on
ACL(a,c)
and
could be used also generally. Let us describe the general situa-
tion.
We investigate the Hardy inequality (8.63)
luwl/g1Iq,(a,b)
For this purpose choose
C
llu'vl/pllp,(a,b)
c E (a,b)
jj
(8.64)
on
ACLR(a,b)
and investigate the Hardy inequalities
uwl/g q,(a,c) 5 CLllu'vl/PI p,(a,c)
on
ACL(a,c)
uwl/gllq,(c,b) < CRl u'vl/pllp,(c,b)
on
ACR(c,b)
II
and (8.65)
The inequality (8.64) or (8.65) holds for
1
< p < q < -
.
if and only if
BL(a,c) = BL(a,c,w,v,q,p) <
BR(c,b) = BR(c,b,w,v,q,p) < -
,
respectively. Consequently, the inequality (8.63) holds for
1
< p < q <
if
max (BL(a,c), BR(c,b)) <
This condition is sufficient for (8.63); moreover, since been arbitrary, the condition
108
c E (a,b)
has
(8.68)
inf
') =
max (BL(a,c), BR(c,b)) <
a
is also sufficient.
This approach forms the basis of the proof of the following theorem, in which it will be shown that a slightly modified version of the condition (8.68) is not only sufficient, but, moreover, necessary for the validity of the Hardy inequality (8.63) on the class
ACLR(a,b)
.
This modified condi-
tion reads 16
_ 63 (a,b,w,v,q,p) < M
where now max (BL(a,c), BR(c,b))
inf
03 =
(8.69)
a_c!Cb
BL(a,c)
with
and
BR(c,b)
BL(a,a) = 0
(8.70)
,
from (8.66) and (8.67), respectively, and with
BR(b,b) = 0
.
The advantage of the condition (8.69) consists among other in its covering also the cases when the Hardy inequality (8.6) holds on the broader class ACL(a,b)
ACR(a,b)
or
.
Moreover, the approach mentioned can be extended to the case 1
q < p < -
8.8. Theorem.
-
Let
see Theorem 8.17.
1
< p < q <
,
v, w E W(a,b)
.
Let the number
63
be
defined by the formula (8.69).
Then there exists a finite constant equality (8.6) holds for every
C > 0
u E ACLR(a,b)
such that the Hardy in-
if and only if
63 < -
(8.71)
Moreover, the best possible constant
C
in (8.6) satisfies the esti-
mate (8.72)
with
2-11p 0 s C< k(q,p) 3 k(q,p)
from
(1.24).
The proof of Theorem 8.8 is a consequence of the following two lemmas
109
(and Theorem 8.2).
8.9. Lemma.
Let
constant
in (8.63) satisfies
C
Proof.
-
v, w C W(a,b)
Then the best possible
.
C < k(q,p) 6
(8.73)
43 <
< p < q = - ,
1
( = -
If .
Let
then (8.73) holds trivially. Therefore, suppose be fixed. Then according to the definition of
> 0
c
see (8.69)
,
there exists a number
-
c E [a,b]
max (BL(a,c), BR(c,b)) < fl + E
03
such that
,
and consequently (8.74)
If
BL(a,c) < c = a
J3 + c
BR(c,b) < Q + c <
<
c = b ), then (8.74) implies that the Hardy inequality
(or
Ikuw1/gllq,(a,b) < k(q,p)( 63 + c) IIu'v1lpllp,(a,b)
holds for
u E ACL(a,b)
u E ACLR(a,b)
If
and
Consequently, the best possible constant
.
has been arbitrary, we have (8.73).
c > 0
c E (a,b)
,
then we will consider two different cases:
Let
1
5_
q <
q E [1,m)
Then
ACLR(a,b) C ACL(a,c)
(or
C ACR(c,b) )
and (8.74) implies the validity of the Hardy inequality on
c
1/q
(
lq w(x) dxJ
S k(q,p) BL(a,c)
[or b [JIu(x)
e
(or
[JIuF(x)IP v(x) dxJ
f1 /p
a
a
(8.76)
(a,c)
c}} 1/p
(c,b) ):
(8.75)
} 1 /q w(x) dxJ
(b
k(q,p) BR(,b)
IIIu'(x)v(x)
dxll
l
JJJ
c 110
satisfies
q=W. (i)
on
C
k(q,p) ( 6 + c)
C
and,since
u E ACR(a,b) ) and a fortiori for
(or
].
Using the fact that
, we immediately obtain from (8.75), (8.76)
q/p ? 1
and (8.74)
b
J
brr
U'(X) Iq w(x) dx = [k(q,P)(
llq/p
{iu'x)P v(x) dx]
+ E)]
a
a
c - 0+ , we get (8.73).
Letting
q = - . Then
Let
(ii)
(8.77)
I
uw1/q Iq,(a,b) =
N
u IW,(a,b)
max `IIuII_,(a,c)'
IIull-,(c,b))
(Ijuwl/gllq,(a,c)' I1uwl/gllq,(c,b))
= max
.
Analogously as in part (i) we arrive at the inequalities BL(a,c)
luwl/qIl q,(a,c)
ju'vl/p11p,(a,c)
(63 + e) 1uwl/qi q,(c,b)
< BR(c,b)
<
since
lu'vl/pIIP,(c,b)
<
90) + E) N'vl/pllp,(a,b)
q =
for
k(q,p) = 1
(
(a,b)
The last two inequalities together with
.
(8.77) imply
uw
Letting
E
1/qI
---* 0+ ,
E)
uv 1/p
+
II
p,(a,b)
we immediately obtain (8.73).
Let
8.10. Lemma.
_ (43 <
q,(a,b)
1
< p,q < °°
,
v, w E W(a,b)
C]
.
Let
B
and
0 be
defined by (8.9) and (8.69), respectively. Then < B
(8.78)
Proof.
If
B
.
then (8.78) holds trivially. Therefore, suppose
B <°°
c E' Ca,b]
such
Obviously, it suffices to show that there exists a point that (8.79)
BL(a,c)
5: B
,
BR(c,b) 5 B
.
111
For
denote
c E (a,b]
Q _ {(c;d) C R2; c E (a,c), d E (c,c)} for
denote
c E [a,b)
(c; d) E R2; c C- (c,b), d C (c,b)}
Qc =
define functions
0
,
by
Y
v-1/ptp,,(a,x)
'V(x) =
IIv-1/PI
p ,(x,b)
,
x C (a,b]
,
4)(a) = 0
,
,
x E [a,b)
,
V(b) = 0
,
and put
co = sup {c E [a,b); '(c) < o}
,
d0 = inf {d E (a,b]; 4'(d) < oo}
Necessarily we have (i)
d0 < co
d0 = c0 E (a,b)
If
since otherwise we would have
,
,
then we take
c = c0
B = m
.
and obtain easily
that
B ?
11wl/gllq,(c,d) min {@(c), Y(d)} _
sup
(c; d) E Qc (8.80)
=
IIw1/ q
sup
sup
a
c
Ilq,(c,d) IP(c)
1/p
Iilwl/qII
=
sup
q, (c ,c)
a
B2
(8.81)
=
=
sup _ (c;d) E Qc sup c
sup c
w1/q
IV-
q,(c,d)
II
r
p ,(a,c)
= B (a, c) L
min {'(c), 'r(d)} =
4'(d) _ sup lwl/qII q,(c,d) c
11v-1/P
q,(c,d)
II
P ,(d,b)
= BR(c, b)
and (8.79) is proved. (ii)
If
d0 = c0 = b
as in part (i) that 112
,
then we take
c = b
and obtain analogously
B =
(8.82)
sup
11w1/gllq,(c,d)
min {'(c), 'Y(d)}
(c; d) E Qb BL(a,b) = BL(a,c)
BR(c,b) = BR(b,b) - 0 < B
Since, moreover
we have again arrived at
,
(8.79). (iii)
d0 = c0 = a
If
and obtain analogously
Ilwl/gllq,(c,d) min
sup
B =
(8.83)
c = a
then we take
,
T(d)} _
(c;d)EQa = BL(a,b) = BR(c,b)
BL(a,c) = BL(a,a) = 0 < B
Since, moreover,
we have again arrived at
,
(8.79). (iv)
d0 < co
If
then it can be easily seen that
liv-1/Pllp',(a,b)
(8.84)
'Y(a) =
a = d0 < co = b
Consequently, (iv-1)
If
is increasing and (a,b)
,
1
< p <- W 'Y
,
then the functions
(0,L)
there exists exactly one point (D (c) _
.
.
is decreasing on
onto the interval
(8.85)
_ '(b) < -
with
(a,b)
,
(P
,
Y
are continuous,
and they map
L = lv-1/pllp',(a,b)
c t (a,b)
the interval Therefore,
'
such that
`Y (c)
and consequently
(8.86)
{(P(c), 'Y(d)} = (P(c) min a
{IP (c), ''(d)} _ Y Y (d) min c
,
B > BR(c,b)
113
which is (8.79). (iv-2)
p =
If
1
then the number
,
I1v-I11.,
S =
(a,b)
according to (8.84). Denoting
is finite
x1 = inf {x E (a,b]; O(x) x2 = sup x1 5 x2
we have
.
{x E [a,b); ''(x) In the case
x1 = x2 = b
we have
min {'(c), T(d)} = min {'(c), S} = O(c) a
c = b
and proceeding as in part (ii), we arrive at (8.82) and,con-
sequently, at (8.79). In the case
min {4(c), 'Y(d)} a
c = a
min a
,
we have
{S, 4'(d)} = T(d)
and proceeding as in part (iii), we arrive at (8.83) and,
consequently, at (8.79). If c E M
xl = x2 = a
M = [x1,x2] (l (a,b) x 0
,
then we choose
and have
min {@(c), 'r(d)} = min a
{0(c), S} = O(c)
min {O(c), 'r(d)} = min {S, T(d)} = T(d) c
8.11. Remarks.
(i)
The assertion of Theorem 8.8 now immediately follows
from the estimates (8.73) (Lemma 8.9), inequality (ii)
(8.87)
2-1/p B
(8.78) (Lemma 8.10) and the
in Theorem 8.2 [see also Remark 8.5 (i)].
-< C
Moreover, the estimate
2-1/p G < C
holds not only for
1
<_
p
q <_ -
but also for
1
< q < p <
Lemma 8.10 as well as in Remark 8.5 (i) we have supposed only
114
since in 1
< p,q < - .
(iii)
We have derived the estimate (8.87) using the inequality (8.78),
the number
i.e.
from (8.9). As will be seen later (see Remark 8.20 (i)),
B
this estimate can be derived also directly, but only for
1
< p < q < w
.
The next subsections deal with the approach described in Subsection 2.6 and Theorem 6.4, i.e. and
w
with formulas connecting the weight functions v
(cf. the formulas (2.13), (2.15), (6.11), (6.11*)).
Let
8.12. Lemma.
< p < q < -
1
w E W(a,b)
,
c - [a,b]
,
and
x
< -
w(t) dt
for every
x E (a,b)
.
J
c
Put x
1-p
p-l+p/q
Twt) dt
wl-P(x)
v(x) = (q
(8.88)
c
for
x E (a,b)
Then
(8.89)
(1;
1
[and, consequently, the Hardy inequality (8.6) holds for every u E ACLR(a,b) Proof. BL(a,c)
with the constant
C = k(q,p)].
It can be easily verified that under our assumptions the numbers and
BR(c,d)
BL(a,c) <
-
1
,
see (8.66), (8.67) and (8.70) BR(c,b) <
1
-
fulfil
,
and (8.89) follows immediately from (8.69). 11
8.13. Remark.
The assertion of Lemma 8.12 remains true if we express the
weight function
v
by the formula x
1-p
(8.90)
v(x) _ (q )
wl-P(x)
w(t) dt + d
J c
with an arbitrary
weight function
d E R .
v
The formula (8.90) sometimes enables us to give the
in a more convenient form.
115
The proof of this modified version is simple: if
b
c(
either 0 < d <
I
d E R is such that
w(t) dt or 0 > d > -
I
w(t) dt
a then it can be shown that there exists a number
c E (a,c)
c E (c,b)
or
such that
J
w(t) dt + d =
I
w(t) dt
C
and thus the proof reduces to Lemma 8.12 with
instead of
c
b other cases it can be shown that either
x estimated from above by
f w(t) dt
or
.
In the
f w(t) dt
can be
a
x
if w(t) dt + dl
c
x
and, consequently, the weight
c
function
from (8.90) estimates from above the weight function
v
from
v
the formulas (2.15) or (6.11*), respectively. This fact implies that BL(a,b)
<_
BL(a,b) <
or
1
8.14. Lemma.
Let
(8.91)
min
1
1
and (8.89)follows immediately.
< p < q < -
v E W(a,b)
,
{IIv-11p1I
liv-1/PEI
p
c,d E (a,b)
and
,
, ,
,
a, c.,
p
,
,
(d , b )
}
< m
.
c
put w(x)
(8.92) for
x E (a,b)
(8.89)
Proof.
,
vl-p'(x)
p
Let
0
1
,
c = c0
if
4'
,
co
d0
,
p ,(a,x)
II
p r ,(x,b)
co = d0
.
be the functions and numbers introduced in c e (a,b)
according to (8.85) if
It can be easily seen that
min 4(x), Y(x)} _ fi(x) a<x
116
ll1
ff
.
the proof of Lemma 8.10. Take and
Lmin
Then
.
2 <_
1J-pl-q
rr
d0
< c0
provided
cxa
,
{0(x), `'(x)} = Y(x)
min c<x
c
we b
.
This together with (8.92) implies that
w(x) = P v 1- P (x)
(x)J
'
(8.93)
for
and
L
-n'-o= 0p, 1-p, (x) v
1-pr
Jv a
I
and
x t (a,c)
1- q/p,
(
w(x) = P v1-pi(x)
(8.94)
I-1-q/p
(t) dt
v1-p' (x)
P
11
J v1-p (t) dt
l
x
for
provided
x E (c,b)
c E (a,b)
Comparing these formulas with the
.
formulas (2.13) and (6.11), we immediately obtain that BL(a,c) <
1
,
BR(c,b) <
1
which implies (8.89). If
c = b
BL(a,b) < If
1
,
c = a
BR(a,b) <
,
then (8.93) holds for
BR(b,b) = 0
which together with ,
then (8.94) holds for
8.15. Remark.
If there is a number
(8.93) and (8.94) are meaningful for
from (8.93), (8.94) (with
c
w(x) = P v I-p'( x ) {
11
such that the formulas and
x E (a,c) w(x)
instead of
[,D(x)]-p'-q
(8.95)
and consequently
implies (8.89).
c E [a,b]
tively, then we can replace the function w(x)
implies (8.89).
x E (a,b)
BL(a,a) = 0
, which together with
1
and consequently
x E (a,b)
[,Y (x)] -p,-q
for
x E (c,b)
,
respec-
from (8.92) by the function c
!), i.e.
x E (a,c)
for x E (c,b)
,
and (8.89) again holds.
The formula (8.95) is meaningful if we assume that there is a number such that
c E [a,b]
fi(x)
course, the weight function function
w
< w
on
(a,c)
and
Y(x) < -
on
(c,b)
.
Of
from (8.95) can be a little worse than the
from (8.92), which actually is the one from (8.95) with the 117
c = c
'optimal' choice
.
2
Let
8.16. Example. x
2
x E [0,11
(a,b) =
,
t 2k
x((
dt
I et
for
< p < q < -
1
w(t) = et. We have x 2k
x 2k+1
dt
I
k0 k!(2k + 1)
k!
_ 11k'0
,
kL0
x2
= e
k!
,
x
x
1
2
2
dt <
et
2
dt +
et
Jr
dt <
t et
e +2 e
x
2
for
x E (1,=)
1110
0
1
2
and since the function c = 0
Lemma 8.12 with
and we conclude that the inequality
2
J
u(x) q ex
holds for every
is even, we have verified the condition from
et
ll1/q dx] < C
11/P Jlu'(x)Ip v(x) dx]
( I
ACLRwith the weight function
u
for
ex2p/q v(x) = ' x2(1- ) e
ex2lp-1+ p/q
(e +
p
for
2 J
IxI
<1
x> 1
with the constant I
-1/P
C = k(q,p) (Q Using the estimate 2
2
2 (e + ex ) < ex
for
IxI
>
1
,
we immediately obtain that the Hardy inequality 2
-1/p
l1/q
<- k(q,P)q
u(x) Iq ex dxJ
(8.96) Jr
holds for every
u C
ACLR(_-,
From (8.96) we obtain for
118
2
fIu/ (x)IP ePx
)
a > 0
the inequality
/q dx 1/P I
(
r
(8.97)
}1/q
eay2
Iu(Y)Iq
1/ P
(
(Y)Ip euPY2/q dy]
Ca[
dy]
l
Cu = [a-1/2(1 + q/p')]1/q +1/p'.
with
u ` ACLRwith
the inequality (6.39) holds for
a>0
8 = ap/q
,
Consequently, we have shown that
.
Compare this condition with the (necessary) condition (6.40) when con-
sidering the inequality on ACLor AC R(--,-) p = q = 2
In particular, for
-
we obtain the inequality (8.2) of
C = 2// .
TREVES mentioned in Example 6.13 with the constant
Now, we will consider the case 1
s q
Our considerations will be based on the approach described in Remark 8.7. Analogously to the number
13
from (8.69), we introduce the number
= r,(a,b,w,v,q,p) =
(8.98)
AL(a,c) = AL(a,c,w,v,q,p)
where
max {AL(a,c), AR(c,b)}
inf
AR(b,b) = AR(c,b,w,v,q,p)
,
are defined
by (1.19) and (6.7). Again, we define (8.99)
AL(a,a) = AR(b,b) = 0
Let
8.17. Theorem.
1
.
_ q < p <
v, w E W(a,b)
Let the number
.
/ct
be defined by (8.98).
Then there exists a finite constant equality (8.6) holds for every (8.100)
such that the Hardy in-
if and only if
u E ACLR(a,b)
A <
The best possible constant (8.101)
C > 0
2
1/q
1/q
-1/p q
r
in (8.6) satisfies the estimates
C <
< C
1/r 2
1/q q
1/q'
,
(P
)
!4
with 1
r
_
1
q
_
1
p
119
and moreover, for A = AL(a,b)
C < ql q (p')1 q
(8.102)
= AR(a,b)
Or
also the estimate
,
iA .
Theorem 8.17 will be proved again via two lemmas. The proof of the first is omitted, since it is an analogue of the proof of Lemma 8.9. [Note that we obtain analogues of the inequalities (8.75) and (8.76) on
(a,c)
and the passage to the whole interval (a,b) is made possible (aq/p + Bq/p)1/q < 21/q -1/p (a + S) 1/p a, 0 by the inequality (c,b)
and
,
since now
q/p <
Let
8.18. Lemma.
constant
.]
1
1
< q < p < m
C < 21 r ql q (,)1/q,
and, moreover, for j4, = AL(a,b)
Let
8.19. Lemma.
1
< q < p
=
<<_
If
C = -
,
or
AT = AR(a,b)
v, w E W(a,b)
,
.
the estimate (8.102).
Then
1/q
2-1/p ql/q (pq)
(8.104)
(i)
Then the best possible
.
in (8.63) satisfies the estimate
C
(8.103)
Proof.
v, w E W(a,b)
,
= C
then (8.104) holds trivially. Therefore, suppose C< m.
In addition, assume that b
(8.105)
L =
J
vl-p'(t) dt
<
a
and on
(a,b)
define functions x
(8.106)
1P(x) = O(x;v) = J vl-p (t) dt a bf
(8.107)
vl-pi(t) dt
'Y(x) = 'Y(x;v) = J
x
Analogously as in the proof of Lemma 8.10, part (iv-1), there exists a point c = c(v) E (a,b)
120
such that
(8.108)
0(c) = Y(c)
(compare with (8.85)). If we show that the following two Hardy inequalities (a,c) and (c,b) hold:
on
(8.109)
< 21/p C
lgw1/qll
lg'v1/P II
for every
p,(a,c)
q,(a,c) gw1/gllq,(c,b)
(8.110)
< 21/p C
jg'v1/pl
g E AC (a, c) L
for every gGACR(c,b)
p, (c, b)
then due to Theorems 5.10 and 6.3 we have 1/q'
(8.111)
(8.112)
2-1/p ql/q
AL(a,c,w,v,q,P)
-<-
C
2-1/p q1/q (
r
)1 q AR( cb,w,v,q,p) _< C
which immediately implies (8.104). Therefore, we will prove that (8.109) really holds (the proof for (8.110) is analogous). We will use the fact that
-
since
C <
the
Hardy inequality (8.113)
Iluwl/gllq,(a,b)
holds for
u E ACLR(a,b)
Take
g e ACL(a,c)
.
` C lu'vl/Pllp,(a,b)
If
I1g'v1/pl '
p,(a,c)
= m
then
(8.109) holds
trivially. Therefore, assume (8.114)
Let
{cn}
llg'vl/Pll
be a sequence in c
For
n E N
p,(a,c)
n
t c
and
such that
n -+ m
for
x E (a,c ] Ig'(t)I
Q0
(a,c)
0
define
for tE (a,cn) for
t E Ccn,c)
x fn(t) dt
gn(x) = J
a
121
,
Then it follows from (8.115) and (8.114) that Ifnvl/pllp,(a,c)
IIg'vl/PIIP,(a,c)
and Holder's inequality together with (8.105) yields (8.117)
fn E L1(a,c) gn
Consequently, (8.118)
is absolutely continuous on
and, moreover,
(a,c]
g(x) = 0
lim
x-*a+ Define the function
for
p(x)
.
P(x) = D-' (T(.))
The continuity and monotonicity of decreasing, maps the interval (8.119)
p(x) = a
lim
by
x E (c,b)
onto
(c,b)
4'
imply that
p
is continuous,
and satisfies
(a,c)
p(x) = c
lim
,
and
x-*c+
x-*bNow define gn(x) (8.120)
for
x C (a,c]
f or
x E
u n (x) gn
p
(
x
)
(
c, b)
From (8.118), (8.119) we have (8.121)
lim
un(x) = lim
The function
un(x) = 0
x-*b-
x-*a+
is absolutely continuous on
u
absolutely continuous also on
since for
[c,b)
.
Moreover,
x E [c,b)
g'(t) dt = gn(P(x)) = un(x)
g'(P(y)) p' (y) dy =
a
x
due to (8.120) and b
c
Jlgn(P(y)) P'(y)I dy = [Ign(x)
c 122
a
dx =
I
a
f (x) dx <
u
we have
p (x )
b - J
(a,c]
n
is
due to (8.117). Consequently, in view of (8.121), un E ACLR(a,b)
(8.122)
.
Obviously (8.123) If
vp'-1(P(y)).vl-p'(y)
P'(y)
< -
p
for
y E (c,b)
then from (8.123) we have
,
P'(y)Ip-l
v(Y) = v(P(Y))
and therefore b
b r
v(y) dy =
lgn(P(Y)) P'(Y)IP v(y) dy = c
c
b -
JIg(p(y)) IP
IP'(Y)IP-1 v(y) P'(Y) dy
b
=
c(
JgP(P(Y))IP v(P(Y)) p'(y) dy =
-
c
v(x) dx
a
This together with (8.120), (8.116), (8.115) implies b(
r
Ju'(x)Ip v(x) dx = 2 Jlg'(x)
(8.124)
a If
y E (c,b)
p =
,
v(x) dx
2
a
a then we have from (8.123) that
p'(y)
1
for
and in view of (8.120), (8.116), (8.115) we conclude
,
lunvl/Pllp,
(c,b) (8.125)
v(x) dx
Ilunll=, (c, b)
llg' (p (x)) P' (x) II-, (c,b) =
= ess sup Ign(p(x))l = ess sup Ign(y)I < ess sup Ig'(y)I = a
1/p llp,(a,c)
Now, it follows from (8.120), (8.122), (8.113) for
un , and (8.124) or
(8.125) that
123
a n wl/q
n Ilgw1/q II
II
= IIun w1/q 1l
q,(a,`c)
q,(a,b)
(8.126)
< 21/P C Ilg'vl/P lip, C IIu'v1/PII n p,(a,b) and
0 < gn(x) <= gn+1(x)
Since
(a,c)
x
lim g(x) = J(g'(t)j dt n-I.. a for
x E (a,c)
we obtain from (8.126), letting
,
x
1JIg'(t)I
Iwl/q(x)
the inequality
n
dtC q,(a,c)
P,(a,d)
a
This implies the inequality (8.109) because
x
x <
Ig(x)I = fJ g'(t) dt a
Let
(ii)
1< p< -
a
W(a,b)
and suppose
define a function
n E N
vn(x) x v(x) + n (1 + Then
dt
be a general function from
v
For
.
Jg't)I
x2/(P
1))
,
x E (a,b)
satisfies the additional condition (8.105)
vn E W(a,b)
Analogously as in part (i) of the proof, we obtain a number
-
cf. (3.13).
cn E (a,b)
such that 4D (cn;vn)
''(cn,vn)
(cf. (8.106). (8.107) and (8.108)). Since the inequality (8.113) implies that
II
holds for on
(a,b)
uw1/qII
uq,(a,b)
< C IIu,vllp n up,(a,b)
u E ACLR(a,b) ,
in view of the obvious estimate
we finally arrive at the following analogues of (8.111),
(8.112): 1/q'
(8.127)
v(x) < vn(x)
2-1/p ql/q tp_g
AL(a,cn,w,vn,q,p) -`-
C
1/qr
r
1/q
2- 1/p
(8.128)
q
Ax(cn,b,w,vn,q,p)
r
From the sequence
n E V
,
{c n}
,
C
.
we can choose without loss of
generality a convergent monotone subsequence {yn}, yn -+ y E[a,b7
Using
.
Fatou's lemma, we obtain from (8.127) or (8.128) the estimates (8.111), (8.112) with
y
Let
(iii)
instead of
,
{an}
and define for
and arrive consequently at (8.104).
(P n(x) =
I
{bn}C (a,b)
,
W(a,b)
bn
x
dt =
vl-p,(t) dt
I
,
Yn(x) =
a1 < b1
,
p =- .
an 1 a,
we have
,
0n(c n) _ 'Yn(cn)
bn
v1 p'(t) dt
dt =
an
an
cn = 2 (ail + bn)
and suppose
such that
x E (an,bn)
x
For
,
be a general function from
v
We choose two sequences bn t b
c
J
J
x
x
The assumption
.
C < -
implies the validity of the inequality (8.113), and consequently Iiuw1/qiIq,(an,bn)
urwl/pu
< C
p,(an,bn)
holds for every
u E ACLR(an,bn)
is satisfied on
(an,bn)
,
.
Since the additional assumption (8.105)
we finally arrive as in part (1) at the
following analogues of (8.111), (8.112): 1/q'
r
(8.129)
(8.130)
2-1/p
rr
q1/q (p_,g)1/q
2-1/p ql/q
Choosing a subsequence proof by letting
``
C
AR(cn,bn,w,v,q,-)
-`-
C
{yn} C {c n}
and
n --' -
AL(an,cn,w,v,q,°°)
yn --)' y
as in part (ii), we complete the in (8.129), (8.130). LI
8.20. Remarks.
(i)
The method used in the proof of Lemma 8.19 enables us
to derive the estimate
(8.131) with
03
2-1/P JB < C from (8.69) provided 1
< p ` q = -
.
125
Indeed: Using Theorems 5.9 and 6.2 instead of Theorems 5.10 and 6.3, we can derive from (8.109), (8.110) the inequalities 2 -1/p
(*)
BL(a,c) < C
1/p
2 ,
BR(c,b) < C
which now replace the inequalities (8.111), (8.112). From
(*)
we immedi-
ately obtain (8.131).
In the case
p =
1
the method fails, and therefore we have to proceed
via Lemma 8.10 and Theorem 8.2. (ii)
The results mentioned in Theorem 8.8, Lemmas 8.12 and 8.14 and
8.17 are due to B. OPIC and the proofs are published here for the first time.
8.21. Examples.
For the convenience of the reader
we will give here not
only the results concerning (for particular inequalities) the case 1
< q < p < m
we will summarize all cases including those handled in
;
Subsection 8.6. .(i)
Let
< p,q < m ,
1
a,
8 EE R .
Then the Hardy inequality
1/q
(8.132)
Jlu(x)lq xa dx
1/p
< C 111u'(x)Ip x8 dxj
0
0
holds for every (8.133)
u E ACLR(0,=)
8 z p- 1,
p < _ q < m ,
l
(with a finite constant
a= 8
P-
p-
Note that the inequality (8.132) does not hold (i.e. This is caused by the fact that c E [a,b] (ii)
C
)
if and only if
1.
C
max {AL(a,c), AR(c,b)} = m
)
if
q < p
for every
(see Examples 6.7, 6.8, 6.9). Let
1
< p,q <
0 < b < =
,
a,
8 E R . Then the Hardy
inequality
u(x)IxdxjC If l/
(8.134)
(
lJ
0
holds for every 126
J
u E ACLR(O,b)
b
[fu'(x)!P
dxJ/P
xs
0
if and only if one of the following two
conditions is fulfilled:
(8.135)
(8.136)
1
<_
Let
1
p
a> -
1
(iii)
8 g -- 1
a >-
q<m,
for
1
a> 8q-p -
ER ,
0
<_ p,q <
8 z p8= p -
for
p
1
.
a, 8ER . Then the Hardy
,
inequality 1/q
a
(8.137)
l 1/p < C
dx
[fiu(x)Iq x
[Jiu' (x )IP x
a
dxJ
a u E ACLR(a,-)
holds for every
if and only if one of the following two
conditions is fulfilled:
8 pa< a
(8.138)
1
,
p_q_
I
1
1
(8.139)
(iv)
1
Let
1
I1
8= p -
1
x1a
1(8.140)
dx1/q
dx]l/P
C
[Jiu'xiP
Iln x18
xP
0
holds for every
(v)
for
a, 8ER . Then the Hardy inequality
0
(8.141)
8 X p- 1,
IR
< p,q < m ,
Iu(x)ln
for
u E ACLR(0,1)
1< p < q < m Let
1
1, a= 8
8xp
,
< p,q < m
if and only if
,
a,
8 e 1R
.
p-
_q_
pI
1
Then the Hardy inequality 1/p
1/q 1
(8.142)
(jrlu(x)Iq
Iln x1a dxJ
x 0
holds for every
< C
()xP-1 Iln xl8 dxj [Jiu'xiP 0
u E ACLR(0,m)
if and only if
127
(8.143)
1
Let
(vi)
r
(8.144)
1
< p,q < m
1
Then the Hardy inequality
Sx
D
jHu'(x)
e
dxJ
1/P
u E ACLRif and only if
1< p < q < m
8.22. Remark.
8 E R .
= C
dxJ
e
holds for every (8.145)
a,
,
11/q
ax
r
I
sP p,
p
8x0
,
,
a= 8 p
The methods used in this section (see Remark 8.7), the re-
sults mentioned in Lemma 7.12, used now separately for the intervals and
(a,c)
for
(c,b)
c E (a,b)
,
and Remark 7.14 (i) allow us to for-
mulate the following theorem which is in fact a complement of Theorem 7.13.
8.23. Theorem. c E (a,b)
(i) (8.146)
Let
v0, v1, w E W(a,b)
,
v0
,
v1
satisfy (7.40), (7.41),
.
If
1
BL(a,c,w,v,q,P) <
px=
, °°
,
,
BR(c,b,w,v,q,P) <
then (8.147)
(ii)
W" P(a,b;v0,v1) Cj Lq(a,b;w)
If
1
< p < q < -
,
(8.146) is satisfied and
lim FL(x;a,c,w,v,q,p) = lim FL(x;a,c,w,v,q,p) = 0 x4cx±a+ lim FR(x;c,b,w,v,q,p) = lim FR(x;c,b,w,v,q,p) = 0 x±c+ x±b-
then (8.148)
WO,P(a,b;v0,v1)
(iii) If (8.149) 128
1
-<_
(; c Lq(a,b;w)
q
AL(a,c,w,v,q,P) <
AR(c,b,w,v,q,P) <
,
then (8.147) holds.
If
(iv)
1
< q < p < m
and (8.149) is satisfied, then (8.148) holds.
9. THE HARDY INEQUALITY FOR
0 < q <
bll
In Subsection 2.10 we have mentioned that the Hardy
9.1. Introduction. inequality
[Ju(x) q w (x) dx]
< C
ACL(a,b)
b(9.1)
[Ju'(x)IP v(x) dx
I
a
a on
1
ACR(a,b)
or
has been investigated by G. SINNAMON [1 ]
for
the case
0 < q < 1, 1< p < m
(9.2)
Here we will follow, again with some modifications, the ideas of SINNAMON. In
the
case
(9.2)
some special methods are used, based on a result
,
of I. HALPERIN [1]. Therefore, let us first formulate this result; the proof, which is rather technical, will be given in Subsection 22.
9.2. Theorem.
Let
f E M+(a,b)
,
p E W(a,b)
be such that
br
f(t) dt <
(9.3)
J a
J
a Then there exists a function
p(t) dt < f0 E M+(a,b)
satisfying
xr
f(t) dt
(9.4)
f0(t) dt
J
J
a
a
for
(9.5)
f0/p is non-increasing on
(9.6)
IIf0/p
P,(a,b),p
<
11f/pl
x E (a,b)
(a,b)
;
p,(a,b),p for every p E [l,')
.
In the main result of this section, which will be now formulated, the 129
number
AL = AL(a,b,w,v,q,p) _
(9.7)
b
b
= If If w(t) dt
r = q1 - p1 I
from (1.19) appears. Note that
holds
dx
and that the number
is now negative due to the condition
q' = q/(q - 1)
(9.1)
I1/ r
dtlr/q'v1-p'(x)
a
x
a
9.3. Theorem.
x x vl-p'(t)
jr/q
0 < q <
Let
< p < -
1
.
v, w = W(a,b). Then the inequality
,
if and only if
u E ACL(a,b)
every
for
1
0 < q <
AL = AL(a,b,w,v,q,p) < -
(9.8)
Moreover, for the best possible constant
in (9.1) the following
C
estimate is satisfied: 1/q(p,)l/q'
(9.9)
q
9.4. Remark.
AL < C ` ql/q AL.
Theorem 9.3 is a consequence of several lemmas, which will be
proved later (see Lemmas 9.7 and 9.8). First, let us go through some auxiliary considerations. In the case on
ACL(a,b)
AL <
,
1
< q < p < -
the validity of the Hardy inequality (9.1)
,
was characterized not only by the condition (1.25), i.e.
but also by the condition
AL <
,
where
the formula (7.33) (see Remark 7.8). The number (9.10)
was introduced by
AL
AL
AL = AL(a,b,w,v,q,p) _ b
x
r /p'
(
(b
If vl-p(t) dt
i J
a
l
r/p
l 1/r w(x) dx}
II w(t) dt] I
a
x
is meaningful also for the case
0 < q <
1
,
1
< p < -
.
Now we will show
that, also in this latter case, the condition (9.8) can be replaced by the condition
9.5. Lemma. 130
AL < -
Let
.
This will follow from the next lemma.
0 < q <
1
< p < -
,
v, w e W(a,b)
.
Let
AL
,
be
defined by (9.7) and (9.10), respectively. Then 1/r
Al (a,b,w,v,q,p) = (q )
(9.11)
AL(a,b,w,v,q,P)
Proof. Obviously, it suffices to prove the following two implications
1/r <
<°' =->
(9.12)
A
(9.13)
AL <
Z'
AZ
AL
1/r
_> (q ) AZ < m
Assume
(i)
AL
AL
and suppose in addition that
r
vl-p E L1(a,b)
w,
(9.14)
.
Then we obtain the identity (9.11) by integration by parts analogously as in Remark 7.8. v
If
,
w
are general functions from
W(a,b)
,
we introduce auxiliary
functions x2/(pr_l))
vn() = v(x) + n (1 +
(9.15)
Then
w
satisfy the conditions (9.14) and, consequently, the identity v n n v instead of w , v (9.11) holds for w n n ,
,
:
1/r (9.17)
Since
AL(a,b,wn,vn,q,p)
AL(a,b,wn,vn,q,P) = (q)
w (x) s w(x) n
and
vl pr(x)
-i vl
n
pl(x) ,
estimate the right-hand side in (9.17) by
for 1/r
x e (a,b)
,
we can
(L)AL(a,b,w,v,q,p)
obtain-
q
ing 1/r
AL(a,b,w ,v n,q,P) n For
n
(ii)
)
q
AA(a,b,w,v,q,P)
Fatou's lemma implies that the implication (9.12) holds. Assume
AL < m
.
Using the functions
FL
from (1.17) and
0L
from (7.34), we can easily verify that the assertions (ii) and (iv) of Lemma 7.7 hold also for
0 < q <
1
< p < m
,
i.e.
that 131
1/r
FL(T) < 1L(E) < AL lim
0
(P
E E (a,b)
,
.
-a+ Consequently,
x
b r
< m
J
x
a
FL(T) = 0
for
x E (a,b)
and
(a,b)
is continuous on
FL(E)
lim
vl-p,(t) dt < W
and
J
the function (9.18)
f
w(t) dt
.
C-O-a+
and for
such that
}, {b } C (a,b) n n define the numbers
{a
Choose two sequences n, k E IN
b
lx
J
n
+ a
,
la
)
b
n
? b,
1/r
r
w(t) dtlr/9 (x vl-p'(t) dtr/q
An'k -
a
v1-p/(x) dx
J
ak
bnx( (J An,k
1r/p/
(b(
J
lx
-1-p'( t) dt
J
ak
J
1/r
r/p w(t) dtj
w(x) dx
a
Integration by parts yields An,k =
-
Letting
k -+ r >
,
-
[FL(bn) - Fl(ak)] + q
FL(b n ) +
lim Fl(ak) = 0 k-
we have
132
(A*'k)r
we obtain
q
since
FL(ak)l + q (A*.k)r
AL ? An,k , we have
and since obviously AL >- P-L
[FL(bn)
lim (An k)r ' k.w
due to (9.18) and
r
lim (An* k)r q
k1-
FL(bn) > 0
.
Letting
n---*
AL ?
(A* )r lim (lira (A*n,k )r) = f'! q
q
k-w
n+w
and the implication (9.13) is proved. 11
In the next assertions we will deal
9,6. Remark.
with the inequality
-
equality (9.1)
b
[J(Hf)(x) w(x) dx j
(9,19)
instead of the in-
-
b
1/ q
1 1/ p
(
J fp(x) v(x) dx
C
a
1
]
a
for
This is possible since the assertion of Lemma 1.10 holds
f E M (a,b)
also for
p
from (9.2). The reader can easily realize that the proof
q
,
of Lemma 1.10 remains valid if we replace the assumption assumption
q > 0
9.7. Lemma.
Let
number
AL
f E M+(a,b)
(i)
(9.21)
1
by the
The same is true as concerns Remark 3.7 and Lemma 3.4.
0 < q <
1
< p < w
v, w e W(a,b)
,
.
Assume that the
from (9.7) is finite. Then the inequality (9.19) holds for every with the constant C = ql/q AL
(9.20)
Proof.
.
q >
Let
.
f E M+(a,b)
and assume in addition that
f E L1(a,b)
and (9.22)
P = vl-p, E L'(a,b)
Consequently, the assumptions (9.3) of Theorem 9.2 are satisfied and there exists a function
f0
satisfying (9.4) - (9.6). From (9.4) we have
((b(
(9.23)
}1/q
IJ(HLf)q(x) w(x) dx 1J
a
x(
b(
(J
a
a
(b(
xr
q
f(t)
I
q
f0(t) dt)
IJ
1/q
w(x) dx
dt)
l
1/q
w(x) dxj
(J
a
and since
p,(a,b),P
lg/pIi=
Igv1/p p,(a,b)
a
due to the definition of
p
-
133
see (9.22)
we can rewrite (9.6) in the form
-
l1/P
br
b(
fP(x) v(x) dxl
Ifa
=
IJ fp(x) v(x) dx l
JJJ
11/p
a
This together with (9.23) implies that it suffices to verify the inequality the inequality i.e. instead of f f0 (9.19) for ,
x
b
f0(t) dt)
(9.24) If
(J
a
a
b
1/q
q
w(x) dx)
q1/q AL IJ
11/p
fp(x) v(x) dx
a
Since b
x(
l f0(t) dtl w(x) dx = 4
1J
1
a
a
)))
b
x
y
[J
l)
a
a
= q
f0(t)
l
a
dtl-1 q f0(y) dyI w(x) dx
the Fubini theorem yields q
0(t) dt)
(9.25)
w(x) dx =
a
a
b lq-
f0(t) dt) a
Let
f0(y)
II w(x)
a
dx)
dy
y
be fixed. The condition (9.5) implies
y c (a,b)
f0(Y) p(t) < f0(t)
t G (a,y)
for every
P(Y)
and consequently (note that
0 < q <
1
!
yq-1
q-1
f0(t)
p(Y)
a Using this estimate in (9.25), we obtain
134
)
lJ
a
p(t) dt)
q
f0(t) dt) a
111/q
w(x) dxJ
a b
Yq-1
(Y)lq-1
q1/q
(f
[P(Y)
1
a
p(t) )
b
f0(Y)
dt1
1/q
l
w(x) dx] dy}
{J
Ij
a
y
11
b(
fb(
(
= q
1/q
J
J
l a
1
dxIJ
w(x)
I
y
p(x) dx
]
la
1/q
Pq/r(y) fq(Y) pi-q-q/r(Y) dY}
and Holder's inequality with exponents
b(
x(
q
l f0(t) dt) w(x) dxJ
If (J
r/q = p/(p - q)
,
p/q
yields
1/q
a
a
b q
1/q
b
i
}r/q
if
1
P(x) dxj
1/r p(y) dy}
J
y
l a
r/q'
y(
w(x) dxl
a
b
b
1/ p
= ql/q AL
fp(y) P1-P(Y) dy}
IJ
fp(x) v(x) dx
lllla
JJl
a
However, this is the inequality (9.24). (ii)
Assume that
f
is a general function from
M+(a,b)
and that
(9.22) holds. Define
fn(x) = min (f(x),n/x2,n) Then
f
n
E L1(a,b)
,
i.e.
(9.21) is satisfied with
Therefore, the inequality (9.19) holds with
1/q
(b
(b
I J (HLfn)q (x) w(x) dxJ
C
a with
C = q1/q AL
J
fn
f
n
instead of
instead of
fn (x) v (x) dx
f
f
:
l I/ P
a .
Letting
n
,
we obtain (9.19) by the monotone
135
convergence theorem. Assume that
(iii)
1-p?
by (9.15). Then v
instead of
is a general function from
v
E. L1(a,b)
.
vn
Define
(9.22) is satisfied with
AI(a,b,w,vn,q,p)
The number
.
i.e.
,
W(a,b)
Vn
is finite since
1/r (
AL(a,b,w,vn,q,p) _
q
AL(a,b,w,vn,q,p)
)
_<
1/r
(9.26) (p
AL(a,b,w,v,q,p) = AL(a,b,w,v,q,p) <
]
q
au
Indeed, the identities in (9.26) are due to Lemma 9.5 (cf. the formula p, <
(9.11)), the sign
follows from the inequality
-pappears
AL , where
definition of
,
(cf. the
in a positive power of an
v
integral) and the last sign
vl-p
vn
is due to the assumption of our lemma.
<
Consequently, part (ii) of the proof implies that the inequality b
1/q 1
J(HLf)q(x) w(x) dxJ
<
a b
1/p
q1/q AL(a,b,w,vn,q,p)
<
(( fp(x) vn(x) dxJ a
holds for
f E M+(a,b)
hand side by
1/q
.
Moreover, we can replace the constant on the right-
AL = q1
/q
AL(a,b,w,v,q,p)
b1/q q
which is independent of
n
due to (9.26):
(9.27)
[J(Hf)(X)
1/p
br
< ql/q AL
w(x) dxj
[J
fp(x) vn(x) dx
a
a ,
Let
g E M+(a,b)
f(x) = g(x)n v(x) . Obviously,
and
we have from (9.27) that b r
J
a
x r
(J
g(t) vn1-p, (t)
dt
q
w(x) dxJ
a b ) 1/p
vn-p,(x)
<
ql/q AL
gp(x) J
a
136
dx
1/q
f E M+(a,b)
and
I
v1-p
vn-p
Since
?
br IJ
a
the monotone convergence theorem yields
,
xr (J
q
I
g(t) v(t) dt)
}1/q < w(x) dx]
a b
1/p
_
,
ql/q A,J gp(x) v1 p (x) dx
<
a
g e M+(a,b)
which holds for arbitrary
.
immediately obtain the inequality (9.19).
0 < q <
Let
9.8. Lemma.
1
< p < - ,
g(x) = f(x) vp'-1(x) , we
Taking
D
. Assume that the in-
v, w E W(a,b)
equality (9.19) holds (with a finite constant
C ) for every
M+(a,b)
f
Then I
(9.28)
AL < C
q1/q (p')1 q
Proof. According to Remark 9.6, Lemma 3.4 implies that the inequality BL 5 C
with
1
,
and, consequently,
x
b
vl-p (t) dt < -
w(t) dt <
(9.29)
x E (a,b)
for
a
x Choose two sequences for
0 < q <
it follows that
C < -
from
from (1.18) holds also for
BL
{a },
n
{b
n
such that
} C (a,b)
a
n
1 a
,
b
n
t b , and
n E N define numbers rbfn
bJ
r/q'
(x
1/r
1
(9.30)
An =
w(t) dtJr/q
LJ
an
IJ
vl-p'(t) dtj
v1-p (x) dxJ
an
x
and auxiliary functions (br
(9.31)
fn (x) =
}r/p
II w(t) dt]
x
,
vl-p (t) dtj an
[J
r/q
v1-p/(x)
X(an,bn)(x)
x
Obviously
fn E M+(a,b)
and from (9.30) we have
137
((
A n
bn
b J
an
1/r
(x
l f [r vl-p,(t) dtJr/q' vl-p'(x) dxJ w(t) dt I1/q [l an an
(9.32)
(b
(L,) 1/r r
w(t) dt]
l
bn
1/q
t] vl-p
U an
an
' (t) dt]
1/p'
due to (9.29). Moreover, b
b {
A
II w(t) dt] dx
fn(x)
J
lll
l
n
x
a
and since
fn(x) > 0
(9.33)
An > 0
a.e. in
(an,bn)
and
IJ w(t) dt
dx1
w C W(a,b)
,
we conclude
.
The Fubini theorem yields (b(
b(
1/q
(
An/q = i
J
fn(x)
=
I
lx
l
a t(
b(
( j
( J
W(t)
II
a
l 1/q l fn(x) dx] dt1
a
and since
x
x
f(t) dt =
fn(t) dt
If [
a
a
1/q
-q' [qJ[J a
we have
r/q
b(
(
(9.34)
q An
J
a
If we denote
138
fn(s) dsJ
f n (t)
a
t1/q' fI
x[In If
a
(s) ds
a
dtj
1/q
q
fn(t) dtI
w(x) dx
gn(t) =
(t
1/q.
l
fn(s) dsJ
IJ
fn(t)
a then we can rewrite (9.34) in the form b
1/q
[J(Hg)(x)
q
w(x) dx J
a
and the inequality (9.19) yields b ((
q An /q - C
II
11/p = gn(x) v(x) dxJ
a rb(
(xr
1/p
p/q,
l
C
J
a
fn(x) v(x) dxJ
U fn(s) dsJ a
bn
x
dsIpli-q1 v(1-p')q(x)
V'-p'(s)
= C {J fn (x) vp'(x) an
x
lJ
an
r
I
lan
1/p
p(q-1) v(1-p')(1-q)(x)
fn(s) dsJ-p/q
dx
V'-p' (s) ds I lJx
an Using Holder's inequality with exponents
1
and the definition
1/(1 - q)
q
of
fn
-
(9.31)
cf.
q
-
we obtain
Ar/q n
bn (
0
=Cj (9.35)
J
bn
x
fn/q(x)
l
vp,/q(x)
llJan V'-p'(s) II( ds
}q/p
1-p,
p/q' v
(x) dx
J
x
p/q
x
fn(s) ds
}(1)/P V'-p' (s) dsJ-p v1-p,(x) dx
=
I
an
an
an
139
bn
x
P/q
rq/P = C An
j
fn(s) dsJ
IJ
I
(1-q)/p
_
w(x) dx
}
flan flan
where
x I-P
(9.36)
vl-pa(s) dsv1-pI(x)
w(x) a
x E (an,bn)
,
n
Since, in view of (9.29), bn
x dsll-P
w(t) dt 5
(9.37)
p
1
<
vl-P(s)
1
J
an
x
we obtain the estimate p/q
xr
r
f(s) ds)
(9.38) I
an
q/p
_
(brn
k(Q, -IJ fn/() v(x) dxJ
w(x) dxJ
J an
fan
with
b
P/q -1 (9.39)
[w(x)
1-p/q ( r n
v(x) _ (q - 1)
IJ
p/q w(t) dtj
x E (an,bn
,
x [cf.
(a,b)
(2.16) and (2.15) with and with
w
instead of
instead of
p/q
p
w , with
and of
(a n,b
n
)
instead of
q J. Using (9.36) and (9.37)
in (9.39), we have
v(x) < p/q -1 (P
<
-
(P -
1)
1)-p/q
rxr
vl-p' (s) ds l-p/4 rvl-p/ (x)] 1- p/q
an This estimate together with (9.31) yields bn r
k1, ) 0 f°/q() an
140
(x) dx
}q/p <
bn q
<
r/q
b
j
L
lj w(t) dtj
an
(x
r
vl-p (t) dtj
lJ
x
r/qr
q/p
r
vl -p(x)
an
dx
=
1
Arq/p n
= q
and using this inequality in (9.38), we obtain from (9.35) that Arq/p)(1-q)/q
Ar/q < C Arq/p (2
q
n
n
q
n
=
(2 )-l/q'
lq
J
C pr/P n
J
In view of (9.32) and (9.33) we have 1/ q
'
(Q
q
An = ql/q (P,)1/qr A n < C
and Fatou's lemma yields (9.28).
9.9. Examples.
C
A comparison with the particular Hardy inequalities in-
vestigated in Section 6 shows that the results derived in Examples 6.7, 6.8, 6.9, 6.10 and 6.12 for the case consider
p
0 < q <
1
for no
a,
,
q
c q < p < m
< p <
,
then the inequalities (6.16), (6.30) and (6.36) hold
bl I/
6 E R (with a finite constant
1/q lJju(x)Iq xa dxJ
C
0
0 < b < W
remain true if we
satisfying (9.2). More precisely, if we suppose that
(b
with
1
C ), while the inequality
[Jiu'xP
P
x6 dxl
0
holds for
u E ACL(0,b)
if and only if
and the inequality m
[jlu(x)I qxa a with
0 < a < m
l1/p
1/q dx)
< C [JIu(x)jP x6 dx]
a holds for
u E ACL(a,00)
if and only if
141
P- P -
S< p 9.10. Remarks.
(i)
p-
or
1
We have dealt only with functions
u E ACL(a,b)
.
Obviously, using the substitutions indicated in Remark 1.8, we can reduce the investigation of the Hardy inequality on
ACR(a,b)
to the foregoing
one. The formulation of the corresponding result is left to the reader; in fact, one has only to replace AL by AR in Theorem 9.3, where AR is given by the formula (6.7). If we consider functions
(ii)
ACLR(a,b)
u
from
,
the condition
then
from (8.100), where
4
is given by (8.98), again characterizes the
validity of the Hardy inequality (9.1) on the class 0 < q <
ACLR(a,b)
for
< p < - . The proof follows word by word the arguments used in
1
the proof of Theorem 8.17, only the estimates (8.101) and (8.102) are slightly changed: For the best possible constant
C
in (9.1) we obtain in
this case the estimates (p,)1/q'A < C < 21/r ql/q
2-1/p
ql/q
with
r = pq/(p - q)
,
and for
AL(a,b)
or u4 = AR(a,b)
we have
the estimate
C ` ql/q.
10. HIGHER ORDER DERIVATIVES
10.1. Formulation of the problem.
Let
k
be a positive integer which we
will write in the form k = m + n
(10.1)
where (10.2) 142
m
,
n
are non-negative integers. Denote
n ACmkl)(a,b)
u(k) = dku/dxk
and let
be defined as the set of all
u(k-1) (a)
u(a) = u'(a) _
(10.3) if
such that
u
(and consequently
m = k
if 1 5 m < k
or
,
n = k ). Thus we have
(and consequently
m = 0
left endpoint and (a,b)
_ ... = u(k-1)(b) = 0
u(b) = u'(b) = ... = u(k-1)(b) = 0
(10.5)
if
... = u(m-1)(a) = 0
u(m-1)(b)
u(m)(b) =
= 0
n = 0 ), or
u(a) = u'(a) = (10.4)
and
u(k-1) E- AC(a,b)
m
conditions at the
conditions at the right endpoint of the interval
n
.
We will investigate conditions on given
p
,
q
)
the (Hardy) inequality
b1l1/q
[TIu(x) lq
(10.6)
w(x) dx]
C
a
bl1/p
[JIu(x) Ipvk(x) dx]
u E AC(k-1)(a,b) m,n
The problem just formulated can be solved using the Hardy which corresponds to the case
inequality (1.11)
-
for the functions
u E
ACR(a,b)....
C A C R(a,b) ,
n - 1
,
k =
u' E ACL(a,b),...,u ,u(k-1
u(m) E
m = 1
under which (for
a
holds for every
10.2. Remark.
w, vk E W(a,b)
.
-
1
(m-1)
successively
C- ACL(a,b)
For example, taking
k = 2
we investigate the inequality
b
b
l
(10.7)
q
Jw(x) dxl l/
l v2(x) dx]
p l
C
a
a for functions (10.8)
u E ACi1i(a,b)
u(a) = 0
,
,
u'(b) = 0
J
i.e. such that ,
and we can solve this problem considering two inequalities
143
b (10.9)
q
1
fj.(.)Iq
blll/r w(x) dxJ
[Ju?(x)Ir vl(x) dxJ
CI
a
(10.10)
[Ju'(x)
r
a
lll/p
brr
vl(x) dxJ
(Jlu"(x)IP v2(x) dxJ
C2
a
a
for
1/r
b
u r ACL(a,b)
u' E ACR(a,b)
and
we consider the case
and a parameter
v1
'intermediate' weight function
r
.
For instance, if
r e [p,q] , we know (cf.
and choose
< p < q <
1
respectively, with a certain
,
Theorems 1.14 and 6.2) that the inequality (10.9) holds for
u E ACL(a,b)
if and only if (10.11)
BL(a,b,w,vl,q,r) < m
,
while the inequality (10.10) holds for (10.12)
BR(a,b,vl,v2,r,p) < -
u' E ACR(a,b)
if and only if
,
Consequently, the pair of conditions (10.11), (10.12) is at least sufficient
for the inequality (10.7) to be valid on
ACili(a,b)
This approach has one disadvantage: the presence of rather undetermined and (from the point of view of the original inequality (10.7)) redundant parameters
r
v1
,
Moreover, the number of these additional
.
parameters will rapidly grow for big numbers
k
.
Therefore, it is our aim to derive (necessary and sufficient) conditions for the validity of the Hardy inequality (10.6), in which only the initial parameters
10.3. Example.
Let
1
p
,
q
, w
< p < q
,
vk
(and
m
n ) would appear.
,
and consider the in-
S L R ,
<
equality (m
(10.13)
(J
u(x) Iq xa dxl
0
on the class Here
m
l/q
_C
ACili (0,o)
w(x) = x a
,
,
dxJ 1/P
0
J
i.e. for
v2(x) = x s
;
introduce an auxiliary parameter 144
[Ju"(x) IP x
u
u(0) =0 and
such that
u'(oo) = 0.
if we use the approach just described, we r
,
p = r = q
,
and an auxiliary weight
function
v1(x) = xY
a = 0
Then the inequality (10.9) holds (with
and
u E ACL(O,=)
b = - ) for (10.14)
.
y < r - 1
-
Cl = Y 9 -
,
,
1
u E ACR(0,0)
and
a = 0
and the inequality (10.10) holds (again with
b = W ) for
,
1, Y=
>p
(10.15)
r- p. -
1
[Cf. Example 6.7, the formula (6.18) with the formula (6.19) with y
we can eliminate
y
and
r
,
y
instead of
instead of
r
,
a
,
q
R
,
p
,
and
.] From (10.14), (10.15)
and we finally obtain that the inequality
r
(10.13) holds for
u E AC1,1(0'-) if
(10.16)
p-1
<
R < 2p - 1
,
a= R
P-
P
- q
1
(Compare with Example 10.16.)
Let us just formulate the main result; for simplicity, we will not deal with the limit values (0,°°)
p ,
q = 1, -
and will consider the interval
only.
10.4. Theorem.
Let
1
< p < q < -
there exists a finite constant
,
m, n E N ,
w, vm,n e W(O,m)
Then
such that the inequality
c > 0
ll1/q(10.17) P < C IJlu(m+n)(x)
[Ju(x) lq w(x) dx]
l1/p v
m,n
(x) dx
0
0
holds for every function
.
u E
AC(m+n-1)(O,-)
m,n
if and only if the following
two conditions are satisfied:
145
kr
sup
(10.18)
0<x<«,
x(
x
vm,n (t)
0
sup O<x<°° lx w(t) 0
dtl
t(n-1)p
tmq dtJ If '
<
vm,n (t) J
x
The necessary and sufficient conditions (10.18),
(i)
10.5. Remarks.
'
dt]
np
1/q
(10.19)
1p
,
,
dt(J
w(t) t(m-1)q
(10.19) can be rewritten in terms of the constants
BL
,
< °°
,
BR
introduced in
(1.18) and (6.2) as follows w(t)t(m-1)q,vm'n(t)t-np,
BL(O, (10.20)
w(t)tmq,
BR(O,
q, p)
vm,n(t)t-(n-1)p,
p)
q,
< -
Similarly as in the foregoing sections, we will deal with a
(ii)
little different problem analogous to Problem 1.7: For
f E M1"(0,-)
,
we
denote (Hm,nf)(x) =
x
(10.21)
t)m-1
1
(m - 1)!(n - 1)!
t)n-1
[
-
J (x -
f(s) ds] dt
1,
t
0
and deal with the inequality p
ll1/q
((
< C
II(Hm,nf)q(x) w(x) dxJ
(10.22)
fp(x) vm,n(x) dxl
II
0
0
instead of the inequality (10.17). Clearly, the function satisfies the conditions (10.4) [with
a = 0
,
b
u = Hm,nf and we have
f = u(k)
10.6. Proof of Theorem 10.4.
Thus, we shall show that the conditions
(10.18), (10.19) are necessary and sufficient for (10.22) to hold.
Let us fix write
H
,
v
,
c
m, n E N instead of
and denote Hm,n '
146
vm,n
cm,n = (m - 1)!(n - 1)!. Further, '
ctt,,n
for simplicity. Then we
have
x
c(f)(x)
ll
t)n-1
[J(s 0
f(s) dsI dt
t
x
x
(
r
J(x - t)m-l [Js
t)n-1
-
f(s) ds +
t
0
J(s -
f(s)
t)n-1
ds
dt =
x
rr
t)m-1
f(s) [J(x -
(10.23)
(s - t) n-l dtI ds +
I
0
0
x
f(s)
+
[(x L
x
- t) M-1 (s - t) n-l dtl ds
=
0
- (Jlf)(x) + (J2f)(x)
.
If we denote s
(10.24)
K1(x,s) =
t)m-1
J(x -
(s _ t)n-1 dt ,
0 < s < x
,
< x < s
,
0
x
(10.25)
On-1
K2(x,s) = I(x - t) m-1 (s -
dt
0
0
then
m + n -
1= <
1
m + n -
x-m+1 s-n
x-m
-n+l
1
K1(x,s)
1
<
K2(x s) <
.
'
m Indeed, in (10.24) we have 1 - tx >
1
_
t s
,
0 < t < s < x
therefore,
;
0 < x - t < x
and
and consequently s
K(x,s) < ( xm-1(s 1 1
-
t)n-1 dt
xm-1 sn n
0
147
S
m-1
(
xm+1 -
s -n K1(x,s) = s
I(1 J0
x
s t
(1 - S
1
=
s
n-1
(1 - 5
dt
m+n-2
dt
0 K2 , where we use the fact that in (10.25) we have
similarly for
0
therefore, 0 < s - t < s
;
and
1
- S>
1
-
-
Thus, the function
(J1f)(x) = J
K1(x,s) f(s) ds
0
is equivalent to the function x
x
J
m-1
s
n
f (s) ds
0
and the inequality (10.26)
f
Jlfllq,(0,-),w < C
p,(0,°°),v
will hold if and only if the inequality x (I
(10.27)
x
(f
n
m- 1
s
q
f(s) ds)
l1/q < C w(x) dxJ
ll1/p
(I
fp(x) v(x) dx] lJ
lJ
0
0
0
holds. However, the last inequality is nothing else than the inequality replaced by
(1.12) with f(s) w(x)
,
replaced by w(x) = x(m-1)gw(x)
v(x)
r
(r
m-1
x
IJ
n s
and with the weight functions
f(s) = snf(s)
f(s) ds
,
v(x) = x-npv(x)
.
Indeed,
w(x) dx = I
0f
0
. =
(
J
lJ
0
and 148
X
fr
°(D
x I
sn f(s) ds q x(m-1)q w(x) dx = 0
I
(J 1
0
U 0
l
f(s) dslq w(x) dx 111
fP(x) v(x) dx = J(xflf(x))P x-np v(x) dx =
I
fp(x) v(x) dx
J
0
0
0
But then, according to Theorem 1.14, the inequality (10.27) holds if and
only if (10.28)
B1(0,-,w,v,q,p) <
which is the first condition in (10.20), i.e. the condition (10.18).
Analogously, the function
K2(x,s) f(s) ds
(i2f)(x) = J
x
is equivalent to the function f xm sn-1 f(s) ds
,
x
and the inequality (10.29)
1
f I
2
q, (0,°°) ,w
< C Ifllp,(0,0),v
will hold if and only if the inequality
r
(r
(10.30)
x J
J
0
x
m
n-l s
f(s) dsI
q
1/P
w(x) dx < C
IJ
fp(x) v(x) dxj J
0
holds. This last inequality is nothing else than the inequality (1.13) with f
, w , v
v(x) =
replaced by x-(n-1)p
f(s) = sn-1 f(s),
w(x) = xmq w(x)
v(x) , and, according to Theorem 6.2, the necessary and
sufficient condition for its validity reads as follows: (10.31)
BR(0,-,w,v,q,P)
<
This is the latter condition in (10.20), i.e. the condition (10.19). Thus we have shown that the conditions (10.18), (10.19) are necessary
and sufficient for the validity of the inequalities (10.26) and (10.29), respectively. However, according to (10.23) we have
149
Hf = c-1(J1f + J2f) H
and since the operators
,
,
J1
,
for
f)(x) s c(Hf)(x)
(J
J2
are positive, we obtain the estimate x e (0,m)
i - 1,2
,
i
This implies that
PifIIq,(0,°°),w ' c {IH0q,(O,-),w IIJ1fIIq,(0,-).w +
<=
IIq,(0,°°),w
and in view of (10.26), (10.29), (10.22), the conditions (10.18), (10.19)
are necessary and sufficient for the validity of the inequality C
i1 f 11
p,(O,°°),v '
In the foregoing proof we did not exploit the particular
10.7. Remark.
form of the necessary and sufficient conditions which guarantee the validity
of the inequalities (10.27) and (10.30). Therefore we can repeat all arguments used in the proof also for the case constants
BL
by the constants
BR
,
AL
,
p > q
AR
,
replacing the
from (1.19), (6.7). According
to Theorems 1.15, 6.3 and 9.3 we immediately obtain
10.8. Theorem. Let
I
0
,
1
,
p > q , r = pq/ (p - q),
E W(0,=) Then there exists a finite constant C > 0 w, v m,n (m+m+n-1) such that the inequality (10.17) holds for every function u E AC (0,°°) m, n E N ,
.
m,n
if and only if the foZLowing two conditions are satisfied: X w(t)
j
0
0
150
m,n
JI
0
m,n
)))
0
J w(t) tmq
1
V'-p'(t) tnp' dtlr/q v1-P' (x) xnP' dx <
J
x
dt}r
I
,
l
t(m-1)q dtlr/q
/q
t(n-1)P'
1, vm,n,(t) x
dt
Irq J
/
vI-P,(x) M'n
x(n-1)P'dx
<
1(). 9. Remarks.
(i)
The pair of necessary and sufficient conditions from AL
Theorem 10.8 can be rewritten in terms of the constants
,
AR
from
(1.19), (6.7) as follows: v
t(m-1)q
(t) t-np, q, p) < - , m,n t-(n-1)P, q, p) w(t) tmq, vm n(t)
AL(0.
w(t)
(10.32)
AR(O,
(compare with (10.20)). (ii)
If we use Convention 5.1 and consider the necessary and sufficien
conditions in the form (10.20) or (10.32), we can formulate analogues of Theorems 10.4 and 10.8 also for the limit values of 1 a p 5 q - -
(Theorem 10.4) and for
< q < p 3 -
1
p
q
,
,
i.e.
for
(Theorem 10.8). It
suffices to use Theorems 5.9, 5.10 instead of Theorems 1.14, 1.15.
[Note that due to our special interval p = q = -
n = 0
equality (10.6) [for u
AC
(10.33)
(0,°')
,
the case
is in fact excluded due to Remark 5.5.
10.10. The case (k-1)
(a,b) _
m = 0
or
.
Let us now consider the Hardy in-
(i)
on the class of functions
(a,b) = (0,m) 1
i.e, satisfying the condition
(0,-)
u(0) = u'(0) _ ... = u(k-1)(0) = 0
If we introduce the operator
(10.3):
.
by the formula
Hk
x (10.34)
for
(Hkf)(x) _ (k
f E 14+(0,5)
,
(10.33) and we have
t)k-1
I(x -
1)i
then the function f = u(k)
u = Hkf , satisfies the conditions
Thus we can again deal with the inequality
.
1/q
(10.35)
f(t) dt
0
jJ(Hkf)q(x) w(x) dx
J fp(x) vk(x) dxl l0
JJJ
1/p
W(
C
JJJ
0
instead of the inequality (10.6). Since obviously x (Hkf)(x)
1),
J
(k
xk-1 f(t) dt
0
151
for
f c M+(0,,,)
,
the inequality (10.35) will hold if the following in-
equality is fulfilled: x
q
r( (
J
lJ
xk-1 f(t) dt)
l1/q w(x) dx] < C
}1/p fp(x) vk(x) dx]
l0
J1
0
(mr
IJ
0
However, this last inequality is the inequality (10.27) for (and
v = vk ), and it holds if and only if BL(0,o,w,vk,q,p) <
,
AL(0,-,w,vk,q,p) <
with
n = 0
m = k ,
w(x) = x
(k-1)q
w(x)
.
p
1
1
<=
q
or
< q < p <
0
1
Therefore, any one of the conditions (10.36) or
(10.37) is sufficient for the validity of (10.35). The conditions (10.36) and (10.37) coincide with the first of the two conditions in (10.20) and (10.32), respectively.
If we consider the Hardy inequality (10.6) on the class (ii) (k-1) for functions u satisfying the conditions i.e. AC (O,m) (10.38)
u(k-1)(-) = 0
u(w) =
Hk
we can proceed analogously using the operator
(Hkf)(x) =
1
J(t - X)
1)!
k-1 f(t)
dt
(k
x
and the estimate
(Hkf)(x)
<_
-(k
tk-1
1
1)! J
f(t) dt
x
Consequently, the inequality (10.35)
-
Hk
with
instead of
Hk
-
if the inequality q JI
lJ
tk-1 f(t) dt)
1/q
w(x) dx1l
1/p C
II lJ0
0
152
x
fp(x) vk(x) dxj
holds
holds. However, this is the inequality (10.30) for v = vk and it holds if and only if
m = 0
n = k
,
,
,
(10.39)
BR(0,-,w,v,q,p) <
°°
AR(0,-,w,v,q,p) <
o°
,
1
5 p 5 q < -
1
< q < p
,
or
(10.40)
x-(k-1)p
,
or
with
v(x) =
i.e.
for the validity of the Hardy inequality (10.6)
v(x)
0
1
These last two conditions, which coincide with the second of the two conditions in (10.20) and (10.32), respectively, are now sufficient for the validity of (10.35) (with Hk instead of Hk .
on
ACpkk1)(0 =)
Let us summarize these results:
10.11. Theorem.
(10.41)
Let
k C- N ,
(flu(x)iq
dx]
I0f
k ? 2
w, vk E '(0,=). Then the inequality
,
1/q lu(k)(x)
< C
w(x)
11/P P vk(x) dx]
[
0
holds (i)
for
1
and
< q < p < - or
0
1
r/q J
lI w(t)
dt]l/P
<
< p < - and (x
lr/q' IJ vk p (t) dt1 vk-P (x) dx ,
dt]
'
0
r = pq/(p - q) (ii)
if
t(k-1)q
x
0
with
satisfying the conditions (10.33)]
x dtll/q [r vl-p'(t) J w(t) t(k-1)q k J J x 0
sup
or if
[i.e.
u E ACkk01)(O,-)
if 1< p < q< -
1
for
u E AC(k-1)(0,-)
[i.e.
satisfying the conditions (10.38)]
and
q<m
x dt]l/p
sup
0<x<=
or if
1
If
< q < p <
w(t) dt ) 1q LI v k p'(t) t(k-1)P'
n
and 153
J
lJ
0
0
p' (t) t(k-1)p' dt]r/q/ vk-pl(x) x(k-1)p' dx w(t) dtJr/q If vk x
10.12. Remarks.
The conditions just mentioned are only sufficient for
(i)
AC(k-1)
the Hardy inequality (10.41) to be valid on ACO(k-1)
k
(0,0°)
.
(0,-)
or on
Necessary and sufficient conditions have been derived by
V. D. STEPANOV [1]. More precisely, it follows from STEPANOV's results that the inequality (10.41) holds for ACkk01)(0,0°)
(10.42)
1
and
< p < q < -
on
k
if and only if max
sup
i=1,2
0<x<°°
B =
Fi(x) <
where
x
0
(10.43)
x dtjl/P
F2(x) =
((
t)(k-1)P'
w(t) dt]l/q [J(x
x
-
1-pt
vk
(t)
0
If
Moreover, he has given estimates of the best constant
C
in (10.41)
of the type (10.44)
with
B
B < (k - 1)! C < 6(p,q,k) B from (10.42), and has shown that the operator
Hk
defined by the
formula (10.34),
(10.45)
Hk : Lp(0,co;v) --+ Lq(o,co;w)
is compact if and only if
B < -
and
lim Fi(x) = lim Fi(x) = 0 x+0+ x-m
,
i = 1,2
Further, analogous results have been derived for the case
154
1
< q< p <
in terms of the numbers
f m A
[J (t -
1= [J
x)
x
0
lr/q
x
(k-1)q w(t) dtlr/4 {JvP'tdt j
1
'
vk P' (x) dxlr
0
(10.46)
lr/PIf
l A2
-
If
Ll
with
r = pq/(p - q)
l w(x) dxr
1
x
0
t)(k-1v k P(t) dt l ]
-
II
J
0 .
The condition
A = max (A1,A2) < m
(10.47)
is necessary and sufficient not only for the continuity of the operator Hk from (10.34), but also for its compactness. (Compare with the situation in Section 7, Theorem 7.5.) For
(ii)
k =
we have
1
F1(x) = F2(x) = FL(x;0,°°,w,vk,q,P) with
from (1.17), and STEPANOV's condition (10.42) coincides with the
FL
if
k = 1
from Theorem 1.14. Analogously for the case 1< q < p <w
BL < -
condition
Al = (p'/q)
then
,
1/r
A2 = AL
(cf. Remark 7.8) and STEPANOV's
condition (10.47) coincides with the condition For or
k >
Ai <- ,
1
,
AL <
°°
from Theorem 1.15.
STEPANOV derived a pair of conditions ( Fi(x) < C. < m
i = 1,2 ) while in Theorem 10.11 only one condition appears
[FL(x;0,°°,w,vk,q,P) S C0 < -
w(x) = x(k-1)q w(x)
-
AL(0,m,w,vk,q,p) < m
or
with
cf. (10.36), (10.37). This single condition is
simpler than the pair of STEPANOV's conditions and can be more easily verified but, on the other hand, it is only sufficient. This is confirmed by the following example.
10.13. Example.
(10.48)
2 sax dx < C2 flu" x) 2 eax J
0
for
u
Consider the inequality
such that
dx
0
u(O) = u'(0) = 0
.
This is the Hardy inequality (10.41) 155
for
k = 2
p = q = 2
,
a<
w(x) = v2(x) = eax
,
with
If we suppose
a C- R .
0 ,
then we can easily show that the functions
from (10.43) are bounded.
Fi(x)
Consequently, the condition (10.42) is fulfilled and the inequality (10.48) holds for
u E AC(1)(0,-)
.
2,0
2
FL(x;O,-,e
On the other hand, we have
ax 2 ax -4r x ,e ,2,2) = a (ax - 1)
2
ax
+
e
the sufficient condition (10.36), BL(O,m; e
and consequently
)
ax 2
x ,e
ax
,2,2)
less than infinity, is not fulfilled.
Since STEPANOV's necessary and sufficient conditions have
10.14. Remarks.
been mentioned without proof, let us show that his conditions follow from our sufficient condition. (i)
(10.49)
with
Let
< p
1
Si
Then the following estimates hold:
<= FL(x;0,°°,w,vkq,p) = FL(x)
Fi(x)
x(k-1)q
w(x) =
q <
w(x)
in the first integral in
for
x E (0,m)
F1(x)
i = 1,2
,
Fi(x)
and
we use the fact that
from (10.43). Indeed, 0 <
x
t(k-1)q w(t) dt1/q
t
and
0
here the right-hand side is (k-l)pf
1
x
FL(x) In order to estimate F2(x) we use < (t/x) (k-1)q in the first integral and the inequality (k-1)p' .
in the second, obtaining (k-1) q
F2(x)
ll
k
lJ
J
the inequality
dtjl/p,
(( vl-p'(t)
x
(x -0
t - x <
(t - x) (k-1)q < t(k-1)q, which immediately yields
consequently
fl w(t)
l1/q
x
X
11/p,
,
vk-p (t) x(k-1)p dtl
dtl
'
=
f
x
0
x
w(t) t(k-1)q lr
dt]l/q
J
x1-k xk-1
1/P,
vk-P,
0 l1
(t)
dtj
= FL W.
JJJ
Consequently, it follows from (10.49) that if the (sufficient) condition (10.36) is fulfilled, then also the (necessary and sufficient) condition 156
l.
(10.42) is satisfied.
The same is true also in the case
(ii)
Ai
numbers
< p
since for the
<
from (10.46) we have A
c
(10.50)
< q
1
i
1
i = 1,2
<= AL(O,-,w,vkq,P) = AL
with appropriate positive numbers ci . follows from the inequality as in part (i) above. For
Indeed, the estimate
x)(n-1)q
(t -
t(k-1)q
<
Al < AL
by the same argument
we have similarly as in part (i)
A2
x r
[JX1V(t)dtJ
1
t
w(t) t)
w(x) dx =
j
J
0
x
0
J w(t) t (k-1) q dtI
r/P
r
J
x
0
1r/P (1-k)qr/p+(k-1)r x w(x) dx =
(t) dt vk J-p,
I
0
r/Pr
x
J(t) t f [JW
(k-1)q
dtlr/p w(x) x(k-1)q (Jvk-PI (t) dt 0
x
0
dx = J
[AL(0)°°,w,vk,q,P)] r = qr AL(O,-,w,vk,q,P) P
(see Remark 7.8), and this is (10.50) for
10.15. Remarks.
(i)
from
ACkk01)(O,-)
case
AC(kkl)(O,-)
,
i = 2
with
c2 = (p'/q)
1/r
In Remarks 10.12 and 10.14 we dealt only with functions i.e.
satisfying the conditions (10.33). Clearly, the
can be handled analogously; conditions analogous to
(10.42) and (10.47) can be stated and compared with the sufficient conditions analogous to (10.39) and (10.40).
The results of V. D. STEPANOV [1] mentioned above without proof
(ii)
are in fact particular cases of a more general investigation. He studied the Riemann-Liouville operator
x (10.51)
(Haf)(x) = r( a)
I(x -
t)a-1
f(t) dt
0
for
a ?
1
and derived necessary and sufficient conditions for the validity
of the inequality 157
Haf,wl/gllq, (0,m) -
GlIf
va/Pllp,
(0,
(or, more precisely, for the continuity and compactness of the operator Ha
:
Lp(0,-;va) - Lq(O,_;w) ). Here we have mentioned the results a - k
concerning the special values
k E IN
,
.
(See Subsection 10.10,
formulas (10.34) and (10.35).)
The results mentioned in Theorems 10.4 and 10.8 are due to
(iii)
H. P. HEINIG and A. KUFNER (see A. KUFNER, H. P. HEINIG [1]). Although only the Hardy inequality (10.17) from (10.21)
i.e. the continuity of the operator
-
H
m,n is considered, it is clear how to obtain conditions for
-
the compactness of the corresponding imbedding and estimates for the best possible constant in (10.17).
Further, it is obvious that also in this case the results can be m, n E N
extended from
instead of the operator
to general
a, B C- R ,
a ?
1
,
S
'_
1
,
if
we consider the more general operator Ha S,
Hm,n
x t),-l[j(s
1(x -
(Ha,sf)(x) =
0
-
t)B-1
ll
f(s) ds] dt
t
and investigate the inequality
G
Ilf
The arguments from the proof of Theorem 10.4 can be used almost literally. Let us conclude this section with some examples.
Let
10.16. Example.
1
< p < q <
a,
a E R , and let us consider the
inequality (
10.52)
I
On the class (i)
form
158
Let
< C ((lu(m+n)
Iq xa dx}1/q
0
Il0
JI
AC
(x) Ip xg dx}1/ P JI
(k-1)
m,n
m ?
1
,
n ?
1
.
The conditions (10.18), (10.19) have the
Co
sup 0<x<- If
to+(m-1)q
dtl1/q
r
J
x
1/p'
ts(1-p')+np' dtI J
0
(10.53) x dt}1/q
(( ta+mq
sup 0<x< o
l
t6(1-p')+(n-1)P' dtJI/p,
If
J
<
11
x
0
and an easy calculation shows that these conditions are satisfied the inequality (10.52) holds for
consequently n 2 1
AC(k-1)(0,W) m,n
and
-
with m ? 1,
if and only if
-
B E (np - 1,
(10.54)
(n + 1) p - 1)
and
a=S P
(10.55)
+
p - 1 - (m + n)q = B P + P - 1 - kq
(Compare these formulas with the formula (10.16) from Example 10.3; in (10.55), we can also write (ii)
k = m
Let
for functions
,
a = 8 P - P - (k - 1)q -
n = 0
,
.)
i.e. let us consider the inequality (10.52)
satisfying the condition
u
1
(10.33). Then the sufficient
condition (10.36) has the form x
ts(1 p,) dtJl/p,
sup
0<x<= LI
x
0
[it is the first condition from (10.53) for
k = m
,
n = 0 ], and it is
fulfilled if (10.56)
and
a
B < p -
1
satisfies (10.55). STEPANOV's necessary and sufficient conditions
[see (10.42), (10.43) have the more complicated form x (r
Osup 11 ta(t -
x
x)(k-1)q
dtl/q
dtll/P
((
< m
ts(1-p') J
J0
,
J
159
ll/q
If ts(l-P)
to dtl
sup
0<x<.
l/p '
,
If
(x - t) (k-1)PP dtJ
JO
111
x
and lead to exactly the same result (10.56), (10.55). m = 0
If we take
(iii)
(10.52) for functions
,
k = n
,
i.e. consider the inequality
satisfying the conditions (10.38), then we can
u
show that it holds for 6 > kp -
(10.57)
again with (iv)
k = 2
.
1
from (10.55).
a
Let us summarize the results for the special case
According to (10.55), we have
x1-2p
(10.58)
lp
J
dx < C
0
a = a - 2p
p = q
and the inequality
p xs dx
f
0
holds
for B < p for B E (p - 1, 2p - 1) for 3 > 2p - 1 1
if u(0) = u' (0) = 0 if u(0) = u'(-) = 0 if u(-) = u'(-) = 0
These results could be obtained also by using the classical Hardy inequality (0.2) successively for
u
and
u'
(cf. Example 10.3). It can be easily
seen that it is impossible to obtain the inequality (10.58) by this approach for
such that
u
u(-) = 0
,
u' (0) = 0
,
since the corresponding intervals of admissible values of
particular classical Hardy inequalities, i.e. the intervals (for
u
)
and
(--, p - 1)
(for
for the
8
(2p - 1,
u') have no common point.
Consequently, the ordering of the 'boundary conditions' in (10.4) has been reasonable.
10.17.
Example.
w(x) = 160
Let us consider the modified weight function ( `
1
0
for 0 < x < a for a < x <
with
a E (0,-)
(this weight function does not fulfil all conditions
mentioned in Definition 1.4, but the foregoing results can be extended also to such modified weights). Using Theorem 10.8 we can show that the inequality a
((
[Ju(m+n)(x) IP xR dxj11/p
11/q
IJ lu(x)lq dx]
0
0
holds for 0 < q if AC (k-1)(0,0) M,n
<m ,
1
qx
a E (np - 1, (n + 1)p i.e. for the same values of
a
1)
1
,
p > q on the class
,
as in the case
p < q
[cf. (10.54)].
11. SOME REMARKS
11.1. A modified Hardy inequality.
The inequality (1.2), which has
initiated the investigation of Hardy-type inequalities, can be rewritten in the form x (11.1)
r( JIX
ll
p
f(t) dt
xE dx < C
J
fp(x) xE dx
]
0
0
f E M+(O,m)
,
E
0
< p -
1
.
If we introduce the operator
HL
x
(H LOW = X
f(t) dt J
0
i.e. a certain integral mean value, then we can rewrite (11.1) in the form
J (HLf)
(x) xE dx < C
fp(x) xE dx 1
0
0
This notation represents only a formal change, but it can be substantially extended if we introduce more general 'integral means'
x (11.2)
(HLf)(x) = R(x)
f(t) r(t) dt J
a 161
for
with
M+(a,b)
f
r E W(a,b)
R,
,
type ((b
dxC 1/q
[f(HLf)q(x) w(x)
(11.3)
a with
1
and consider inequalities of the
b1/P fP(x) v(x) dx}
Ij a
< p,q
v, w E W(a,b)
.
This approach, in which in fact four
weight functions occur, has been used by various authors, sometimes in a little more general situation. Let us mention N. LEVINSON E1] (for v(x) = w(x) E
), K. C. LEE, G.
1
S. YANG C1] ( w(x) = v(x) = xa ), B. G.
PACHPATTE C1] and others. Mostly
they derive sufficient conditions in p = q
the form of differential inequalities for the case
.
Let us show that the investigation of inequalities of the type (11.3) can be reduced to the investigation of our fundamental inequality (1.12), i.e. bj
(11.4)
q
x(
l1/q w(x) dxJ
f(t) dt
(J
b C
IJ
a
a
l1/p fP(x) v(x) dxJ
a
For this purpose, rewrite (11.3) in the form
bx(
[J(--R(x)
(11.5)
J
a
with
1/q
q
(b
f(t) r(t) dt)
w(x) dxJ
C
,
IJ
fP(x) v(x) dx,
a
a
f 6 M+(a,b)
1/p
}
denote
f(x) = f(x) r(x)
and introduce new weight functions w(x) = w(x) R-q(x) Obviously,
f E M+(a,b)
,
;
v(x) = vv (x) r-P(x)
and we obtain (11.4) from (11.5). Moreover, (11.5)
will hold if and only if (11.4) holds (with the same best possible constant C ). Since the necessary and sufficient condition for the validity of (11.4) is well-known (see Theorems 1.14, 1.15 and 9.3), we immediately obtain the necessary and sufficient conditions for the validity of (11.5). For instance, for the case
if and only if
162
1
< p 5 q < m
the inequality (11.5) holds for every fE M+(a,b)
b
l1/p,
x
vl-P'(t)
}}
sup a<x
(11.6)
1l w(t) R q ct) dtJ l/q
x
11
a
If we take
11.2. Example.
r(t) = v(t)
R(t) =
and
x
(J
(J
a
q
111/q
f (t) v(t) dtw(x) dx'C
1
,
we obtain the in-
b1/P
equality b
<
dtl
If
fp(x) v(x) dx
I
If
a
a
and the condition (11.6) has the following simple form: b
sup
x
dtIl/q
If v(t) dt
If w(t)
a<x
a
x
l/p'
< W
,
I
Let us note that G. SINNAMON E21 investigated the following generalization of the inequality (11.7): dp(t))q
l J
(
dv(x)J 1/q
J f(t)
I
with general Borel measures
p
11.3. Some imbedding theorems.
weighted Sobolev space
J fp(x) dp(x)J
C
,
v
1/p
-'
.
In Subsection 7.11 we have introduced the
W1'p(a,b;v0,v1)
and its subspace W0'P(a,b;v0,v1),
emphasizing the connection between the Hardy inequality (11.8)
lluwl/gliq,(a,b)
C
IIu'°1/Pllp,(a,b)
and the continuity and compactness of the imbedding (11.9)
W0l'p(a,b;v0,v1) C L'(a,b;w)
(see Theorem 7.13). Here only the weight functions important, the weight function
v0
vl
,
have been
w
has not played any role.
The Soviet authors M. 0. OTELBAEV, K. T. MYNBAEV and
R. OINAROV have
investigated the inequality (11.10)
1luwl/gllq,(a,b)
s C(jluv0/PIp,(a,b) + Ilu'vl/Pllp,(a,b) 163
for
u E W1'p(a,b;v0,v1)
as well as for
u E W1'p(a,b;v0,v1)
.
They
establish necessary and sufficient conditions for its validity, i.e. for the continuity of the imbedding (11.9) as well as of the imbedding (11.11)
W1'p(a,b;v0,v1) C. Lq(a,b;w)
,
and also for the compactness of these imbeddings in terms of all three v0
weight functions 1
<_
q < p < -
Vi
,
,
w
The cases
.
1
< p
<_
q
p x 4
,
and
are considered.
The results mentioned are rather complicated and some of them appear without proofs. Here we will reproduce some recent results mentioned in the paper by R. OINAROV [1l. It is supposed that (11.12)
w, Vol v1 E 1"
v- 1/p
1
loc(a'b)
l
'
Lp
E
'
loc(a'b)
(compare with our conditions (7.40), (7.41)). Further, the following numbers are introduced: C (
ha = J v0(t) dt +
vl
p'(t) dt
1
a
(11.13)
b
b
(
(
v0(t) dt + J
hb
vi p (t) dt '
c
c with
c E (a,b)
,
and four cases are distinguished:
(i)
ha <
hb
(ii)
ha <
hb
==,
(iii)
ha =
hb
< m ,
(iv)
ha = hb = m
<
.
In the case (i) we have (11.14)
v0
L1(a,b)
,
vI/p E Lp(a,b)
,
v1 p,E L1(a,b)
i.e.
(11.15)
164
1/p v1-
C.
Lp,(a,b)
However, then the numbers
Bi(a,b,v0,vl,p,p)
i = L
,
and
i = R
(see
(1.18) and (6.2)) can be estimated from above by the number
v0/p'Ip,(a,b)'IIv11/pl1p',(a,b)
which is finite due to (11.15). Consequently, the Hardy inequality 1/pl
Iuv0
u'vl/PII
p,(a,b)
holds for every
C
1
p,(a,b) W1O'p(a,b;vO,vl),
and on the space
u C- CD(a,b)
the norm
(7.39) is equivalent to the norm given by u,v1 1/pll
p,(a,b)
Thus, the inequality (11.10) reduces to the Hardy inequality (11.8) (with instead of
vI
v ), and the conditions of its validity have been studied
in detail in the foregoing sections. The reader can easily see that the condition w E L1(a,b)
(11.16)
,
i.e.
wl/q C Lq(a,b)
,
guarantees the validity of (11.8) and consequently, the validity of the inequality (11.10) for
u C WO
p
(a,b;vO,vl)
-
i.e. the continuity of the
imbedding (11.9). OINAROV has stated that the condition (11.16) is necessary and sufficient for the inequality (11.10) to be valid for u E W1'p(a,b;v0,vI)
,
i.e. for the imbedding (11.11) to be continuous.
In the cases (ii) - (iv), the integrability of the weight v0
,
1-p' vl
(iii)
-
is violated, either at one of the endpoints or at both ends
-
-
functions
the cases (ii),
the case (iv). Then it is necessary to
introduce some new notation and define some modifications of the numbers AL ' AR '
BL
'
BR , which have been used to characterize the validity of
the Hardy inequality in the foregoing sections. Following R. OINAROV [1]
we define some functions, intervals and numbers, which have been introduced and extensively used For
x F (a,b)
by
and
M. 0. OTELBAEV for the case y ? 0
such that
v1(x) =
[x, x + y) C (a,b)
1
.
define
165
x
x+(y
f
vi p/(s) ds <
a(x,y) = sup {d > 0;
vi-p'(s) ds
J
J
(x - d, x] C (a,b)}
,
d+(x) = sup {d > 0;
1/p
x+d (
x+d
/
(
v0(s) ds
J
vi-p (s) dsJ
I
J
x-d(x,d)
l/p 1
x-d(x,d)
[x, x + d) C (a,b) } d -(x) = 6(x, d+(x))
,
A -(x) = [x - d(x), x]
A+(x) = [x, x + d+(x)]
,
A(x)
= A (x)t) A+(x)
a0
= inf {x E (a,b); x - d -(x) >
b0
= sup {x E (a,b); x + d+(x) < b} (a,B) C (a,b)
Further, for
,
a}
denote
1/q
t ((
(11.18)
B+
sup
(a, 6) =
p'q
x C (a , S)
t
sup
IJ w(s) ds E A+( x ) l
(compare with the definition of the numbers
_p'
vi
l
1
1/p'
(s) ds J
111
Bi(a,s,w,vl,q,p)
,
i = L,R ).
Then the following assertion holds: Let (11.10)
1
< p < q
holds for every
,
p
and
h
a
= h
u E W0l'p(a,b;v0,v1)
166
Bp,q = max {Bp,q(a,b),
BP,q(a,b)}
=
.
Then the inequality
[and consequently, the
imbedding (11.9) is continuous] if and only if (11.19)
b
<
Moreover, the best possible constant
in (11.10) is equivalent to
c
B
.
p,q
The last assertion concerns the case (iv). In the cases (ii), (iii),
similar assertions hold with the following change in (11.19): in the case B
(ii),
(a,b)
p,q is replaced by
is replaced by B+ (a,b ) p,q 0
B
p,q
(a ,b)
case
< q < p <
1
are replaced by the numbers
Bp,q(a,s)
to our former numbers
AL
(a,b)
p,q
.
An analogous result holds for the numbers
in the case (iii), B
;
0
,
AR
Apiq(a,6)
only the
which correspond
(7.33)) and are defined by
(cf.
A p,q
l
a
ds r/p w(t)
(t)w(s)
r/p'
vi-pl(s)
IlJ
d sl
l
A-(t)
dt
1/r
t
A+ (a,s) _ p,q
Here
J
a
L+(t)
are functions inverse to the functions
,
x -; x - d -(x)
,
1/r lr/p w(s) ds] w(t) dtJ
J
+(t)
and
r = pq/(p - q)
x --r x + d (x)
t
r/p' dsr/pt
,
vi -p (s)
J
,
respectively,
x C- (a0,b0)
.
These results concern the imbedding (11.9). As concerns the imbedding (11.11), i.e.
the validity of the inequality (11.10) for
u E W1'p(a,b;v0,v1)
every
the corresponding necessary and sufficient conditions
,
are derived only for the cases (ii) and (iii) and have, roughly speaking, the form B
p,q
+
F(a,s) <
where
1/p'
S
1/q
F(a,s) = IJ w(s) ds] a
and
(a,s) = (a,a0)
J
1-p'
vl
a
in the case (ii),
This concerns the case
(s) ds]
1
(a,s) _ (b0,b) ,
px
;
for
in the case (iii). 1
<_
q
,
the 167
number
Ap,q
has to be added to the corresponding number
F(a,S)
p = q = 2
For the case
vl(x) =
,
1
,
a necessary and sufficient
condition for the inequality (11.10) to be valid on
obtained by E. T. SAWYER [2 1
.
ACLR(a,b)
has been
His condition has the following form:
.
36r }
J w(s)ds < C
v0(s) ds + 2(( - a)-1J
I
L
3a
CL
for every interval
(a,B)
such that
(3a,3S) C (a,b)
We have mentioned here results concerning the continuity of the imbeddings (11.9) and (11.11). Results concerning the compactness can be found in the book of K. T. MYNBAEV, M. 0. OTELBAEV [1].
11.4. Hardy inequalities with fractional derivatives. In Remark 10.15 we have mentioned the inequality
11/p
1/q
((rr
(11.20)
(Haf)q(x) w(x) dxy
1
C
IJ
0
l0
on the class
fp(x) v(x) dx
M+(0,-)
,
where
x (x -
(Hf)(x) = r(a)
t)a-1
f(t) dt
0
and
a '_ For
1
.
0 < a
<
1
,
H f a
integration; conditions on
is the so-called operator of fractional w, v E W(0,-)
which guarantee the validity of
(11.20) have been investigated by several authors. Here, let us mention K. F. ANDERSEN, H. P. HEINIG [1]; H. P. HEINIG [1] (sufficient conditions); K.
F. ANDERSEN, E. T. SAWYER [1] (necessary and sufficient conditions). If we formally denote by
Da
the operator inverse to
Ha
,
then we
can rewrite the inequality (11.20) in the form
11/p
1/q lll
(11.21)
[fu(x)
w(x) dx
C
11Dau(x) Ip v(x) dx
1
0
0
which represents a 'Hardy inequality for fractional derivatives'. Such
168
inequalities have been investigated by P. GRISVARD [1] for the special case p = q
:
he has shown that the inequality
(11.22)
I
p p
_'P
dx
lu(x) - u(y) IP c
Jr
J
u E C0(0,') I
< p <
provided
0< A< 1,
oo
dx dy
Ix - y
00
0
holds for
1
X X
1
,
P
[note that the right-hand side in (11.22) is the p-th power of the norm of the 'derivative of order
A
of
u
Sobolev spaces of fractional order
which appears in the definition of
'
W
X.P -
cf.
K. A. ADAMS [1], Chapter
VII, Theorem 7.48, or A. KUFNER, 0. JOHN, S. FUCfK [1], Definition 6.8.21A. KUFNER, H. TRIEBEL [1] have shown that the inequality (11.22) holds also for
0
weights
< p <
1
w
v
,
,
and extended (11.22) for
p >
1
to the case of two general
The proofs of the results just mentioned are based on the theory of interpolation.
169
Chapter 2. The N-dimensional Hardy inequality 12. INTRODUCTION
12.1. Some definitions. Let For
u = u(x)
(12.1)
be a domain in RN
Q
defined (a.e.) on
Q
with a boundary
aD
denote by
supp u
the support of the function
u
,
i.e.
the closure (in the Euclidean norm)
of the set
lx `=
S2;
u(x) x 0}
For a subset M C Q = 2 0 aQ denote by (12.2)
CM(S2)
the set of all infinitely differentiable functions (12.3)
u
on
Q
such that
supp u (l M = 0
Further, denote by
(12.4)
C0(S2)
or
C0(S2)
the set of all infinitely differentiable or continuously differentiable functions
u
(12.5)
supp u() as2 = 0
on
D
such that
and, moreover, the set b ounded
domains
supp u
is bounded.
[Note that
C0(S2) =
D .1
Finally, denote by
(12.6)
W(S2)
the set of all weight functions on
Q
,
measurable, positive and finite a.e. on
170
i.e. D
.
the set of all functions
for
Let
12.2. Formulation of the problem.
1
w,v1,v2,...,VN C- W(S2)
< p,q < - ,
We will deal with conditions which guarantee the validity of the inequality
N
ll1/q
(f
(12.7)
1/p u(x)Iq w(x) dxi ()Ip fl3u vi(x) dxj 1i=1
11
on a certain class
K ,
K D CM(Q)
with an appropriate set
M C S2
Instead of the inequality (12.7) we will also investigate some of its modifications, for example, the inequality
[JUX)I
(12.8)
1l1/q
i11 uw(x) dxl p vi(x) (
dxl
I
i
11
where
S
is a subset of the set
{1,2,...,N}
11111
or the (equivalent) in-
,
equality 1/q
(12.9)
f
Ju(x)Iq w(x) dx`= C
12.3. Remark.
L
ff
iE S
I
f
(x)Ip CC
[jIBx
i
l1/p
vi(x) dx j
We will deal with the inequalities mentioned above mainly
for the special case
p=q The reason is in its simplicity which enables us to explain the fundamental ideas more clearly and without disturbing technical details. The passage from the case
p = q
to the case
is easy if we
p > q
content ourselves with estimates which are sometimes rather rough: we need only Holder's inequality, as will be described in the following example. The case
p < q
needs more sophisticated considerations
-
cf. Subsection
12.13. A general approach to both cases is made possible via imbedding theorems for weighted Sobolev spaces investigated in Chapter 3 (cf. Lemma 16.12).
A complete answer was given also by V. G. MAZ'JA
C1]
who used the
notion of capacity and investigated the inequality (12.7) on the class C0-
(0)
;
we will mention his results in Section 16.
171
.
12.4. Example.
Suppose that the inequality (12.7) is fully investigated p = q
for the case
that the inequality
i.e,
,
N
111/p
(x) dx
[1u(x)IP
(12.10)
C J
holds for every Let
1
u E K
l1/P vi(x) dx]
a
i=1
i
.
5 q < p < -
.
Using Holder's inequality with the exponents s = q
l4
lux>Iq w(x) dx = ju(x)I
[Jiux)Ip w(x)
dx
1
q/p
W4/P(x) w-q/P(x) w(x) dx
(I(o,w,w,q,P))
(p-q)/p
J Q
(12.12)
I = I(0,w,w,q,p) =
f
w4/(q-P)(x) wP/(P-q)(x) dx
J
0
The estimates (12.11), (12.10) imply dil/P
(12.13)
[Jiux) q w(x) dxjl/q < C I1/r i=1
0
where
1/r = 1/q - 1/p
q < p
with
(i.e. for
fax.(x)P
vi = vi
,
.
0
i
vi(x) J
However, (12.13) is the inequality (12.7) for
and consequently
p = q ) provided the number
we have derived (12.7) from (12.10) I
from (12.12) is finite.
12.5. Some special weights. In the theory as well as in applications, weight functions appearing most frequently are of the type
(12.14) with
M L S2
(12.15)
with
and
a E R or, more generally, of the form
v(x) = v(dist (x,M))
v = v(t) E W(0,-)
notation
172
v(x) = [dist (x,M)]a
(see Definition 1.4). Sometimes we will use the
(12.16)
dM(x) = dist (x,M)
for
x (E
12
.
Investigating the inequalities from Subsection 12.2 for these special weight functions, we can use with advantage the 'one-dimensional' approach, exploiting the results derived in Chapter 1. Let us illustrate this approach by some examples.
12.6. Example.
Let
be the cube
12
N
x = (x1,x2'...,xN) E R
,
M
and let
Further, let K = CO(Q)
for
;
dM(x) = xN
(12.19)
for
w(x) = w(xN)
(i)
Q
,
RN-1 ,
i.e.
.
w,v1,...,vN E W(0,1) p = q
For
= (x11x2,...,xN-1) E
'
xN = 0}
and consider the inequality (12.7) on
with the weight functions
w(x) = w(dM(x)) Since
x
be the 'basis' of
M = {x E Q
(12.18)
.
denote
with
x = (x',xN)
(12.17)
Q = (0,1)N
,
x E Q
vi(x) = vi(dM(x)) ,
we obtain
vi (x) = vi(xN)
Under the notation from (12.17) we have 1 11
(12.20)
[Ju(x',xN)Ip w(xN) dxN] dx'
JIu(x) S2
P w(x) dx = 1 M
0
Using the one-dimensional Hardy inequality, we can estimate the inner integral on the right-hand side of (12.20) obtaining 1Iu(x',xN)l
1(
w(xN) dxN
(12.21)
Cp IIau (x',xN)IP vN(xN) dxN
0
0
N
provided (12.22)
with
43 (O,1,w,vN,P,P) <
r given by the formula (8.69).
Note that
u C Cp(Q)
and 173
u(x',xN) E ACLR(0,1)
consequently
x'E M .]
for every
M , we obtain in view of
Integrating the inequality (12.21) over (12.20) that
P w(x) dx < Cp
(12.23)
dx
J3XvN(x)
Q
(
which is the inequality (12.8) for
p = q
and
S = {N}
.
This immediately
implies the following assertion: Let
(0,1,w,vN,p,p) <
<- p <
1
1/p
(x)IP vl(x) C(.L a 1i=1 QJlx, i
Q
holds for every functions (ii)
.
dx)
without any further assumptions on the weight
u E CO(Q)
vi(x) = vi{dM(x) Let us write
x" = (x2,...,xN) Q
}l/p
au
NC
[1(u(x)Ip w(x) dx
(12.24)
Then the inequality
.
,
x E RN
be the face
M1
and let
(x1,x")
x
in the form
{x
E Q; x1
where of our cube
0}
Then 1
(12.25)
[TIuxi,x") P dx1J w(xN) dx"
IIu(x)Ip w(x) dx =
.
J
Ml 0
Q
Since
0 (0,1,1,1,p,p) < -
,
1
we can estimate the inner integral on the
right-hand side of (12.25) using the ('non-weighted') Hardy inequality 1
1
(lu(xlpx") IP dx1 < CP 1!(xl,xt')iP dx1 0
1
Multiplying this inequality by
(12.26)
Jlu(x)lp w(x) dx
The symbol Q (0,1,1,1,p,p)
Q
we obtain
1
denotes the number
(8.69) for the particular weights
174
M1
s_ Cp Jlp w(x) dx
Q
1
and integrating over
w(xN)
w = v =
1
.
(0 (0,1,w,v,p,p)
from
which is the inequality (12.8) for
S = {1}
and
p = q
and with the
v1(x) = w(x)
special weight function
This procedure can be obviously repeated with the role of the variable X1
being played by
xi
i x N
,
and we immediately obtain the following
,
assertion:
Let
and let
1 < p < -
i &_ {1,2,...,N-1}
equality (12.24) holds for every
instead of
C
with
vi(x) = w(x)
and
c 1
without any further assumptions on the weight functions
VI .(x) = vi (dM(x))
j xi
,
Now we can combine the inequalities (12.23) and (12.26) (with
(iii)
Xi
u c- C'(Q)
be fixed. Then the in-
instead of
x1
i = 1,2,...,N-1)
,
and obtain immediately the following
assertion:
Let (12.27)
with
<-
1
p < - . Let
w, vN E W(Q)
w(x) = w(dM(x))
w, vN E W(0,1)
vN(x) = vN{dN(x))
,
Suppose that
.
6 (0'1,w-,V-N,P,P)
Then the inequality 28)
be given by the formulas
<
ll(12.
[JIu(x)I p w(x) dx]
<
Q
N-r1
Ilax
i=1 Q
i
(x)IP w(x) dx + 1I - (x)Ip vN(x) dxj j au N Q
C0.
holds for every
u E
with the constant
(Q)
C = N max (C,C1)
12.7. Remark.
In the last example we in fact derived three types of
estimates of the form (12.7) (i.e. (12.24)), providing rather big possibilities of choice for the admissible weights on the right-hand side: (a)
vN
connected with
i = 1,2,...,N-1
w
via the condition (12.22),
V.
arbitrary,
;
175
vi = w
(B)
for some
i E {1,2,...,N-1}
v1 = v2 = " ' = vN_1
(Y)
w
,
vN
j z
arbitrary for
vj
,
i
satisfying (12.22).
Clearly, the special form of the domain
0 as well as our special weights (12.27) have played an important role. Let us illustrate the influence of the 'geometry' of 0 by another example.
12.8. Example.
Let
Q = {x = (x1,x2,...,xN) E RN ;
xi > 0
i = 1,2,...,N}
,
,
M = {0}
(12.30)
and consider again the inequality (12.7) for
on
p = q
with weights
C0(0)
of the type (12.15), i.e. (12.31)
where
w(x) = w(dM(x))
w, vi E W(O,=)
.
,
Q = (0,m) x G
= 1,2,...,N-1}
.
then the image of G = {0 = (0
,
S2
0
is the infinite
) 0 < 0
For simplicity, we will consider the case
x1 = r cos O1 (12.32)
,
If we introduce the spherical coordinates
(r,0) = (r,01,02,..,,ON-1) 'cylinder'
vi(x) = vi(dM(x))
,
<
n
i
N = 3
.
Then
,
x2 = r sin 01 cos 02 x3 = r sin 01 sin 02
G
is the square
(0,
2) x (0, 2)
,
and the Jacobian of the corresponding
mapping is (12.33)
Here
J(r,0) = r2 sin 01
dM(x) = r
and
.
w(x) = w(r)
,
vi(x) = vl(r) .
(i)
(12.34)
Using the spherical coordinates (12.32), we have
Ju(x)lp w(x) dx
=
f
[Iuro)
p
w(r) r2 dr] sin 01 dO
.
1
52
G
0
Making use of the one-dimensional Hardy inequality, we can estimate the 176
inner integral on the right-hand side of (12.34) obtaining
(12.35)
Ia=(r,0)Ip v(r) r2 dr
JIu(ro) P w(r) r2 dr s CP
(
0
0
provided
JO (0,°°,w(r)r2,v(r)r2,p,P)
(12.36)
sin 01
Multiplying (12.35) by
to the Cartesian coordinates au (x) p
with v E W(0,-)
<
then integrating over
,
G
,
going back
and using the obvious estimate
x
P
3p-1
x
(x)
ax
a -r
i=1
i
we obtain 11
1/p
3
31/P , <
[Ju(x) Ip w(x) dx
(12.37)
with
v(x) = v(dM(x))
with
v1(x) = v2(x) = v3(x) = v(x) (ii)
ax
l i=1 3
]
0 .
p
C
i
ll1/p
v(x) dxJ
(x)
However, (12.37) is the inequality (12.7) for
p = q
.
Obviously,
J u(x) IP w(x) dx = 0
¶/2 (1r2 1 J J
J
0
Since
Iu(r,01,02)IP dO2J w(r) r2 sin 01 dO1 dr
l
0
0
'3 (O,7r/2,1,1,p,p) < -
,
we can estimate the inner integral with help
of the ('non-weighted') Hardy inequality
72 J
rr2 Iu(r,01,02)IP dO2 < Cz J
0
0
Multiplication by
w(r)r2 sin 01
,
P
dO2
(r,01,02)I
a0 2
integration with respect to
r
,
01
and
the obvious inequality P
a02
(x)
<
2P
(x)p Ixlp +
(
p
l
aX
2
Ixlpl
aX (x) 3
J
177
finally yield
11/p
p
ll1/p
(12.38)w(x) dx] [11ux
21/PC2(
( aX (x)
w(x) IxIPdx]I
S = {2,3}
with
i=2
p = q
This is the inequality (12.8) for v2(x) = v3(x) = w(x)
and
lxlp
Consequently, we can assert that the inequality (12.7) holds (for p = q ) with the above mentioned weights
v2
,
without any further
v3
assumption on
v1
12.9. Remark.
In the last example we in fact derived two types of estimates
.
of the form (12.7): (a)
v1 = v2 = v3 = v , where
is connected with
v
w
via the
condition (12.36); (8)
If we put for
arbitrary,
v1
a
lxla
w(x) =
v(t) = to+p
v2(x) = v3(x) = w(x) lxlp i.e.
,
,
W_(t) =
v(x) =
i.e.
then (12.36) will be fulfilled
0,
lxla+p,
.
provided
a
3
,
and the in-
equality (12.37) has the form
1/p (12.39)
[1iu (x) lP lxla dx]
<
2 P
31/p?
(x) P
f
ia + 3l
Ul
lxla+p
1I/p dx
axi
]
0
On the other hand, for w(x) = lxlo we obtain in (12.38) again +p = v3(x) , but now the inequality (12.38) holds for all
v2(x) =
lxla
with the constant
(2p)-
1/p it
a E R
Combining these two results we show that
.
the inequality 3
111/p
(12.40)
[IiuiP
Ixlo dx]
< C
P
fl}-(x)
I
I
xi
i=1
l1/p lxla+p dx]
Q
holds for every
u E CO(52)
and
a E R
1/p C = min
178
p in + 31
3
;
(2p)-
1/p
with the constant n 1
12.10. Example.
M = {x E St;
(12.41)
be defined by (12.29), but now put
x2 = x3 = .. =
= 01
X N
is the edge of our polyhedron
M
Since
0 L RN
Let
0
.
we can use with advantage the
,
and (xi,r,0) with r = dist (x,M) cylindrical coordinates For simplicity, we restrict ourselves to the case 0 = (Olp.... 0N-2
N = 3
Then
.
xl = xl , x2 = r cos 0 x3 = r sin 0
,
and
J(xi,r,0) = r and
p = q
Again we consider the Hardy inequality (12.7) for
u E COO)
with a weight of the type (12.15), where now dM(x) = r (i)
We have n/2
{ IIui,r,0)Ip w(r) r drI dO dxi
lu(x)Ip w(x) dx =
g
1
J
0
0
0
and the one-dimensional Hardy inequality yields p
au
Ju(x1,r,O)Ip
v(r) r dr
ar(xl,r,0)
w(r) r
0
0
provided (12.42)
with
63 (0,",w(r)r,v(r)r,p,p)
v E W(0,°')
Integration with respect to
.
xi
,
and the obvious
0
estimate au
p
p
p
2p-1
(x)
ax (x) 2
leads to the inequality (
(12.43)
3
IIH(x)lp w(x) dxl/p
p lau
Jax.(x)
v (x) dx
11/p
i=2 179
with and
v(x) = v[dr1(x)J
p = q ) with these
(12.7) holds (for tion on
v1
(ii)
However, (12.43) is the inequality (12.8) for
.
v2(x) = v3(x) = v(x)
S = {2,3 } with
v2
,
p = q
Consequently, the inequality
.
without any further assump-
v3
.
On the other hand, we can write
== r
Jlu(x)lp w(x) dx =
j
r/2 r
u(x1,r,O)lp dOJ w(r) r dr dx1
,I
J 0
111
0 0
0
and since
Z,
1,
we can estimate the inner integral
1, p, p)
('non-weighted') Hardy inequality
with help of the
r2
n/2 P
1 lu(x1,r,O)
p
au
dO < Cp
dO
a0(x1,r,O)
0
0
w(r)r
Multiplication by
,
integration with respect to
x
,
r
and the
1
obvious inequality < 2P-1
p
DU axe (x)
O x)
P
Ixlp +
ax (x) 3
finally yield the inequality (12.38).
Consequently, we have shown that the inequality '1/P lu(x)lp w(x) dxJ J
(au
3
C 111i=1
J
1/P
P
(x)I
v (x) dxJ
axi
si
holds for our special domain with an arbitrary weight v(x) = w(x)
or
IxIP
12.11. Remark.
0
from (12.29) and for
and with
M
from (12.41)
v2(x) = v3(x) = v(x)
is connected with
w
express the integral
Jw(x) dx
in the form
1r/2 lu(r,01,02)Ip sin 01 dO111 w(r) r2 dO2 dr
180
1
J
0
0
I
,
where either
via the condition (12.42).
In Example 12.8 we have not used the possibility to
(i)
- 7/2
v(x)
v1
1
0
and estimate the inner integral by means of the one-dimensional
Hardy in-
equality
it/2 1 lu(r,01,02)lp sin 01 d01 < 0
r/2 < C 1
p
lao (r,01102
j
v(01) sin 01 dOI
0
provided J (0,
r/2, sin 01, v(01) sin 01, p, p)
This approach is possible but leads to weight
<
functions
vi(x)
which are
not necessarily of the form (12.15). (ii)
In Example 12.10 it was possible to express our integral also in
the form 7/2 ( I
[JJu(xi,r,e)IP
0
w(r) r dO dr dx1
J
0
0
Here the inner integral cannot be estimated by some 'non-weighted' Hardy inequality since the corresponding number
' (0,-,1,1,p,p)
is infinite, while
a weighted analogue leads again to weight functions which are not of the form (12.15). (iii)
In the foregoing examples we have dealt with the inequality
(12.7) for the case
p = q
.
In the next two sections we will illustrate
how to modify the approach described if we omit the assumption
12.12. The case
1
< q < p <
p = q
.
This case was mentioned in Remark 12.3.
Here, we will illustrate how the one-dimensional Hardy inequality can be used, and deal with the special domains and weights from Examples 12.6 and 12.8. (i)
Under the notation from Subsection 12.6 we can write 1
(12.44)
Ju(x)lq w(x) dx =
(xN) dxNJ dx'
I
J
0
181
[compare with (12.20)] and estimate the inner integral by means of the one-dimensional Hardy inequality
lu(x',xN)Iq w(xN) dxN < Cq
IJ
(x',xN)I
ax
0
0
q/P
p
(r
f
(12.45)
vN(xN)J
N
provided A!(O,1,W-,VN,q,P) <
(12.46)
with A given by the formula (8.98). Integrating the inequality (12.45) over
and using Holder's inequality with the exponents
M
p/q
,
p/(p-q)
we obtain in view of (12.44) that 1
lq/P dx vN(xN) dxN]
P
Jlu(x)lq w(x) dx < Cq J
Q
[flaxN W
,xN
0
M
1
jj
< Cq [N-1(M)] (P-q)/P
(12.47)
P
Cq
(f
au ax
q/p
p_ vN(xN) dxNl
(x,
,xN)
J
N
dx'I
dx1q/P
vN (x)
(x)
Q ax N
lJ
J
S = {N} , and consequently the inwithout any further assumption u E C0(Q)
which is the inequality (12.8) for equality (12.7) holds for every on the weight functions
vi(x)
,
i = 1,2,...,N-1
.
We only need that the
condition (12.46) be satisfied. (ii)
If we use the notation from 12.6 (ii), we have
Jlu(x)lq w(x) dx
M1 0
Q
Since
,
[Jlu(x1,x")lq dx1J w(xN) dx"
= J
(0,1,1,1,q,p) < -
,
we can estimate the inner integral on the
right-hand side by means of the ('non-weighted') Hardy inequality 1I
1
u(x1,x11)1q 0
182
P
dx1 < Cq 0Jlax (x
l,xrr)
114/P
dx1J
,
w(xN)
Multiplying this inequality by
integrating over
,
M1
and using
Holder's inequality we obtain similarly as in (12.47) that 1
q/P
P
w(x) dx <= Cq
(
QJ I"(x)
f
("l,x")
ax
M1 hJ0 J
dx1J
w(xN) dx" _
1
1
P
Cq J I
(xx")
8 x
I
M1 0
lq/P dx" WP/q(xN) dx1Jl
1 1
(M )] (P-q) /p
< Cq Cm N-1
1
1
f ( LJ
8u
Tax
x")
(x
p WP/q(XN) dxl] dx "
1
M1 0
= Cq1
fJQ
q/p
P
Fl au
(x)
@x
wp/q(x) dx I
This is the inequality (12.8) for function every
v1(x) = wp/q(x)
u E C0(Q)
.
S = {1}
and with the special weight
Consequently, the inequality (12.7) holds for
without any further assumptions on the weight functions
v2
In the case of the domain
(iii)
S1
and set
M
from Example 12.8
we can proceed similarly if we use the expression
u(x)lq w(x) dx =
J
C
[Jlu(r,0)Iq w(r) r2 drI sin 01 dO 0
[compare with (12.34). If we use the expression
flu(x)
q w(x) dx =
0
n/2
it/2 u(r,01,02)lq d0 2J w(r) r2 sin 01 dO1 dr
0
0
0
[see Subsection 12.8 (ii)], we can estimate the inner integral by means of
the ('non-weighted') Hardy inequality since ft (0,n/2,1,1,q,p) < Holder's inequality with the exponents
p/q
,
p/(p-q)
°°
.
Then
yields
183
Ilu(x)Iq w(x) dx (12.48)
o
P
waP/q (x) dxj
(x)
q/p [I((i,w,q,P)]
(P-q) /P
2
where I(a,w,q,p)
W(1-a)P/(P-q) (r) r 2 dr
_ 0
a E R such that
Consequently, if there exists a number
S = {2,3}
then (12.48) yields the inequality (12.8) for v2(x) = v3(x) = waP/q(x) IxIP
weight functions 12.8 (ii)
I(a,w-,q,p)
<
with the special
[similarly as in Subsection
cf. the formula (12.38).
-
12.13. The case
Here we will deal only with the cube
< p < q < m
1
M
from Example 12.6 with
Q
given by (12.18). Then we have the following
assertion:
The inequality 1/q
q w(x) dxl
[Iux)
X(x) N
(
(12.49)
C
QJ
p
l1/p wp/q(x) dx]
i
Q
holds for every
(12.50)
u E C0(Q)
provided
N- N+ 1'_ 0 q p
w(x) = w(xN)
w E W(0,1)l AC(0,1)
with
non-zero a.e. in (0,1)
such that the derivative
, and (0,1,;0,Wp/q,P,P)
(12.51)
<
where ,
(12.52)
v0(t) = w P/4 (t) Iw'(t)P
Indeed: Let
u E C0(Q) P
(12.53)
aX (x) J
Q
184
1
.
We can assume that
wp/q(x) dx < W for
i = 1,2,...N
,
w'
is
since otherwise the inequality (12.49) holds trivially. If we denote U(x) = u(x) w1/q(x) = u(x',xN) wl/q(xN)
(12.54)
,
then P
au
(12.55)
for
au
dx =
(x)
x Q
ax.
wP/q(x) dx
(x)
1
i = 1,2,...9N-1
and
,
p
jau
dx
(x)
(12.56) N Q
2p-1
p wp/q (x) dx + (q)p
(x) ll axN
Q
Q
where
Jlu(x)lp vo(x) dxl
v0(x) = v0(xN)
.
Moreover, in view of (12.51), we have 1
[u(x'xN)p v0(xN) dxNJ dx'<
u(x)lp v0(x) dx = J
J
M
Q
0
1
(12.57)
<_
J
M _ p - C0
P
au
co (J 0
ax
au
,
,x N)
(x
dx' _
wp/q(xN) dxN
N
) IP
xN(x
wp/q(x) dx
and consequently we conclude from (12.55) and (12.56) in view of (12.53) that au
E LP(Q)
Moreover, since
i = 1,2,...,N
for
supp U C Q
,
.
we can use the classical imbedding theorems
for Sobolev spaces which imply that
U E Lq(Q)
with
q
from (12.50) and
that
1/p
1/q
(12.58)
U(x)Iq dxl
= C2
S J i=1 Q
au (x) ax.
Ip
dxj
i
(see, e.g., A. KUFNER, 0. JOHN, S. FUCIK [1], Theorems 5.7.7, 5.7.8, or
185
R. A. ADAMS [1], Theorem 5.4 ). Now, (12.49) follows immediately from (12.58) in view of (12.54), (12.55), (12.56) and(12.57).
12.14. Remark.
The main tool we have used in the foregoing subsections
was the one-dimensional Hardy inequality from Chapter 1. We have used here the special weights (12.15) and the considerations have been relatively simple due to the special choice of the domain
and of the set
1
M .
Let us mention that the approach described in Examples 12.4, 12.6, 12.8 can be used also for more general sets
D
M
and
using local
coordinates, but the considerations are rather cumbersome and, sometimes, restrictive additional assumptions on the weight functions are needed. We will not deal here with this approach, which is described in A. KUFNER [2]. In the next section we will again deal with the approach via the one-dimensional Hardy inequality but for more general weights than those mentioned in Subsection 12.5.
13. SOME ELEMENTARY METHODS
13.1. Some useful notations. The weight functions from Subsection 12.5 have been very special and, in fact, have depended only on one variable (either
directly or after an appropriate change of variables). Here we will deal mainly with weight functions independent of one variable (or depending on it in a special manner). Therefore, for the point
N
x
we will use also the expression (13.1)
x = (xi,xi)
xi = (xl""'xxi+1'""xN) E
where
For
S2 E RN
RN- 1
i E {1,2,...,N }
denote
Ii = Ii(52) = inf {xi; x = (x',xi) E H} (13.2)
,
S. = S.(Sl) = sup {x.; x = (x'.,x.) C- H}
D. = D. (R) = S. (Q) - I. (Sl) (i.e.
186
D.
is the 'diameter of
3
in the direction of the
xi-axis').
Further, let us denote by
(13.3)
Pi(S2)
the projection of let us put
52
and for x!1 E P.1 (0
onto the hyperplane x.1 = 0
C(S2;xi) _ {t E R; (x!,t) E S2}
(13.4)
(the Cut of
by a line orthogonal to the hyperplane
S2
xi = 0
;
draw a
picture!). Obviously, there exists a (finite or infinite) sequence of open and mutually disjoint intervals
Jj W) = (aj (xi) ,bj W))
(13.5)
such that
C(SZ;x!) = U Jj (xi)
(13.6)
.
j
u = u(x) defined on
For
f(t) = u(xi,t)
(13.7)
,
x = (xi,x1)
,
S2
xi E Pi(S2)
,
,
denote
i
t e C(c;x')
The symbol
(13.8)
AC i(S2)
denotes the set of all measurable functions corresponding
f
on
u
S2
such that the
from (13.7) satisfies
f E AC(J.(x1'))
for every
and for
j
N-1-a.e.
n'
xi E Pi(0) .
Finally, (13.9)
ACi,L(ci)
denotes the set of all
such that the corresponding function
u E AC .(12) 1
f
from (13.7) belongs to
xi E Pi(12)
.
AC (J. L
J
(x'.))
1
for every
j
and for
mN-1-a.e.
Consequently,
lim u(x',t) = 0 t+a.(xi )+ 1 J
for
mN-1-a.e.
xi E Pi(S2)
and for every
j
.
Similarly, we can introduce
187
the set AC
i,R (52)
.
The following lemma generalizes the approach described in Example 12.6, part (ii).
Let
13.2. Lemma.
<_
1
Let
.
DP
r
au ax.
JJu(x)IP w(x) dx p
holds for every
Further,
Then the inequality
x
u E C0(0)
u E C0(0)
Let
.
.
0
0
Proof.
xi
and suppose that
Di = Di(0) < -
such that
be independent of the variable
w E W(Q)
(13.10)
be a domain in RN
0
i E {1,2,...,N}
there exists a number let
p < -
and define
u(x) = 0
x e RN \ 0
for
.
Then
obviously Xi
u(x) = u(xi,xi) = J
(xi,t)
ax
dt
i
I. i
and consequently
we have Si
(13.11)
Ii)P-1
ax.(xi't)
(we have used Holder's inequality if
and integrating over
depends only on
x'
dt
1
I. i
w(x)
p
au
J
lu(x)Ip < (xi -
p
> 1
). Multiplying (13.11) by
we immediately obtain (13.10) since
0
w(x)
.
The proof of the following assertions, which are simple generalizations of Lemma 13.2, are left to the reader.
13.3. Lemma. I. = Ii(S2)
w E W(0)
188
> -
Let '
1
[or
s p <
.
Let
Si = Si(0) <
and suppose that
0
be a domain in RN for some
such that
i E {1,2,...,N}
.
Let
(13.12)
vi(x') _
w(x',x1)(x1 - aJ(x')1P-1
(
J
J
J
dx1 <
J. W) 1
3
[or v(xi)
(13.13)
j JJ J(x')
w(x',x.)(b (x')
i
j
1
1
I
1
xi e Pi(S2)
mN-1-a.e.
Then the inequality (13.14)
flu(x)
P w(x) dx
W
au JI aX. 1
<_
0
)
p
vi(x') dx
holds for every u E C0(c2) 13.4. Corollary.
In addition to the assumptions of Lemma 13.3 let us
suppose that
w(x) = w1(xi) w2(x')
x = (x',xi) C S2
,
and
xi)(xi -
Ii)P-1
dx
< C < m 1
[or xi)P-1
J
wl(xi)(Si -
dx
< C < m I.
.
1
Ii
Then the inequality
J u(x) Ip w(x) dx <-
holds for every u E Remark.
ax (x)
i
Q
13
C
p w7(x1;) dx
C0W
(s2)
In Lemma 13.3, the assumption
u E C0(1)
can be obviously
weakened; it suffices to suppose that
u E AC
(Q)
189
[or
u C- ACi,R(Q) I.
Let
(1)
13.6. Examples.
for every u E AC1,L(9)
N=2 we have
eax2
dx =
lu(x)lP
and
(0, ")
x
,
aER.
Then
Ip eax2
1
dx
ax (x)
p
J
0 = (0,1)
,
1
0 2
ax
P
1e
dx
C(a,p)
e
lax
2
(x)
dx
1
R 1
2
eax1
where
x1P-1 dx1
C(a,p) = J
.
0
(ii)
Let
and
N = 2
00 = (0,-) x (0,m)
a x 1
0
0
for every u E AC1,L(0)
and 2
ax
P
C(a,p) f u(x)( p ealxl 0
dx
e
(x)
Then we have
.
2 ax')
p
au
Jlu(x)lp ealxl2 dx < C(a,p)
a < 0
,
e
ax (x)
dx <
2 1
dx
2
W
for every u E AC2,L(Q)
eat2
(
where
tP-1 dt
C(a,p)
.
Consequently,
0
the inequality 1ealxl2 dx <
0 ax 2
P
<
C(a,p) I [ 0
ax (x)I 1
e
au (x) P
ax2
2 +
holds for every u E AC1,L(St) n AC2,L(O)
axi 1 dx
e J
.
In Lemma 13.2 we have dealt with a weight
w
independent of
following assertion extends the corresponding result to functions independent of some curvilinear coordinate.
190
xi
w
.
The
137.
Theorem.
1,2,...,N
.
Let
1=p<
Let
F
N Let R, Q be domains in R ,
be a regular one-to-one mapping of
Q
w, vj E W(S2) , onto
1
with the Jacobian D(F1,...,FN) DF
D(Y1,...,YN)
independent of ci
for
for some
D. = Di(Q) <
Let
yi
and let
i E {1,2,...,N}
Suppose that there exist positive constants
.
j = 1,2,...,N , and a measurable function
,
Y = (Y',Yi) E Q
(13.15)
cdF(yi)
(13.16)
w(F(Y))
w(F(y))
dF
:
be c
,
C
Pi(Q) --* R+ such that
,
CdF(Y')
IDF(Y)l
<_
P
aF,
2y
(Y)I
C. vj(F(Y))
<=
,
j
= 1,2,...,N
.
Then the inequality
fu(x) Ip w(x) dx < Cp
(13.17)
X
holds for every function
u = u(x)
vj(x) dx
(x)
a j
j=1 on
Q
such that
u(F(y)) E C01 (Q)
with
the constant C0 =
(13.18)
[Di(Q)]p p
Np-1
C c
max C. J
Proof.
Let
u
be such that
coordinates x = F(y)
u(F(y)) E C1(Q)
J
Q
Q
w(F(y))
(13.19)
w(F(y))
The transformation of
yields
jlu(x)Ip w(x) dx =
and since
.
))I' .(F(Y))
is independent of
IDF(Y)I dy
y.
i
= w(Yi)
we have in view of (13.15) fu(x) lp w(x) dx `= C Ju(F(y)) P w(yi) dF(y ) dy S2
Q
Using Lemma 13.2 with the weight function
w(y!) dF(yi)
we obtain from
formula (13.10) that
191
fju(x)IP
(13.20)
p
w(x) dx <
p
w(Yi) dF(yi) dy
I
0
Q
Since obviously
l au F(
p
)
r
ax,(F(y))
ayi
we obtain (13.17) with
p
p
pNNC
N-1
from (13.18) in view of (13.19), (13.20),
C0
(13.15) and (13.16). 11
Obviously, we can extend the assertion of Theorem 13.7 by
13.8. Remark.
weakening the assumptions on
Q
w
and
in such a way that Lemma 13.3 can
be used.
13.9. Example.
Let
N = 2
and let
Q C. R2 \ {(x1,0); xl > 0}
For
be such that
Q
(0,0) e 3s2
,
put
x E S2
w(x) = w(Ixl)
with
w E W(0,-)
xl = yl cos y2
,
.
Introducing the polar coordinates x2 = yl sin y2
(Y1,y2)
we obtain from Theorem 13.7 that the in-
,
equality (2n)P
J u(x) lp w(x) dx
=
Du
2p-l
j-1
p
S2
holds for every
u E C1(0)
In particular, for
.
J
w(x) = Ixls-p
x1p dx
we obtain the inequality au
(2n)P
JI
w(x)
J
(Cf. Example 12.8.)
2P-1
u(x)lp lxl9-p dx _<
(13.21)
p
aX.(x)i
p
=
0
which holds for every a E R (and u E
1
C01
(Q)
Jr
axj
(x)
).
The following lemma is an application of the one-dimensional Hardy inequality. We proceed similarly as in Section 12, but now without any
192
assumptions concerning
the special structure of the weight functions. We
will use the intervals
Let
13.10. Lemma.
1
Jj(xi)
from (13.5).
p<
Zet
be a domain in RN
o
,
w, v E W(Q)
penote (13.22)
BL,j(xi) = BL(aj(xi), bj(xi), w(xi,.), v(xi,.),
p, P)
BL
is given by the formula (1.18), and suppose that for some i E {1,2,...,N}
where
Ci =
(13.23)
ess sup
sup B
x'E i Pi (Q)
(x')
i
L,j
<
Then the inequality
Ju(x)IP w(x) dx <_ C0 fl"
(13.24)
0
with
C0 = p1/P (p')1/PIC.
(13.25)
Proof.
u E ACi,L(9)
v(x) dx 1
0
holds for every
(x)IP
Let
equality and
u E ACi,L(Q)
condition
ju(x)IP
Fubini's theorem, the one-dimensional Hardy in-
.
(13.23)
(cf. Theorem 1.14) yield l1
w(x) dx =
J Pi(tt)
Q
C
J
Iu(xi,x)Ip w(xi,xi) dxi] dxi C(c,x,i)
bj (xi)
ji
P1 0t)
a (x?) SS
{
<
Pi(9)
u(xi,xi)IP w(xi,xi) dxi
Lp1/p
L
(P') 1/p
1
BL,J (xi) J
p
au
ax
lp
(xi'i x i
v(xi,xi) dxi }dxi <
193
P
au
v(xi,xi) dxi
f
I
CO
C(N,x')
Pi(0)
au
axi
13.11. Remark.
dxi,
1
P
(x)
11
Assume that the weights
w
,
from Lemma 13.10 have the
v
special form
w(x) = w(x
w(xi) xi t Pi(O)
v(x) = v(xi) w(xi)
xi E (Ii,Si)
,
Then obviously (13.26)
BL (Ii,Si,w,v,P,P) = BjpL
BL,j(x') =
and the condition
Bi,L <
°°
implies (13.23). Consequently, we can formulate
the following corollary of Lemma 13.10.
Corollary.
13.12.
Let
Suppose that for some
let
1
0
there exist positive constants
i E {1,2,...,N}
and positive measurable functions
k
,
K
w
:
P1(0) -, R+ such that
be a domain in RN, w, v E W(0). w, v
:
R
and
u E AC. L(0)
with
(Ii,Si)
w(x) 5 K w(xi) w(xi) (13.27)
v(x) = k v(xi) w(xi)
If Bi L = BL(Ii,si,w,v,P,P) < then
the
inequality
(13.24)
CO - (K) I/P
on the class even with
194
holds for every (P/)1/PI
P1/P
13.13. Remark.
°°
Bi,L
In Lemma 13.10 we have dealt with the inequality (13.24) ACi,L(0)
u E ACi,LR(O)
.
Obviously, we can deal with ACi,L(0) n ACi,R(O)
,
u E ACi,R(0)
or
if we replace the numbers
BL,j(xi)
BR,j(xi)
from (13.22) by the corresponding numbers
defined on the basis of
r from (8.69).
from (6.2) or
BR
oj(xi)
or
An analogous remark can be made concerning Corollary 13.12.
inequality
N = 2
Let
(i)
13.14. Examples.
(ax e
2
(
dx _ CO
0
a E R . Then the
f2 = (0,1)
,
ax2
P
au
(x)I
dx
e
flax 2
holds for u E AC2,L(0) if a < 0 , for u E AC2,R(0) if a > 0 u E AC2,LR(S2) if a x 0 . Here we have =
C
0
and for
,
Plal
according to (13.25).
A comparison with Example 13.6 (i), where
au/ax1
appears, leads to
the inequality eax
u(x)IP
2 dx
J
au (x) axI
for u E
P
e
ax2
dx+ 2
E_
1
au p
I
P (x)
ax2 dx
e
ax2
with a x 0
C1(S2)
Theorem 13.7 has extended Lemma 13.2 to the case of curvilinear coordinates. Similarly, we can formulate the 'curvilinear extension' of Corollary 13.12 (and Remark 13.13). The proof, which is obvious, is left to the reader.
13.15. Theorem.
w, v . E W(S2)
Let
1 = p < -. Let
j = 1,2,...,N
,
.
Let
S2
,
Q
be domains in RN
be a regular one-to-one mapping
F
J
of
Q
onto
with the Jacobian
0
i E {1,2,...,N}
and
w , v
,
DF
.
Suppose that for some
there exist positive constants
K
and positive measurable functions
w
dF
:
I1(Q), S1(Q)) --. R+
,
c
AF
such that for
,
C
,
Cj
j =
Pi(S2) --r I
y = (yi,yi) E Q
195
w(F(Y)) < K w(Yi) w(Y!) P
aF
i
i
v(Y.) w(Y!)
Cv(F(Y))
j j
(Y)
ayj
cdF(yi) AF(Yi)
< CdF(y.) AF(yi)
IDF(Y)I
Let
161 = i (Ii(Q), Si(Q), wdF, vdF, P, P)
< m
Then the inequality (13.17) holds for every function
CO = Pl/P(P')l/p'
IKNP-1
6
max
c
13.16. Example.
R = sup
Let
0 C R2
on
such
0
C,)1/P
j
j
be the domain from Example 13.9. Denote with
w(x) = w(IxI)
and put again
IxI
u = u(x)
with
u(F(y)) E C1(Q)
that
.
w E W(0,R)
.
Suppose
xC0 that there exists a function
such that
v E W(0,R)
G = 3 (0, R, w(t)t, v(t)t, p, p)
(13.28)
y = (yl,y2)
Denoting by
<
the polar coordinates, we obtain from Theorem
13.15 that the inequality 2
J u(x) Ip w(x) dx
P
C
C0
Jax.(
L
j=1
0
v(IxI) dx
x)
J
J
holds with (13.29)
C0 = 2(P-1)/P pl/P (P')l/P/ 0
In particular, for
w(x) =
.
with
IxIs-P
6 x p - 2
we obtain the in-
equality 2
r
IxIs-p
u(x)P
(13.30)
dx < CPO
ax
j=1 2
where
j is-p
is given by (13.29) with
C0
v(t) = is
We can easily obtain
.
C
196
au (x)
0
< 2 (P-1)
/P
P
Ig + 2 - pI
03
from (13.28) where
w(t) =
Combining the inequalities (13.21) and (13.30), we arrive at the inequality (13.31)
J(x) Ip Ixls-p dx Q
(2n)P 2p-1 min { l
(
p
J
P
1g+2-pI p
au (x) ax1
l
) p 1
.
J
au (x) ax2
s2
1
and for every
u t C 1(S2)
which holds for every
B G R
The condition (13.23) was sufficient for the validity of the Hardy inACi,L(0)
equality (13.24) on the class
.
The next lemma shows that, for
Some special domains, this condition is also necessary.
13.17. Lemma.
Let
< p < -
1
and let
S2
be the cylinder
52 = {(xr,x.); x' E G, x. E (a,b)}
with
G
number (13.32)
a domain in C>0
RN-1
.
Let
w, v E W(S2)
.
Then there exists a finite
such that the inequality au
I(x) lp w(x) dx < Cp
J
ax.(x
s2
holds for every function (13.33)
u L ACi,L(12)
if and only if
C. = ess sup
x'E i G Moreover, the best possible constant
in (13.32) satisfies the
C
estimate C.
Proof.
< C < p1/P
(pr)l/Pr
For our special domain
and consequently
the number
C.
we have
S2
C.
P .(S2) = G 1
,
C
.(2;x1r.) = (a,b)
,
1
from (13.33) coincides with the number
Ci
i
from (13.23). Therefore, according to Lemma 13.10 the condition (13.33) is sufficient for (13.32) to be valid.
197
Suppose now that there exists a number every
u E ACi,L(0)
such that (13.32) holds for
C
Assume that for such a function
.
u
,
the integral
on the right-hand side of (13.32) is finite, and rewrite this inequality in the form b
P [ci
(13.34)
3x.i,xi)
v(xi,xi) dxi -
J
J
a
G
1
b 1l
JIu(xx)lp w(xi,xf) dx1J dxi
-
0
a
We will show that (13.34) implies that the inequality b
br
CP (lf'(xi)lp v(xi,xi) dxi - Jlf(xi)lp w(xi,xi) dxi > 0
(13.35)
a holds for every
a f E ACL(a,b)
and for a.e.
xi E G
.
According to Theorem
1.14 we then have BL
for a.e.
C
xi E G
and consequently
,
the condition (13.33) is satisfied.
The proof of the validity of (13.35) will proceed by contradiction. If (13.35) does not hold for every there exist a function 'N-1(M) > 0
,
f E ACL(a,b)
f e ACL(a,b) ,
0
f
and a.e.
and a set
xi e G
,
then
M C G
such that b(
1
r
Hf(xi) =
I
LCP
f'(xi)lp v(xi,xi) -
lf(xi)Ip w(xi,xi)J dxi < 0
a
for every
xi E M
Consequently, there exist a number
.
Mj C M ,
0 < mN-1(Mj) <
(13.36)
Hf(x') < - -
,
such that
for every
xi 6 M. J
If we denote b
F(xi) = fIf'(x1)Ip v(xi,xi) dxi a
198
j E N and a set
then
xi E Mi
f10
is measurable on
F
F(x!) < 1
F(xi) > 0 for a.e. xi E M1 since
. Moreover, Put
.
and,by virtue of (13.36),
C
XM (xi)
[F(xi)]-1/P
v E- W(S2)
for
and
xi E G
,
J x = (x',xi)
u(xi,xi) _ $(xi) f(xi)
Then
u e ACi,L(0)
0
i
and P
v(x) dx = mN-1(M ) < m
Further,
b
b
u
[GP J
i
a
G
v(x',xi) dxi - J Iu(xi,xi) P w(xl,xi) dxil1 dx'
P
ax (xi'xi)
J
L
a
J(xi)I
Hf(xi) dxi
- 1 J
dxi < 0
J
G
M. J
which contradicts (13.34). 11
In the foregoing Subsection we have dealt with the case next
two
13.18. Lemma.
Let
1
1
The
< q < p < -
< q < p <
1
,
=
r
(13.37)
.
lemmas are the analogues of the assertions from Lemmas 13.10
and 13.17 for the case
w, v E W(S2)
p = q
.
1
-
q
1
let
,
0
be a domain in RN
P
Denote
AL,j(xi) = AL
is given by the formula (1.19), and suppose that for some
i E {1,2,...,N} API
Ci = j
J
l
[ 7 L
LJ
}1/r
r/p'
,
(
(13.38)
(xi)
W
dx' J
Pi(0)
Then the inequality (rr
(13.39)
1/p
p
ll1/q
fJlu(x)Iq w(x) dxj
< CO
(J
ax
(x)
v(x) dxj
199
=
u(=- ACi,L(Q)
holds for every
q,
CO = ql/q (p,)1
Proof.
Cl
Fubini's theorem, the one-dimensional Hardy in-
u E ACi,L(Q)
Let
with
.
j cq
equality (see Theorem 1.15), the inequality
(
j cj)q
,
Holder's in-
equalities for sums and for integrals and the condition (13.38) yield
111/q
[Ju(x) Iqw(x) dx]
bj
(xi) 1/q
(
u(xi,xi)Iq w(xi,xi) dxil dx' J
J L X aj(xl)
J
Pi(0)
J
l
[ql/q(p,)1/q'A
J
pio)
L,j(x1,)
L
j b. (x!)
p
ax (xi,xi)
v(xi,xi) dxi)
1/p
q,
J
aj (xi)
l 1/q
x d
i
}
ql/q(p,)1/q' <
j
Pi '(52)
bj (x f) au
ax .(x)
p
v(x) dxi
l
J
l
Pi(g)
j
,
(
p/j (x/q/p 1
x
J
200
J
r
bj (xi) .
p J
aj (xi) q1/q(p,)1/q'J
111/
p
Du
a jf(xl)
ax.
1
(x)
llq/p v (x) dx iJ]
1/q dx'1
}
(q/pr)(p/(p-q))
Ar
[
ql/q(pr)1/qr
i
l
L,J(xi)
1
Pim
l(p-q)/(pq)
dx'
f
J
b (xi) 111/p
Let
1
<_
11
q < p < m
that for some (fixed)
w, v E W(0)
1 ,
=
1
-
q
1
i E f1 ,2,...,N}
(a(xi),b(xi))
Let
.
p
equality (13.39) holds for every
0
the cut
for mN_1-a.e.
Then there exists a finite number
.
=
111/p
r
of only one interval
i
dxiJ
v (x) dxJ
1
0
in RN
aX._(x)
a(xi)
p
Du (x) ax.
13.19. Lemma.
J
[
J
Pi(w
v(x) dx.
p
u E ACi,L(O)
be such a domain C(O;xl)
xi E P1.(S2)
CO > 0
.
consists Let
such that the in-
if and only if
1/r
(13.40)
C. = l
AL(xi) dxiJ
J
P
i
(0)
where (13.41)
AL(xi) = AL(x';w,v) =
Moreover, the best possible constant
C0
in (13.39) satisfies the
estimate I
(13.42)
Proof.
qr
r
C1 < CO S ql/q (pr)l/q
ql/q (p rq)
For our special domain, the number
C.
i
the number
C. i
C
from (13.40) coincides with
from (13.38) and,therefore, the condition (13.40) is suffi-
cient according to Lemma 13.18. Suppose now that there exists a finite number holds for every (i)
(13.43)
C0
such that (13.39)
u e ACi,L(Q)
Assume in addition that
J
w(x) dx < -
vl-pr(x) dx
,
<
j S2
12
201
xi E Pi(c)
Let us fix
and choose two sequences
real numbers such that
an(x') 1 a(x') i
Then there exist non-negative functions
,
{an(xi)}
{bn(xi)}
,
for
n
b (x') t b(x') i n i gn E ACi,L(S2) such that
b(xi) p
v(x',xi) dxi =
(13.44)
1
and b(xi) 1/q gn(xi,xi) w(xl,xi) dx1j 1/q > q
(13.45) l
1/q,
P r
i A n (x)
J
a(xi)
where b(xi)
bn(xi)
r A(x)
w(x',t) dtl
J
an(xi)
v-P (x',t) dtl
I I
J
l
xi
' r/q
r
(
xi
an(xi)
vl-p r (x',xl)
dx.
(cf. Lemma 3.11, formulas (3.34), (3.35)). Further, let
be a sequence of domains in RN-1
{Qk}
Qk C Qk+1 C Pi(e)
,
mN-1(Qk) < m
and
kEN For
n, k E N
Qk = Pi(S2) define
Ar/k (xi) g n (x',xi)
(13.46)
x = (x',xi)
,
where
min {An(xi),k}
for
xi E Qk
0
for
x E Pi(Q) \ Qk
,
An'k xi r
The formulas (13.44), (13.45) and (13.46) yield
202
such that
of
v(x) dx]
n,k(x) axi
if J
1/p
p
au
b(xi) (13.47)
f =
ag
f
An,k(xi')
L
J
Pi (1)
p
ax n("i
J
i) i v(x,x
x
i
a(xi) 1/p
dxiJ1
f
An,k(x')
J
Pi(Q) and
l1/q
[IUfl,(X)I q w(x) dxI ((
=
Q
b (x') r
(
Arq/P(x') n,k
i[
i J
a(xi)
p i(S2)
ril( 1/q'
(13.48)
> ql/q
w(xi,xi) dxiJ dxi
9q(x',x i) i n
Arq/p(xi)
11/q An(x') dxiJ
l
J
J II
p i (0)
1/g/ gl/q Er
f
P
1/q
r
An,k(xi) dx' A
i (s2)
Using now the inequality (13.39) for the function
un,k E ACi,L(Q)
,
we
obtain in view of (13.47), (13.48) that (2
ql/q
1/q
r
r
)
An, k(xi) dx1
An,k(xi) dxiJ
1/q'
< COl
J
P
i
J1/P
(0)
and since 0 <
Ar,k(x') dx' < kr mN-1(Qk) <
f
P
i
(Q)
we have
r
r r
ql/q
l1/q
(P )
11/r
Ar,k(x') dx1I L
< Co
,
J
p i(2)
203
The monotone convergence theorem, applied first for
n -+ W
,
yields r
1/ ' (13.49)
(ii)
and
q
r
ql/q
L
Let now
w ,
r
1
A L(x';w,v) dxi] /
J
Pi (0)
from (13.41) for
AL(xi;w,v)
with
and then for
k
w
,
<
satisfying (13.43).
v
be general functions from
v
CO
W(O)
,
and for
x E 0
n E N put
n
wn(x) = min wn
Obviously,
and
vn
Ixl(N+1)/(p'-1)1
(1 +
v n(x) = v(x) +
(w(x),n/IxIN+l,n)
.
fulfil the additional assumptions (13.43) and
consequently, the inequality (13.49) implies that 11/r
r
q
q
AL(xi;wn,vn) dxil
lpr )
CO
j
(E2)
Since
wn(x) ? w(x)
and
vn P,(x) T vl P(x)
convergence theorem yields for w, v E W(D)
.
n -- m
for a.e.
x E 0 ,
the monotone
that (13.49) holds for all
(13.49) is the first inequality in (13.42).
However,
14. THE APPROACH VIA DIFFERENTIAL EQUATIONS AND FORMULAS
14.1. Introduction.
In this section we will extend to the N-dimensional
case the approach described in Subsections 2.2 and 2.4. Some attempts in this direction can be found e.g. in the paper by D. C. BENSON [1], where the special case
p = q = 2
has been considered. For the same values, R. T.
LEWIS [1], [2] has shown that the inequality
Ju(x)2 w(x) dx < 4 IVu(x)2 v(x) dx
(14.1)
S2
SZ
holds for u E Cp(0) 204
with
, au ' Vu = grad u = rau lax1 ax2
(14.2)
if the weight functions
w
,
' au
ll
axNJ
are chosen according to the formulas
v
2
(14.3)
w(x) = IAg(x)I
v(x) = Ivg(x)I
,
with an appropriate function
Ag =
g
.
(Recall that
? ag )
L
i=1 ax. 1
The following assertion is a direct (N-dimensional) extension of the result mentioned in Theorem 4.1 (but here for Let
1
< p <
Let
.
w,v1,...,vN E W(St)
be a bounded domain in RN
S2
P w(x) dx <
Ju(x)
i=1
holds for every
,
let
Then the (Hardy) inequality
.
N (14.4)
p = q ).
u 6 C0(0)
p au vi(x) dx (x) ax. 1
if there exists a solution
y
of the (partial)
differential equation P
N``
(14.5)
iLl axi
such that for a.e. (14.6)
y(x) A 0
sgn x + w IyJp-1 sgn y = 0
ax
[vi
in
S2
x E s2 and
ax (x) / 0
,
i = 1,2,...,N
i
14.2. Remark.
The assertion just formulated is a corollary of the more
general Theorem 14.4. The precise formulation needs some assumptions concerning the domain
0
and the introduction of some function spaces.
This will be done in Subsection 14.3. Note that the equation (14.5) can be viewed as the Euler-Lagrange equation of the functional
205
as
NCC
J(Y) = J [
L
The domain
.
0
For
(i)
.
1 < p < -
and
in RN , denote by
0
(14.7)
1
vi - IYIp wJ dx
W1'p(Q)
14.3. The Sobolev space a domain
p
Lp(0)
11
on
u
the set of all measurable functions
1/p
lu(x)Ip dx] (14.8)
cull
p, 0
=
with the finite norm
0
for
1 < p < -
for
p =
,
0 ess sup lu(x)I
.
xE0 Further, denote by
(ii)
W1'p(0)
(14.9)
the set of all functions derivatives
2x
u E Lp(0)
belong to
Lp(0)
such that their distributional ,
i = 1,2,...,N
.
The space
W1'p(S2)
1
is called the Sobolev space and is normed by N
(14.10)
Ilull1,p,2 = (Ilullp,2 +
iLl
ou
1/p
p 1
u
axill
p,0
In the sequel, we will use some properties of functions from
(iii)
Sobolev spaces; in particular, the existence of the trace
°lasi of a function
u E W1'p(0)
assumptions about the domain
on the boundary 0
.
30
.
This notion needs some
We will not go into details here; let us
only note that we will use domains of the class (14.11)
C
0,1
which are, roughly speaking, bounded domains whose boundary can be locally described by functions satisfying the Lipschitz condition. A precise definition of this class will be given in Section 19. Here we will use the fact that for this class of domains Green's formula can be used, since
206
(a)
the notion of the trace
(we will write simply
ulaS2
instead of
u
is meaningful
in the sequel),
ula0
at the point
0
the outer normal to
u e W1'p(I)
for
x 6 aS2
V(X) = v = (v1,v2,...'"N) mN-1-a.e.
is well defined for
X E aO
[For details, see, e.g., J. NECAS [1] (Chapters 1, 2), R. A. ADAMS [1]
(Chapters II - V, VII), or A. KUFNER, 0. JOHN, S. FUNK [1] (Chapters 5, 6).
Let
14.4. Theorem.
< p <
1
Let
0
be a domain in RN
,
S2= lim S2
,
n0nC Stn+l C Q and
where
defined on
y = y(x)
0
0n e CO' 1
.
Let
w,v1, ... ,vN E W(Q)
and
fulfil the following conditions:
The derivatives
(i)
3 ax
.
exist a.e. in
avi
Ip-1
ax' axi ax.l 1
'
0
sgn
.
.
Dx
.
The function
(ii)
vay
a
is a solution of the differential equation
y
(14.5) and satisfies the conditions (14.6). Further, let (iii)
be such that
u = u(x)
gulp vi yi E W
1,1 (Qn)
i = 1,2,...,N
,
,
and N
(
lim sup
(iv)
X vi yi vni) dS '_ 0
lulp
n
i=1
3Q j
n
where ax.1p-l
sgn (14.12)
and
vni
yi
,
lylp-1
i = 1,2,...,N
outer normal to
0
ax.
sgn y
,
are the components of the unit vector of the
n
Then the inequality (14.4) holds for
u
satisfying (iii), (iv).
207
Proof.
is such that
u
If
N
J(u) =
lau
`
ax.
. t
i=1
1
is infinite, then the inequality (14.4) holds trivially. Therefore, assume that
J(u) < -
.
The following well-known inequality SP + (p - 1)tp - pstp-1 holds for
< p <
1
N
(14.13)
-W
.Ip + (p
E
Hs1
we have
and consequently
s,t <
0
,
0
.Ip`1)
.Ip - pls1 .I
>-
0
It1
1It1
i= 1
for
i = 1,2,...,N
si, ti E R ,
If we put
.
aY__
axi
S. _ au vi1/p ax.
,
1/p
ti = u - vi y
1
then we have p
au
NNC
vi + (p - 1)
ax.
lulp
v.
1 p
1
1
IYI
p-1
- p
au ax.
lu
p-1
Ip-1
1
and consequently
Ad i+ vi]
0
IY
the inequality p
NN
(14.14)
iLl
[
au axi
p
ax.
v. + (p - 1)
Iulp
1
IYp
1
v.
I 1
p ax.
Iul-1 sgn u
1
with
y
,
1
Since
208
from (14.12) holds a.e. in
11
.
vi yij
0
au
a
sgn u vi Yi
lul
p axi
uIp vi Yi] - lulp aX-(viyi)
axi
the inequality (14.14) implies ay
i=1
ax.l
p
au
N
(14.15)
vi + (p - 1) Iulp
axi
p
vi + lulp ax (viY1)] >
1
lylp
i
l
N
a = ix axi [luip vi yi ]
Further, we obviously have _
a
axi(viyi)
a
(
axi
l
vy
ax
p-1
ay axi
[vi
sgn
1
ax.J
p
1
ay ax
Y g n ax.]
ax.
IYIp-1 sgn y
and since
lyll-p sgn y]
1
lylp
is a solution of the equation (14.5), it follows that
y
p N
i=1
N
a
ax
w + (1 - p)
(viyi)
axil
ic
L V. =1
i
IYIp
This formula together with (14.15) yields p
NC
au
N
vi
axi
C
i=1
Integrating this inequality over
r
C
lulp w >
i
n
ax. 1
pulp vi Yi)
and then applying Green's formula to
the right-hand side (which is possible by virtue of the assumption (iii)), we have N
(14.16)
au
N
p
CC
u
uIp
1=1IaXi
as2
n
(
L vi Yi vni)
dS
i=1
n
Due to the assumption (iv), we finally obtain the inequality (14.4) from (14.16) letting
n -- m
.
LI
14.5. Remark. M If the domain
S2
belongs to C° '1
,
then we can choose 209
Stn =
n t N . Nonetheless, sometimes this choice need not be the best
for
S2
one. For instance, the assumption W1,1(Q)
luIp v i yi E
imposes conditions on the function w , vi
while the assumption (iii),
,
w
or
consider functions
u
small set
mN(M) = 0
taking
M C S2
,
such that
Stn
the unboundedness of
,
allows to
Indeed, we can eliminate the set
.
Stn ri M = 0
.
by taking
S2
gulp vi yi E W'''(S2n)
which exhibit 'bad behaviour' on some
vi
,
as well as on the weight functions
u
M
by
Analogously, we can sometimes eliminate Stn = S2 n {x E RN;
lxI
< n}
.
See also
Example 14.13 and Remark 14.14. (ii)
If
u E CO(S2)
,
then the condition (iv) of Theorem 14.4 is
fulfilled automatically and the assertion from the end of Subsection 14.1 follows immediately. If
u E C(S2)
1 E C O'1
,
,
and if we suppose that
P-1 sgn az vi U. = 0
(14.17)
on
3Q
1
then again the condition (iv) is fulfilled and we can assert that the Hardy inequality (14.4) holds for y
u E
provided there exists a solution
C_(s1)
of the boundary value problem (14.5), (14.17). [Note that the boundary
condition (14.17) is a non-linear Neumann-type condition.] If we consider the inequality (14.4) on the class CM_
for some set
(Q) _ {u E C-(S2); supp u n M= o} M C 2S1
,
then we can show that the inequality (14.4) is
satisfied if there exists a solution
y
condition (14.17), but this time only on
of (14.5) satisfying the boundary 2S1 \ M .
Naturally, the condition (iii) is supposed to be satisfied.
The result contained in Theorem 14.4 is due to B. OPIC, A. KUFNER [1], [2]; for
p = 2
,
see also A. KUFNER, B. OPIC 111. A modification which
is due to V. P. STECYUK [1]
will be described in the following theorem;
see also A. KUFNER, B. OPIC, I. V. SKRYPNIK, V. P. STECYUK [1].
210
14.6. Theorem.
where
Let
and
Stn C Qn+l C S2
defined on (i)
l
Let
< p <
1
Stn E
be a domain in RN
S1
C°'1
Let w, v E W(S2)
.
,
and
S2= lim S2
n-
n
y = y (x)
fulfil the following conditions:
The derivatives av
axi
exist a.e. in
a
axi
'
(ii)
IVYl
i
= 1,2,...,N
, where
S1
ay axi
CN
(14.18)
IvyIP-2
v
axi
'
=
(
G i= 1
The function
211/2
y
is a solution of the differential equation
IVYlp-2 a ., + w IYIp-2 y = 0
(14.19)
in
st
i
and satisfies the conditions (14.6). Further, let
u = u(x)
be such that
(iii)
lulp v yi E W1'1(S2n)
(iv)
lim sup n im
,
i = 1,2,...,N
,
n e N ,
and r
( N 1 lulp vl X yi vni))) dS ? 0 l
1i=1
J
o
fn
where
vYlp-2 axi 1_ (14.20)
y. = IYIp-2
y
1
Then for
u
satisfying (iii), (iv) the following inequality holds: N
(14.21)
Jlu(x)lp w(x) dx <
p
Np-1
L
i=1
lax(x) Q
v(x) dx
i
Suppose that the right-hand side in (14.21) is finite (otherwise (14.21) holds trivially). Analogously as in the proof of Theorem 14.4, we
Proof.
start from the inequality (14.13) where we put
211
si
1/p' au
= N
1/P
N -1/p
-
t
,
ax. v
i
IVYI
u
vl/P
y
We obtain N
au
Np-1
i=1
p
v+ p
uIP
1
i
N
Vylp v IYIp 1
au P
lulp-1
Ivy Ip
a
> 0
v
IYIp-1
i
and consequently
the inequality p
r
N
LNP-1
(14.22)
N 1 gulp IQYIP
v +
lax.l
v
IYIP 1 Iulp-1
p ax.
with
yi
from (14.20) holds a.e. in
sgn u v yi] ? 0
12
Since au u p-1
p ax.
sgn u v y
ulP
=
a aX.
1
v Y1)
Iulp ax.
(v Y1)
the inequality (14.22) implies N
(14.23)
au
Np-1
ax,
i=1
v + (p -
p
1)
lulp
IVYIP
IYIP +
C
i=1
Using (14.20) and the fact that
lulp ax.(vyi)
Z:
C
i=1
i
(1u1p v yi)
ax
.
i
is a solution of the equation (14.19),
y
we have
ax (vyi)
L
i
[,Iv Ip-2 N
ax,
L
1
y
ax
i=1
Y
y ax
P-2
ax.
r
YI1-P sgny]I
=
1
-
2
NC
IvYlp-2
1=1
212
IYIp-2
y
i
+
C
i=1
(1 - P)IYI-P ('Y ) axi
v =
w+(1 -P)
ivy
P I
v
and this formula together with (14.23) yields N Np-1
p
au ax
/
v - gulp w
L
axi
pulp v yi)
-
i=1
i==1
The rest of the proof repeats the arguments used in the proof of Theorem 14.4 (integration over
Stn
Green's formula and the application of the
,
condition (iv)). C]
14.7. The approach via formulas.
The foregoing theorems have shown that
the investigation of the Hardy inequality is closely connected with the
solution of a certain boundary value problem. Now we will prove a theorem
which is due to B. OPIC [1] and represents an analogue of the approach described in Subsection 2.4. Before formulating the general result, let us illustrate it on a certain simpler case.
It can be shown that the Hardy inequality 1/p
(14.24)
i?1
J
v.(x) dx
ax.
sz
St
holds for every
1/p
p
au (x)
N
[Ju(x) P w(x) dx
if the weight functions
u E C - (St)
w,v1,...,vN
are given
by the forrm1ias (14.25)
w(x) = div g(x)
(14.26)
vi(x) = pP lgi(x)lp [div
where
g(x)]1-p
g = (g1,g21...1gN) N
(14.27)
,
i=1
i = 1,2,...,N
,
is an appropriate vector function such that ag.
div g(x) _
,
aX
i
(x) > 0
a.e. in
Q
The formula (14.26) can be rewritten in the form ,
(14.28)
div g - pp
1-PI
vi
lgilP
r
= 0
,
i
and the formulas (14.25) - (14.28) can be exploited in two ways:
213
If we suppose that
(i)
w
function
are given, then the weight
v1,v2,...,vN
for which (14.24) should hold can be determined by solving
the system of non-linear differential equations (14.28) (for the unknown functions (ii)
gi ) and using (14.25). If
is given, then we have to solve the equation
w
div g - w = 0
N = 1
If we take
by (14.26).
vi
and then determine the weights
0 = (a,b)
,
g1 = g
, write
,
v1 = v
and assume
in addition that x
1 vl p'(t) dt <
for every
°°
x E (a,b)
a
then the function x (P')-p
= 1
1
f(
vl-p (t) dtl
p
1g(x)
a
is a solution of the ordinary differential equation (14.28). Moreover, g'(x) >
0
for
the function
and consequently
x E (a,b)
w
from (14.25)
is given by
x
w(x) = g'(x) = W)
-P
vl-p, (xW
vl-p'(t) dtJ-p
J a
This formula coincides (except for a multiplicative constant) with the formula (2.6), and thus
the approach just described is a natural extension
of that in the one-dimensional case.
The formulas (14.25), (14.26) are also extensions of the formulas (2.11): the function
g
from (14.25) is connected with the function
from (2.11) by the formula
A = (1 - p') In g
.
14.8. Example.
Let a function
x E 0
g = grad G
214
and put
C = G(x)
be such that
in (14.25), (14.26). Then
4G(x) > 0
for
A
(14.29)
aG p aXi
(AG) 1-P
and the inequality (14.24) assumes the form
11/p f(Ilu(x)lp
(14.30)
4G(x) dx]
<
0 au
aG
p
ax
(x)
u E CO(0)
Let
14.9. Theorem.
0n C. 0n+lC Q
,
.
Let
< p <
1
0nE CO'1
.
p
ax.(x
.
and holds for every
dxljl/P
(4G(x))1-p
)
Let
0
be a domain in RN ,
the functions
gi '
0 = lim 52n
n-W i = 1,2,...,N, satisfy
gi E W1'l(On)
(14.31) and
where
g = (g1,g2,...,gN) (i)
x E 0
for a.e.
div g(x) > 0
(14.32)
.
Let us define the weight functions
w,v1,...,VN
by the formulas
(14.25), (14.26). Then the Hardy inequality (14.24) holds for every u - u(x) (14.33)
defined on
such that
0
n E N
for every
u E C(11 n)
provided N
r
lim inf
(14.34)
n i* W
(14.35)
J
ao
l
l = i=1
g.v niJ dS < 0
n
Let us define the weight function
(ii)
function
IuIP
v
w
by (14.25) and the weight
by
v(x)
N
I v
]=1 ]
1P-1 (x)1 J
215
with
vj
from (14.26). Then the Hardy inequality
(14.36)
u(x)P w(x) dx <
ax (x)
i=1
holds for aZZ
Proof.
u
satisfying (14.33), (14.34).
Again suppose that the right-hand side in (14.24) or (14.36) is
finite (otherwise the corresponding Hardy inequality holds trivially). Using Green's formula, we obtain for
gi
,
satisfying (14.31),
u
(14.33) that N
P div g dx =
gulp
J
(
i=1
gvni i
l
dS J
aQ N
_
(14.37)
i=1
fp
sgn u
au
dx
12
n
J luiP
`
asp
p-1
dS +
(
p
1-1
n
lu
au axi
n
M Using the formula (14.26) and estimating the last integral by Holder's inequality we have
ulp-1
p
ax.l
Igil dx = J IuIP-1
au ax,I
(div g)1/P vi/p dx t
i
12
n
SZ
n
dx]l/pI
p
au
IJ IuIP divg
`-
n
ll1/p
vi dx]
aX,
JS
n
which together with (14.37) yields N
IuIP divg dx < J
(14.38)
J f Iulp 312
0n
IuIP div g
JI
i
n
11/p' N +
1
Y givni) dS +
dx`
Du ax.
J
1 dL
216
n
1
n
p
1/p
vi dx I
Denote
(14.39)
J(Q) =
lu(x) Ip div g(x)
dx
J
Q
Without loss of generality we can suppose that J(SZn)
> 0
(14.31),
for
n
> 0
and, consequently,
sufficiently large. Moreover, by virtue of the assumption
J(O ) < -
[J(Qn)]1 /p
J(S2)
for
u
and letting
n
satisfying (14.33). Dividing (14.38) by in view of (14.34) we obtain the desired
,
inequality (14.24). (ii)
Holder's inequality and the formula (14.26) yield au Igil
ax
L
N
Du
p
pJ1/p IiNjlg1!P']1/p, L
ax.
i=1
i
i= 1
P-1
(div
I
i=l[axi Ip)1
g)1/p Ii=1
vl/(P-1)1/P 1
1
Using also the formula (14.35), we have N Iulp-1
P
J
f
lulp-1
Igil dx = J P
axiI
i=1
N (1Lllax gi
12
n
n N
lulp-1
(div g)1/P
= J R
i=1
au ax.
pl1/P v1/P dx
.
1
n
Now we estimate the last integral by Holder's inequality and obtain from (14.37) the following analogue of (14.38):
iNjgi' N
gulp divg
dx 5
J Q
asp
n
+ lJ Iulp divg
n P
i
N
dx I
au
J
p
ax.
11=1 12
n
iJ dS +
=
111
1
p
u
J
ll1/p
v dxI
n
From this inequality we derive the inequality (14.36) by the same arguments as we have derived (14.24) at the end of part (i). 11
14.10. Remarks.
(i)
Let
Q
be a domain in RN
and denote by 217
C1(Q)
(14.40)
on
which are bounded and uniformly continuous
u = u(x)
the set of functions
au/axi
together with their first derivatives
Q
i = 1,2,...,N
,
.
Obviously, the assumption (14.33) can be weakened to
for every n E N ;
u E C1(Qn)
this last assumption together with (14.31) again guarantees that Green's formula can be used and that (ii)
J(S2n)
(14.39)) is finite.
(cf.
By the same arguments we can show that Theorem 14.9 holds if the
pair of assumptions (14.31) and (14.33) is replaced by
gi E
(14.41)
C1(S2n)
uE
,
for every n E N .
W1'p(S2n)
Let us consider the weight functions (14.29) from Example
14.11. Example.
14.8. Using the formula (14.35) we have ( N
v(x) = pp (AG(x))1-P
I
Y
1=1
ac
ax.
x
P/(P-1)lp-1
)
1
and instead of (14.30) we obtain the inequality (14.36), i.e.
(14.42)
AG(x) dx
flu(x) S2
N
p
< PP
(
i=1
S2
L
i
p = 2
then
,
dx
ax (x)
.
j=1 N
If we set
P/(P-1) lp-1
N``
r (AG(x))1-P
ax (x)
p/(P-1)P-1
8G x
2
and the inequality
= IVG
1111=1a
j
J
(14.42) is exactly the inequality (14.1) with
w
and
14.12. Some applications of Theorems 14.4, 14.6, 14.9.
v
given by (14.3).
Let us check the
important condition (iv) of Theorem 14.4 for some special weight functions. For
1
< p < -
,
x0
(x01,x02""' x ON
) C RN
p -
(14.43)
218
w(x)
(IE - p + Nil l
p
E-p I
x - xO
and
E E R ,
E
p - N
,
put
(14.44)
vi(x) = Ix - x
Ix i - x Oi1
le
2-p
(
lIx --X01
G
It can be shown that the solution
of the differential equation (14.5)
y
has the form (14.45)
y(x) = Ix - x0la
with
a = 1 -
N
e
,
P and the condition (iv) reads 0 < lim-sup
aQ
alp-1 sgn a
x0IE-p
Iu(x)Ip Ix -
J
n N
(xi - x01) vni(x) I ds L
i-1
This rather complicated condition
will certainly hold if
N
(14.46)
Iu(x)lp sgn a
Y (xi i=1
for
n
(14.47)
vni(x)
- x01)
>=
0
on
BQ n
sufficiently large. If we denote
h(x,xO,Qn) _
(xi - x01) vni(x) i=1
then obviously the sign of this function for
x ` aQ
n
will be important
since, for instance, if
sgn a h(x,xo,1 n ) < 0
for
x E F n C 30n
then the condition (14.46) will be satisfied provided
u(x) = 0 for x E P n Therefore, let us introduce some special sets which will be exploited in the following examples:
For
G C RN
,
G C- C 0, 1
,
x0 E
RN
denote
8G+(x0) _ {x E 8G; h(x,x0,G) > 0} (14.48)
@G-(x
0
)
_ {x E 8G; h(x,x0,G) < 0}
219
[Of course,
h(x,x0,G)
is defined by (14.47) where the i-th component of the outer normal to G .]
Let
14.13. Example.
1
< p < W
S0EC0'1
vni
x0 E 0
,
.
is replaced by
Then the inequality
f Iu(x)IP Ix - x0IE-p dx
(14.49)
R
p iN
P <
[IC
- D + N -) i=1 f 0
2-P
Ixi - x0il
au
)
Ix - xO IE
P
aX.(x
i
Ix - X01
dx J
holds provided one of the following two conditions is satisfied: (i)
and
u E W1'P(0)
supp u (1{x0} = 0
(14.50)
(ii)
(14.51)
Here
< p - N
t
e> p - N
and
u = 0
aSZ(x0)
on
2n = 0 \ B(x0,1/n)
(14.52)
u E W
(i)
on
aS2 (x 0)
1'p(S2n)
n N
is such that
where
B(x0,r) = {x E RN;
14.14. Remarks.
u = 0
,
is such that
x - x0
< r}
.
The conditions 14.13 (i) and/or 14.13 (ii) guarantee
that the assumptions of Theorem 14.4 are fulfilled. Before we show it let us
insert some geometrical considerations. Taking (14.53)
with
52n =
S2
B(x0,1/n)
\ B(x0, n ) the ball from (14.52) for
(draw a picture !) that for
(14.54) Suppose
220
aQn = [a S2 (1 S2 n] S2 E
CO'1
.
x0 E S2
and
U [aB(x0' n
Since obviously
)
r = 1/n n
n SI ]
,
we can easily verify
sufficiently large,
[2B(x0, n )1'' -St] C a0n(x0) we obtain that
(14.55)
aO+(x0)
(14.56)
30-(x0) = [act-(x0) n gin] U [aB(x0, n ) n
=
ast+(x0) n sn
(for the notation see (14.48)).
Moreover, if the domain
ast+(x0) = ast
(14.57)
,
is strictly convex then
0
a0-(x0) = 0
.
Now, let us go back to Example 14.13. In the case (i) we have a> 0 (cf. (14.45)), i.e.
sgn a =
1
.
If we put
0n = 0
for every
the condition (iv) of Theorem 14.4 will be satisfied if
u = 0
while the condition (iii) of Theorem 14.4 will be satisfied if
n E IN
on
,
then
aQ (x0),
x0 0 supp U.
So, we obtain the conditions (14.50).
In the case (ii) of Example 14.13 we have and consequently
u=0
(14.58)
a < 0
,
i.e.
sgn a = - 1,
the condition (iii) of Theorem 14.4 will be satisfied if
aStn(x0
on
aQn(x0)C a0+(x0)
However, according to (14.55) we have
,
and (14.51)
implies (14.58). The condition (iii) of Theorem 14.4 is satisfied automatically due to the fact that Stn = 0
x0 It 0n
.
On the other hand, if we took
as in the case (i), the condition (iii) could be violated.
If we suppose in addition that
is strictly convex, then the
0
conditions (i), (ii) from Example 14.13 are simpler:
(ii)
c
,
u(= W1'P(St) ,
e>p-N
,
u E W1'P(0n)
For
p = 2
,
x0 it supp u, u = 0 on
,
aQ .
the inequality (14.49) assumes the form
f ,u(x)12 Ix -
x01c-2
dx
0 4
le - 2 + N
N
au (x)
2
i=1
1
axi
2
jx - x01
dx
221
This inequality is proved e.g. in R. T. LEWIS [1] under a little stronger assumption
on
u
For
(iii)
consequently
.
< p = 2
1
(Ixi - x0.I/lx - x01 2-P
we have
1
and
the inequality (14.49) implies
Ix - x016 p dx <
p
(14.59) J
0
p N
p
I6 - p+NIJ i 1 I
au (x) ax.
P
Ix - x0le dx
.
Compare this inequality with the classical Hardy inequality (0.2)
(N = 1)
and with the inequalities (13.30), (13.31) (N = 2).
The foregoing inequalities have been derived by using Theorem 14.4. Now we will use Theorems 14.6 and 14.9.
14.15. Example.
Let
1
< p < -
x0 E S2
.
Then the inequality
JJu(x)JP Ix - x016-P dx
(14.60)
0
Np-1
p
P
p + NI
N
i=
au (x) 1 axi
p
x - x016 dx
holds if one of the conditions (i), (ii) from Example 14.13 is satisfied. This result can be derived by using Theorem 14.6 where the solution of the differential equation (14.19) is again given by the formula
y
(14.45).
Compare the inequality (14.60) with the inequalities (14.49) and (14.59). In (14.60)
the constant is worse, but in (14.49) we have a little
more complicated weight
functions
v,
while (14.59) was derived only for
i 1
< p < 2
.
14.16. Example.
equalities
222
Let
1
S E C 0, 1
x0 C- H
.
Then the in-
ll1/p
(14.61)
[Jiuxip
<
Ix - xOIE-p dx1
SI
P
p
axi(x)I
Ic - p + NI i=1
Ix -
- x
xOle (Ix I
l
1/p
p
Oi
Ii
dxl
x - x0
i
Q
and
(14.62)
u(x)Ip ix - xOIE-p dx f
J
p N
P
Ile - p + Nil
p
X - xole-p
ax(x)
)
iL1
.
ci
P/(P-I) lp-1
- x
Ix.
1
j=1
JJ
dx
Oi
again hold if one of the conditions (i), (ii) from Example 14.13 is satisfied.
The inequalities (14.61), (14.62) are the inequalities (14.24) and (14.36) from Theorem 14.9 (see also Remark 14.10) where we set gi(x) = sgn (c - p + N)
Ix - x0IE-p xi
(Ixi - xOil/Ix - xOI)p s
Since
1
,
we obtain from (14.61) the in-
equality
(14.63)
x0IE-P [JIu(x)i p
I x -
ll1/p
dx)
0 p Bxi(x)
IE - p + NI iLl
Ix - x0I
e
1/p
dx
lJ
which is an inequality of the type (12.9). Using the estimate N
1
a.
i=1 1
N
aP
Np
i=1
1
,
a
i ?0
we obtain the inequality (14.60) directly from (14.63). The same inequality can be derived from (14.62) [using the fact that while for
1
< p < 2
Ixi - x0.j
<_
Ix - x01 1,
we can derive the inequality (14.59) again from
223
xO.Pr
(14.62) [using the fact, that 1
X
-
Ix
s
(
X
Ix. - x012
p'/2
for
< p < 2 ]. If we take formally
N =
and
1
S2 = (0,m)
,
then all the foregoing
inequalities reduce to the classical Hardy inequality (0.2). Consequently, one can expect also that the inequalities mentioned remain true for more C 0,1 ). This is really the
general domains (unbounded, not belonging to case for certain classes of domains.
14.17. Definition.
will say that
(i)
Let
x0 E Q
be a bounded domain in RN
52
.
We
belongs to
Q
/1011 llim(x0)
if there exists a sequence
{rn}
rn 1 0
,
such that
,
Stn = 0 \ B(x0,rn) ECO'1 Let
(ii)
0
be unbounded,
x0 E RN
We will say that
.
belongs to
0
(0,1
lim(x0'm)
if there exist two sequences
{R n}
{rn}
,
Rn t
rn 4-
0
,
such that
CC0,1
0n = 52 (l [B(x,,Rn) \ B(x0,rn)
(14.64)
,
For both types of domains we define
(iii)
852±(x0)
= lim [80(x0) (1 852 ]
.
n±.=
14.18. Remark.
The inequalities derived in Examples 14.13, 14.15, 14.16 is replaced by the assumption
remain true if the assumption
o Cr C0'1
0 E C °im(x0)
0 E C Oim(x0''o)
.
If we consider
,
we only have to add in
l
(14.51) the assumption supp u and
0
n
is compact in RN
has to be given according to (14.64).
Finally, let us present two examples with a little different weight
224
functions.
14.19. Example.
Let
< p < m
1
S2 L r00
lim
,
x0 E
1(x0,m)
RN
Let
.
be
Stn
given by (14.64) and let one of the following two conditions be satisfied:
5> 0
(i)
is compact in RN
supp u
B<0
,
x0 0 supp u
.
(ii)
and
Y> p - N -
,
uE
,
1
W1,P(Sl n)
u=0
,
on
852+(x 0) ,
;
y
uEW1,P(Qn)
,
1
u = 0
,
81 (x0)
on
Further, denote
-p+N
IY
1 p-1
p
1
C = C(R,Y,P,N) =
I
lI
Then the following inequalities hold:
S x-x l0 f u(x)IP
x0 Y-p+1 dx
Ix -
e
8u
5Ix-x01
p e
8x.(x)
Y
2-p
Ix - x0I
xi
xUil
dx
,
i
x-x
H
J u(x)Ip
Ix -
01
0lY-p+l dx <
x
e
S2
p
NCC
6Ix-x 0
8u
NP-1 C
i=1
8x .
I
Ix - x0IY-p+2 dx
e
(x)
i S2
y
These inequalities follow from Theorems 14.4, 14.6 where the solution
nIx-x01
y(x) = e
of the corresponding differential equation is the function with
a = B/(1 - p)
14.20. Example.
.
Let
1
< p < m
S2 E C 0im(x0,")
,
,
x0 E RN
,
N > 2
.
Let
one of the following two conditions be satisfied:
(i) a < 0
u E Wl'P(S2n)
,
u=0
,
on
852+(x0) ,
supp u
is compact
in RN
(ii)
a>0
,
u E Wl'P(S2n)
,
u = 0
on
8U(x0)
,
x0
supp u
.
Then the following inequality holds: 225
aaIx-x012-N
II
Ix -
x
u(x) I P
[j
p ealx-x0I
au (x) axi
p
IaI(N - 2) i=l
12 (1-N) 0
1/P <
dxJ
2-N p(N-2)+2(1-N)
lx - x
1l
Ixi -
II
0
1/p
xoil P dxJ
This inequality follows from Theorem 14.9 (and Remark 14.10) where we have N
aIx-x012-N
set
gi(x) = - a e
(xi - x0i)
Ix - x01
15. THE HARDY INEQUALITY AND THE CLASS
Let
15.1. The Muckenhoupt classes.
1
,
i - 1,2,...,N
.
Ar
w E W(RN)
< r <
.
B. MUCKENHOUPT
[2] introduced a class of weights denoted by Ar - Ar(RN)
.
This class plays a very important role in the theory of weighted norm inequalities (cf., e.g., J. GARCIA-CUERVA, J. L. RUBIO DE FRANCIA [1]) and is now commonly called the Muckenhoupt class. It is defined as the set of all weights
w
sup
such that
mu(Q)
11/r'
l1/r J w(x) dxJ
w1-r' (x)
dx
< m
,
I'4a (Q)
JJ
Q
Q
the supremum being taken over all cubes QC RN
with sides parallel to the
coordinate axes.
This definition can be extended to pairs of weights
being a domain in RN (15.1)
.
Denoting for
Q = Q(y,R) = {x E RN;
the open cube with centre at
(15.2)
< R,
Ixi - yi
,
,
R > 0
k
,
we will say that
0 < k < W
,
by
i = 1,2,...,N}
(w,v) C A(S2)
if there exists a number
226
y
y e RN
w, v E W(Q)
such that
,
S2
1/r'
(15.3) f
l mN(Q)
(
dxJ
J
(Q)
I
QnR for every cube
'I-r' (x)
w(x) dxJl/r
<_
k
Qr) Q
Q - Q(y,R)
y E RN
with
R > 0
,
.
In this section we will show that the (Hardy) inequality ((
u(x)I
(15.4)
holds for all
li=l
1
where
u E C O(Q)
such that
r
with
1/q
( au w(x) dxCJ Nax.(x)
q
1
v (x) dx]
i
Q
l/p
p
is a fixed cube, provided
Q
< r < p < Nr
1/q = 1/p - 1/(Nr)
,
(w,v) E Ar(Q)
The proof uses
.
certain estimates for Riesz potentials and maximal operators. Therefore, let us start with some definitions and auxiliary assertions.
15.2. Definitions. W For
p >
1
a domain in
R
,
RN
,
w E W(R)
define the weighted Lebesgue space
(15.5)
Lp(S2;w)
as the set of all measurable functions
f = f(x)
on
2
such that the
number l/p
If(x)Ip w(x) dxl
(15.6)
< p < -
for
1
for
p =
,
R
IIfIIp,R,w =
ess sup If(x)I
xCS2 is finite. For space
,belongs to (15.7)
1
we obtain the classical (non-weighted) Lebesgue
introduced in Subsection 14.3 M.
Lp(12)
Since
w =
f
belongs to
LP(R)
if and only if the product
Lp(12;w)
fw1/p
and
I0p'R'w =
it follows immediately that
Ilfwl/pilp'R
LY(R;w)
is a Banach space with respect to the
norm (15.6). (Recall Convention 5.1!) (ii)
For
f
measurable on
R
and
a
>_
0
,
denote
227
E = E(f,a) = {x e 0 ;
(15.8)
f(x)I
> a}
and
w(E) = w(E(f,a)) =
(15.9)
w(y) dy
J
E(f,(j)
Define the weak Lebesgue space (Marcinkiewicz space) Lp'*(2;w)
(15.10)
as the set of all measurable functions
f = f(x)
on
S1
such that
1/p
sup a [w(E(f,a))] a>0
(15.11)
(iii)
For
measurable on
f
Mf
the maximal function (15.12)
(If)(x) =
define the Riesz potential
and
If
by
If(Y)N-
r
dy
,
x E RN
Ix - YI
RN (15.13)
RN
(Mf)(x) = sup mN(Q)
Jf(y)I
xERN
dy
Q
the supremum being taken over all cubes containing the point
(iv)
For
x
Q = Q(z,R)
zERN
,
R> 0
,
.
defined on 51C RN we put
f
If = If
Mf = Mf
and
where
f(x)
for x E Q for x E RN \
f (X) jl
0
15.3. Theorem
Sl
(B. MUCKENHOUPT [2], Theorem 8; J. GARCIA-CUERVA, J. L. RUBIO
DE FRANCIA [1], Chap. IV, Theorem 1.12). Let
there exists a constant
K > 0
such that
JjMf1Ip,d,w <= K IIf1Ip,dl,v for all
f E LP(d;v)
if and only if
(w,v) E A(d) 228
.
.
1
< p <
w, v E W(d)
.
Then
15.4. Theorem
(the Marcinkiewicz interpolation theorem; cf., e.g.,
A. ZYGMUND [1], Theorem 4.6; J. BERGH, J. LOFSTROM [1], Theorem 1.3.1). Let
p0'p1'g0'g1 E [l,-) such that
IlTfll*
p0 x p1
,
i
Ilfll
Let
.
T
be a subZinear operator
pi,0,v
Pi
for every
f E L 1
=
(0;v)
- 0
1
and assume that
pl q
p
For
.
0 E (0,1)
put
1= 1- 0+ 0
+ 0
p0
p
i = 0,1
,
q
q0
ql
Then
(w,v) E A (Q) p
Then
.
Tfllq,2,w < c
lfllp,0,v
for every f E Lp(0;v) and 1-0 c0 c0
c
Let
15.5. Lemma. Proof.
0
c1
.
(w,v) E A
r (0)
.
Holder's inequality with exponents
for every
(p - 1)/(p - r)
p E (r,W)
and
yields
(p - 1)/(r - 1)
vl-p (x) dx =
1
l
(Q)
Qr.Q r
(-1)
Ip-r)/(p
1
(r-1)/(p-1)
1 r
_N(Q)
<
j
1
v
mN(Q)
for every cube
Q
}
w(x) dxJ
J
N(Q) 111
J
Qr)Q
w(x)
vl-p'
f
p1
(x) dxJ
<
Qn0
Q.(Q
mN (Q)
(r-1)/(p-1)
(x) dxJ
Using this estimate in (15.3), we obtain
.
(Q)
1-r
(x) J dx
v
[mNQ)]
dxllmN (Q) J
l
j
Q^i
vl-r(x) dx}
r-1
< kr
J
229
i.e.
(w,v) E Ap(S2)
. 11
The proof of our main result is based on the following assertion about the continuity of the Riesz potential.
15.6. Lemma.
Let
r < p < Nr
<
1
,
1
=
q z C-
RN
R > 0
,
Let
.
W(Q) =
(15.14)
(w,v) E Ar(Q)
1 - Nr . Let P
Q = Q(z,R)
with
and denote
w(y) dy
J
Q
Then there exists a finite constant
(15.15)
CR[w(Q)]-1/(Nr)
IIIfIIq,Q,w `
holds for every f E LP(Q;v) Proof.
For
e >
QC = B(x,e) f where
B(x,e)
,
Q
I
such that the estimate
If,Ip,Q,v
.
x E RN
and
0
C > 0
denote
= Q \ QC
QE
f G LP(Q;v)
is the ball from (14.52), and for
(Ief)(x) =
Put
If(y)N-1 1
dy
Ix - yl
Qe
If(y)
(Ef)(x)
=
L. -
J
N-1 dy
QE
Then (15.16)
If = Ief ± IEf
,
and Holder's inequality for three functions with the exponents p/(r - 1)
,
p/(p - r)
p
implies
(r-1)/P (JE(x))(P-r)/P (15.17)
v'-r
(1Ef)(x)
_<
IIfIIp,Q,v l
(y)
dy)
Q
where (15.18)
JE(x) = J QE
230
x - y1(1-N)P/(P-r) dy
r c
Nr
(p-Nr)/(p-r)
the last estimate can be derived using spherical coordinates; measure of the unit sphere
c
is the
IxI = 11 ]. From (15.17),
S(0,1) = {x E RN;
and (15.18) we have (15.19)
(IEf)(x) s <
Y)(r-1)/P
(- p - r 1(P-r)/P E1-Nr/p (f v1-r'(Y) d l Nr - pJ lJ
If
.
IP,Q,v
Q
kn = E2
put
(IEf)(x)
In order to estimate
I
(15.20)
(IEf)(x) _
n=
Ix - yI
Qkn
n n=0
.
Then
If(Y)
N-1 dy
<_
L
J
N-1 °
r
n=0 B(x,kn)(kn+1)
Qkn+1
(2k )N <-
n = 0,1,2,...
,
f(Y) I
J
=0
-n
1
I f (Y) I dy
J
N-1
(2kn)N
(k n+1)
Q(x,kn) (2kn)N
< (Mf)(x) n=0
(kn+l-
N-1
=4
N
E(Mf)(x)
In view of (15.19) and (15.20), we have from (15.16) that
(15.21)
(If)(x) <
v1-r
1-Nr/p
(Y)
If1p,Q,v + E(Mf)(x) I
dY,
Q
with r
kl = max
c
XT
L
(p-r) /P
r
p
-
4N]
j
p
The estimate (15.21) holds for every
E > 0
the right-hand side in (15.21) over all (15.22)
(If) (x) < k2 [(Mf) (x)] 1-p/ (Nr)
c
.
Evaluating the infimum of
> 0 , we obtain
f
Ip/(Nr) 'p,Q,v
v1-rr
11/(Nr') (y) dY
Q
where k2 = kl
(pr
_1)P/(Nr)-1
Pr
231
Since
(w,v) E Ar(Q)
v1-r'
IlQ
lJr
we obtain from (15.3)
,
1/(Nr') < kl/N
(Y)
[m(Q)]
dyj
1/N [.(Q)]-1/(Nr)
_
J
=
with
w(Q)
kl/N 2RLw(Q)]-1/(Nr)
from (15.14), and the last estimate together with (15.22)
yields
(If) (x) < k3R Lw(Q)]-1/(Nr) [(Mf) (x)J 1-p/(Nr) Ilfllp/(Nr)
(15.23)
P,Q,v with
k3 = 2k Let
1/N k2
rI E (p,=)
(w,v) E Ar1(Q)
be fixed. Since
> r
rI
Lemma 15.5 implies that
,
which together with the assumption
(w,v) E A(Q)
and
with Theorem 15.3 leads to the estimates for every
11MfIIr,Q,w
Mf1irl,Q,w `= K1
i
p
constant (15.24)
K > 0
f
r
r
for
f E Lr(Q;v)
r
r
we conclude that there exists a
such that
I;Mfl5 K P,Q,w
fiIP,Q,v
for
f E LP(Q;v)
,
which implies that
(1-p/(Nr))q
1l 1/q
w(x) dx]
((
ll1/q
[JI(Mf)(x)J p w(x) dx]
=
Q
Q
= 11MfIlp/q
KP/q
<
P,Q,w
(note that
=
1/q = 1/p
I1fI1p/q
P,Q,v
=
KP/q IfIj1-p/(Nr) p,Q,v
1/(Nr) ). Moreover, from (15.23) we have 1ll/q
IIfiiq,Q,w
III(If)(x) q w(x) dxj
<
Q
k3R [w(Q)]-I/(Nr)
(1-p/(Nr))q IIf
(M f)(x)I
P,Q,v p/(Nr)
If
Q
232
ll1/q
w(x) dx]
and consequently
15.7. Lemma.
C = k3Kp/q
(15.15) holds with
Let
< p <
1
be a domain in RN and u E
0
Let
11
C1 (0)
.
'Mien the inequality c-1 N1/P'I(lVulp)(x)
5
Iu(x)I
(15.25)
x E 0
ppZds for every
being the measure of the unit sphere in RN
c
,
and N
x
IVul
(15.26)
(
proof.
Let
a {zE RN;
i
u E CI(0)
supp u C B(x,R)
that
ax.
1=1
p
Iz - xI
p 1/p
au
L
x E 0
and
Then there exists a ball
.
u(y) = 0
Consequently,
.
for
B(x,R)
such
y E S(x,R) _
and from the formula
= R}
R au
-. u(x) = u(y) - u(x) _
(x+
t y-x
Yi - xi l dt Y - xI J
y - xI)
0
we obtain by Holder's inequality (for sums) that R au axi
(x + to)
IP 1/P
lyi - xiI P' 1/P'
N
)
( 1
1i=1
dt
Iy - xI Rr
N
N1/PI
i=1
0
ax, (x 1
where
0 = (y - x)/Iy - xI
to
over the unit sphere
0
Iu(x)I <
N
pl1/P
au
+ to)
p
0
Integrating the last inequality with respect
.
S(0,1)
1/pt
,
we obtain
Rf
r
dtJ d0
IVu(x + t0)l J
S(0,1)
and the substitution
dt
Vu(x + t0)
dt = N1/p/
z = x + to
I
p 0
(with the Jacobian
tN-1 =
Iz _ xIN-1
)
yields
233
I_ N
Iu(x)
1/p'
Az
Iou(z)Ip
c 1
N N -1
Iz - x
B(x,R)
Ir
1
-p
c
1
IIvuPJ(x) 0
Let
15.8. Theorem.
< r < p < Nr
1
1
,
q
z E RN
,
R > 0
Let
.
(w,v) E Ar(Q)
Then there exists a constant
> 0
c
=
-
1
p
1
Q = Q(z,R)
Nr w(Q)
,
and Zet
with
be defined by (15.14).
such that the inequality
1/q
(
ljrlu(x) Iq w(x) dxJ
(15.27)
s
Q
p
< cR [w(Q)]-1/(N=) (
holds for every Proof.
u E CO(Q)
(x) dx
f
i=1
Ql
1
.
According to (15.26), N
i=1
P
au Jr
v(x) dx
ax.1 (x)
II IouIPIIp,Q,v
Q
Let
u E C1(Q)
and assume that
IVuIpIIP,Q,v < Vulp ) imply II
and Lemma 15.6 (with
f =
which is (15.27) with
15.9. Remarks.
SZ
= Q)
N1/P' III(Ioulp) Ilq,Q,w <
Ilullq,Q,w <
< Z-1 N1/P'
Lemma 15.7 (with
CR
[w(Q)]-1/(Nr) U IVuIpIIp,Q,v
c = c-1 N1 Pr C
.
,
0
(i)
The foregoing results are due to P. GURKA, A. KUFNER [1]. This paper generalizes some results of E. FABES, C. KENIG, R. SERAPIONI [1] and F. CHIAREN7_A, M. FRASCA [1], who have considered the inequality (15.4) for w = v .
(ii) Using some covering lemmas, one can extend the foregoing result
from cubes to more general domains P. GURKA, B. OPIC [1].
234
S2
(including unbounded ones)
-
cf.
(iii)
The condition
(w,v) E Ar(Q)
represents a criterion for the
choice of admissible weights, i.e. of such weights that the corresponding Hardy inequality holds. If we compare this criterion,e.g., with the criterion
'via solvability of differential equations' (cf. Theorems 14.4, 14.6), then the former is relatively easier to verify in a general situation. on the ether hand, it is rather restrictive, which can be illustrated by the 'following example (see P. GURKA, A. KUFNER [1]): If we consider the special w(x) = [dist
case
(x,3Q)]a
,
v(x) = [dist
(w,v) E Ar(Q)
can be shown that
a >- 1
a,
B E R , then it
if and only if
,
(x,3Q)]
a >_ B,
while, for instance, the results based on imbedding theorems derived in Chapter 3 (see Theorem 21.5) allow a substantially bigger set of admissible a
value
B
,
described by the conditions
a >_ B
Nr
Np(r - 1)
Nr - p
Nr - p
(draw a picture in the
B x P - 1'
(a,B)-plane !).
16. SOME SPECIAL RESULTS
The conditions derived in the foregoing sections which guarantee the
16.1.
validity of the N-dimensional Hardy inequality 11/q
[JIU(X)I q w (x) dxl
(16.1)
are mostly sufficient.
.
I
au
J---(x) ax
P
11/P
vi(x) dx]
V. G. MAZ'JA [17 has derived necessary and
8uffieient conditions on n E CO(ST)
( N C
w,v1,...,vN
under which (16.1) holds for every
His conditions are expressed in terms of capacities and are
difficult to verify. In our opinion, the advantage of MAZ'JA's results lies
in the possibility of obtaining some information about the capacity of a set, once we have derived some information about the validity of the corresponding Hardy inequality by another method.
On the other hand, MAZ'JA considered also inequalities of a more complicated form, for example with right-hand sides of the type
235
11/p
[JL(x; Vu(x))]p dx] Q
(for
4
see Definition 16.2), and admitted also (weighted) Orlicz norms
,
on the left-hand side. Here we will mention some of MAZ'JA's results for a
particular function
(D
namely
,
N
(16.2)
1/p
p
(D(x;E) =
Icil
,
J
without proof.
Let us start with the definition of the capacity.
16.2. Definition.
Let
negative continuous function on respect to
E
capacity of
.
K
Let
1
< p < m
D x RN
,
.
Let
with respect to
be a non-
O(x;E)
homogeneous of degree 1
For a compact set
.
K C 0
,
with
the (p,f)-
is defined as the number
0
(p,O)-cap (K,Q) = inf
(16.3)
RN
be a domain in
0
[(A(x; Vu(x))]p dx
{
;
u E M(K,0}
where M(K,Q) = {u E CO(52); u =
(16.4)
1
on
K}
Here we will consider the particular function
v E W(S2)
vi continuous on
,
i
16.3. Theorem
w E W(52)
,
(V. G. MAZ'JA
SZ
[1i'
,
i = 1,2,...,N
from (16.2) with
.
Let
Corollary 2.3.3.1).
continuous on 0 ,
viE W(Q)
(D
-<
u E C0_(S2)
if and only if there exists a finite number
c
p < q <
ti
given by (16.2).
(D
Then the inequality (16.1) holds for every function a finite constant
1
B
with such
that )
I(
w(x) dx
(16.5)
l
1/q <=
B [(p,(D)-cap
J
K
for every compact set
K C 52
Moreover, the best possible constants estimate
236
C
and
B
are related by the
1)1-p
B < C <= pp (p -
B
.
16.4- Remark. The inequality (16.5) is a so-called isoperimetrie inequality and connects the measure
w(K) =
of an arbitrary compact
w(y) dy
K C Q
J
K
with the (p,O)-capacity. Using (16.3) and (16.2), we can rewrite (16.5) ,jn the form dy)1/q
(16.6)
w(Y)
lJ
K
B
I
inf
NJ {
p
au
ax.(x
1/p
vi(x) dx
u E M(K,SZ)}
;
L
To find conditions on
w,v1,...,vN
under which the Hardy inequality (16.1)
'holds means to investigate (16.6) for any compact set
K C St
,
which
represents a rather complicated task.
V. G. MAZ'JA
considered also the case
p > q
Let us formulate a
.
particular result which illustrates the complexity of the problem.
16.5. Theorem
w E W(S2)
,
(V. G. MAZ'JA [11
v. E W(1)
Theorem 2.3.5).
continuous on
the inequality (16.1) holds for every function
constant C if and only if A<
where [I IlJ
A =
sup (Qi}
Let
1
< q < p <
be given by (16.2). Then
0 ,
A
u E C0'(S2)
with a finite
is defined by p/q
(p-q)/q
w(x) dx I
L )]q/(p-q)
,Q i+1 .
i
Here the supremum is taken over all sequences whose boundaries are
{Q.}
C--manifolds and which satisfy
of open sets Q. C Qi+1
Q, C S2
i E 2
.
'
Less complicated necessary and sufficient conditions have been derived for the special case S2 = RN
,
vi -
1
for
i = 1,...,N
237
16.6. Theorem
(V. G. MAZ'JA [1] , Corollary of Theorem 1.4.1.2, Theorem
1.4.2.2). Let
1
, or 1 = p < q <
p
,
Let wE W O R N) ,
Then the Hardy inequality (16.7)
IJ
lu(x)1
1N
11/q w(x) dxj S C
8u
i-1
If
RN
holds for every function
(x)
with a finite constant
C'(RN)
u (Z-
i
ax.
RJ
c
> 0
if and
only if (16.8)
B =
sup R
sup
1/q
1-NIP
xERN R>0
16.7. Remark.
In the case
< W
w(y) dyJ
.
J
lB(x,R)
< p < q < w
1
p < N
,
the proof of Theorem
,
16.6 is based on the estimate for the Riesz potentials
(16.9)
if 11
<
N
c
N
11f1l
p,R
q,R ,w
This analogue of (15.15) (cf. Lemma 15.6) is due to D. R. ADAMS [1] and states that the inequality (16.9) holds if and only if the condition (16.8) is satisfied.
Moreover, D. R. ADAMS [2] extended the result just mentioned to the case of two weights. Assuming that
v E W(RN)
is such that
he showed that the inequality
(16.10)
if 11
holds for every
(16.11)
B=
N
N
f11
p,R ,v
q,R ,w
f E LP(RN; v)
if and only if r
sup [J (R)]I/P'
sup
xERN R>0
x
w(y) dy
J
[
1l1/q <
)
B(x,R)
where J (R) = ( t(1-Np)/(p-1) x
[Note that
238
r
vl-P (Y)
J
p E W(RN)
belongs to
A_(RN)
if
dt
.
v1-p/
E A_(RN
1
sup
[exp
J P(Y)
(Q)
dy]
In P 1(Y) dYJ < m
J
mN(Q)
Q
Q
Using the inequality (16.10) and Lemma 15.7, we can prove the following assertion in a similar way as we have derived Theorem 15.8.
Let
16.8 Theorem.
v1_p'E A-(RN)
p< N
< p < q <
1
w, v E W
Let
N
l/q
[JIU(X)I q w(x) dxJ
(16.12)
.
(RN)
,
Then the inequality
.
< C
1/p
p J
f
aX (x)
v (x) dx I
i-1 RN
RN
u E CO(RN)
holds for every
16.9. Remarks.
if the condition (16.11) is satisfied.
There are several other results in the direction
(i)
mentioned above. Let us present a result due to K. A. DZHALILOV [1] who investigated the inequality 1/q
((
[Ju(x)dx
(16.13)
[Jvu(x)2
C
l1/2 v(x) dx]
I
0
on the class
0
with
CO(U)
a bounded domain in
0
RN
,
i.e. the special
case of (16.1) with
p=2
q>
,
2
w-
,
1
,
v1 = v2 = ... vN = v
.
He has shown that the inequality (16.13) holds if (16.14)
sup
sup
R
J
0
v E AZ OR
N )
1/2
l+N/q l
<
B(x,R)
(2,0)-capacity of a single point is zero
and the
O(x,E) = vl/2(x)IEI,
v(y) dy]
x -_ 0,
E RN ). Compare the sufficient
condition (16.14) with the condition (16.8). (ii)
In the one-dimensional case we substantially exploited the
properties of the operators
HL
and
HR
from (1.6), having shown that
the Hardy inequality (1.12) [(1.13)] holds if and only if space
Lp(a,b;v)
continuously into
Lq(a,b;w)
.
HL [HR ] maps the
E. T. SAWYER [1] extended
this approach to the two-dimensional case: he has given necessary and 239
sufficient conditions on the weight functions operator
(Hf)(x,y) =
under which the
w , v
f(s,t) ds dt
f J
0
is bounded from 1
< p < q < m
0
Lp((O,-) X (O,m); v)
16.10. Weighted Sobolev spaces. Let
v0,v1,...,vN E W(O) (16.15)
into
Lq((0,-)
X (0,"); w)
.
1
<<
p <
,
let
0
be a domain in
RN,
Denote
.
S = {v0,v1,.... vN}
and define the weighted Sobolev space
(16.16)
W11p(S2;S)
as the set of all functions derivatives
au/axi
set we define the norm of
Lp(2;vi)
u E W1'p(SZ;S)
[uiv p'0'v
[for
,
vil p
(16.18)
l,p
i = 1,2,...,N
,
i=1
0
On this linear
by the formula p
au
+
.
ax
11/p
i lip, S2,vi
see (15.6)]. If, in addition, i
E Lloc(0) vil
[which means that W
such that their distributional
N``
p
(16.17)
space
u E Lp(O;v0)
belong to
(S2;S)
i = 0,1,...,N
,
r
p E L(K)
for every compact set
K C 0 I
is a Banach space.
The inclusion
CO(S2) C W1'p(O;S) holds if and only if (16.19)
vi E L1
loc (0)
,
I =
Consequently, under the assumptions
(16.20) 240
W0'p(S2;S)
(16.18), (16.19), the space
then the
defined as the closure of the set
with respect to the norm (16.17)
C0-(12)
is also a Banach space. Therefore, let us point out that in the sequel we vi
Will always assume that the weight functions
satisfy the conditions
(16.18), (16.19).
In Chapter 3 we will deal with weighted Sobolev spaces in detail. Mostly we will consider special collections S = {v0,v,v,...,v} i.e.
V1 = v2 = ... = vN = v
of weights, namely
S
,
.
In this case the corresponding spaces (16.16)
and (16.20) will be denoted by (16.21)
W1'P(2;v0,v)
W0I,P(Q;v0,v) ,
and the norm (16.17) will be denoted by (16.22)
.
1,p,Q,v0,v
Further, let us introduce the seminorm
(16.23)
N
IIIuII1 1,p,Q,S
which will be
-
[j1
(x)
111/p
vi(x) dx]
for the spaces from (16.21)
(16.24)
IIuIII1,P,S2,v
[in fact,
IIuWWI1,p',1'v
16.11. Remark.
p
au
-
denoted by
see (15.6) and (15.26)].
=
The Hardy inequality (16.1) can be rewritten in terms of
the seminorm (16.23) as
(16.25)
ullq,Q,w <_ CIIIuIII1,p,Sj,S
If (16.25) holds for every
W1'P(Q;S)]
Wp'P(Q;S)
[or
(16.26)
WI'P(Q;S)
y
u E CO(Q)
[or for every
u E W1'P(c;S) ]
is continuously imbedded in
LP(P;w)
,
then
i.e.
Lq(Q;w)
241
[or
(16.27)
W1'P(2;S) (j Lq(Q;w) ].
Moreover, the following assertion holds
16.12. Lemma.
Let the expressions
equivalent norms on
W1'P(Q;S)
[or on
and
111.1111
be
W1'P(c;S) ]. Then the imbedding
(16.26)
[or (16.27)] holds if and only if the Hardy inequality (16.25) is valid for every u E C'(Q) [or for every u E W1'P 0
242
(Q;S) ]
Chapter 3. Imbedding theorems for weighted Sobolev spaces 17. SOME GENERAL NECESSARY AND SUFFICIENT CONDITIONS
W1'p(Q;S)
Let
17.1. Introduction.
W0'p(D;S)
and
be the weighted Sobolev
spaces introduced in Subsection 16.10. Our aim is to establish conditions on
the collection
of weight functions
S
weight function
vi
i = 0,1,...,N
,
,
and on the
which guarantee the continuity or the compactness of
w
the imbedding of the weighted Sobolev space into the weighted Lebesgue Lp(S2;w)
space
.
More precisely, we will deal with the continuous imbeddings
(17.1)
W1'p(52;S) (. L4(Q;w)
(17.2)
W01'p(S2;S)
y
,
Lq(Q;w)
and with the compact imbeddings
(17.3)
W1'p(D;S) Ca ( L9(Q;w)
(17.4)
WD'p(D;S)
L4(D;w)
(for the notation, see Subsection 7.1). These problems have been investigated
by many authors. Here, we will mention only some of them, mainly by giving references to results contained in books and survey articles. One of the first authors who studied imbedding theorems for weighted Sobolev spaces was L. D. KUDRYAVTSEV [1]; he considered unbounded domains
1-1)a
a c R . J. NECAS [1]
and weight functions of the type
(1 +
investigated weights of the type
v(x) = [dist (x,aQ)]a
bounded domains
12
.
,
a E R ,
on
Spaces with these weights have been considered by
S. M. NIKOL'SKII [1] (who is mentioned here as a further representative of the numerous Soviet school) and by H. TRIEBEL [1] who, moreover, has investigated also weighted Besov spaces. The case of weights of the type
v(x) = v (dist(x,M)) with
v E W (0, °_)
and
M C a1
is analysed in detail 243
in A. KUFNER [2] . All these authors consider the case
p = q
;
a survey [1]
of results till 1977 can be found in A. KUFNER, 0. JOHN, S. (Chapter 8, Section 8.10).
A rather general approach for the case
1
s p
<_
q < - ,
0
a bounded
domain, can be found in the extensive papers by P. I. LIZORKIN, M. OTELBAEV [1].
The results established in the sequel are mainly due to B. OPIC and P. GURKA. The exact references will be given at the respective places.
17.2. The basic method.
Let
0
be a domain in RN
and let
{Qn}
denote
its special covering, i.e. suppose that (17.5)
Qn C Qn+1
CO
n E N,
,
U Qn = 52
.
n=1
Further, denote (17.6)
Qn
=
\ Qn
We will suppose that there are local imbeddings of the type (17.1) - (17.4) (i.e. on the subsets
instead of
Qn
0 ) and look for additional conditions
which guarantee the validity of the global imbeddings (i.e. on the whole
domain
0 ).
17.3. Lemma.
Let
and suppose that
p,q < -
1
W1,P(Qn;S) r
(17.7)
yEc Lq(Qn;w)
Assume that for every
> 0
ilulQQ,w
for every
u E W1'p(Q;S)
(17.9)
WI'p(O;S) C Gi Lq(52;w) Let
{un }
n E N
such that
EIIuIqP,2,s + MI,,Qn,w
.
Then
be a bounded sequence in
IIunII1,p,S2,S
244
n e N
there exists an
(17.8)
Proof.
for
<=
C
for every
n E IN
W1'p(52;S) .
,
i.e.
yor a given
e > 0
choose
(0, Eq/(2Cq + 1)]
E1
Then there exists a
.
n E N such that
number
(17.10)
Ilullq,Q,w
E W1'P(2;S)
.
funk} C fun}
gubsequence
The imbedding (17.7) implies the existence of a which is a Cauchy sequence in
q hunk - unz q,Q-,w
<
Lq(Qn;w)
k0 E N such that
consequently, there exists a number
for k, k ? k 0
E 1
phich together with (17.10) yields
Ilun
k
- un £Ilqq, s,w < Elllun k - un Ilq , k
+ Ilun
l:
- un R,Ilqq,Qn,w
<
< E12Cq + E1 < Eq {un
Thus
in a Cauchy sequence in
}
Lq(0;w)
11
.
k
Let
17.4. Lemma.
< p,q < m
1
` Lq(0;w)
W1'P(f,;S)
(17.9)
Then for every every
u E W1'P(S2;S)
.
there exists an
c > 0
c > 0
(17.11)
As
vnllqq
{v
n
, ,
p,
,
n
such that
} C W1'P(Q;S)
IunIIq,Qn,w
S x 0
;
taking
v
n
= u /c
n
n
we obtain
q
subsequence vIN } .
(u n lll
{u
a sequence
and
,2,w > E + IIv nq,Qn,w
is bounded in
n}
Lq(Q;w)
c
holds for
contrary, that the statement of our lemma is false.
Ilun q,Q,w > Ellun 1,p,0,S +
This implies that
such that (17.8)
n c N
.
Proof. Assume, on the Then there exist
and suppose that
W1'P(0;S)
and a function
and (17.9) holds, there exists a
v E Lq (S2;w)
such that
vnk
-' v
in
Now, (17.11) yields
245
1v11q,Q,w - E + li"11,,12,w
which is a contradiction since
E > 0
.
11
17.5. Remark.
The condition (17.8) is equivalent to the condition sup
lim
(17.12)
ull 1
Qn
where
= 0
n
q,Q ,w
is given by (17.6).
Indeed, according to (17.6), the inequality (17.8) can be rewritten in the form jjujj
(17.13)
q Qn,w
<
and since IluIq n q,Q ,w
for
'lulq
n >_ n
, we have (17.12) from (17.13).
q,Qn,w
Conversely, suppose that (17.12) holds. Let Then there exists a number
n E N
q,Qn,w
11U1113P,S22 51
and denote
E1 = El/q
such that < E
sup
c > 0
n > n
for every 1
and, consequently, N11
for every
n
<=
q,Q ,w
E111u111,p,1,S
u E W1'P(Q;S)
1IUjjq,Q'w
and
n > n
.
In view of (17.6) this implies that
` E q)IUJI1,P,S2,S + uIJq,Qn,w
i.e. the inequality (17.8) holds
(
E1 = E
0
Summarizing Lemmas 17.3 and 17.4 and using Remark 17.5 we have
17.6. Theorem. (17.14)
and
246
Let
W1'P(Qn;S)
1
< p,q < m . If
y y
Lq(Qn;w)
for every
n E N
lim n--
(17.15)
sup
' 0
11u11
q,Qn,w
11u111,p,0,s 51
then
Lq(Q;w)
W1,p(c1;S),-
(17.16)
Conversely, if (17.16) holds, then the condition (17.15) is satisfied.
X7,7. Notation.
Let
be a closed subspace of
X
W1'P(R;S)
and for
n E N
denote
Xn = {u; u = V I Q
v E XI
,
n on
Xn
,
l
we consider the norm
Further, we denote
I
.
1,P,Qn,S
1,p,1,S
X
The next theorem can be proved analogously as Theorem 17.6.
17.8. Theorem. Let
1
< p,q < m . if for every
Xn ; c Lq(Qn;w)
lim
sup uEX,lluIX<1
n C IN
=0
ull
,
q,Qn,w
then (17.19)
X c (; Lq(p;w)
Conversely, if (17.19) holds, then the condition (17.18) is satisfied.
17.9. Remark.
Let us again point out that Theorem 17.6 implies that under
the assumption (17.14) the condition (17.15) is necessary and sufficient for the compactness of the global imbedding. An analogous result holds also for continuous imbeddings:
17.10. Theorem. (17.20)
Let
1:5 p,q
Xng Lq(Qn;w)
<
for every
If n L N
247
and
(17.21)
lim
sup
(lull
<
n
q,Q ,w
then (17.22)
X ; Lq(Q;w)
.
Conversely, if (17.22) holds, then the condition (17.21) is satisfied. Proof. As lu
q,S2,w
`
flu
n
I
q,Q ,w
+ 11u11q,Q ,w n
we have
llullS2,w < q,
sup uEX, ullx<<<1
sup <
+
Ilu ucX,11OX<1
q,Qn,w
sup ue-Xn,lIu
1Iullq,Qn,w X<1
n
This inequality implies (17.22) since the expressions on its right-hand side are bounded due to (17.21) and (17.20).
The converse assertion follows by a contradiction argument from the inequality S
ll u ll
q,Qn,w 17.11. Remarks.
(i)
ll u
q ,Sl,w 11
If we take
X = W1'p(Q;S)
in Theorems 17.8 and 17.10,
we obtain assertions about the imbeddings (17.3) and (17.1), while for X = Wl'p(c;S) we obtain assertions about the imbeddings (17.4) and (17.2). (ii)
The foregoing results can be found in B. OPIC C2] together with
further necessary and sufficient conditions for the compactness of
the
imbeddings mentioned. (iii)
The limits appearing in the conditions (17.21) and (17.18)
always exist since they concern monotone sequences in
R+. A comparison
with the one-dimensional case (cf. Theorem 7.13) indicates a certain relationship. For instance, for the compactness of the imbedding, we
248
require in both cases ( N =
N >
as well as
1
1
)
that a certain limit
should be zero.
17.12. An application. Q
and subsets constants
n
cn c
C 1
such that for a.e.
Cn
,
< w(x) < C
n
Let us consider weight functions
w,v0,v1,...IVN
with the following property: there exist positive
n
,
n
cn f vi(x) < Cn Then the weighted spaces
x E Q
i = 0,1,...,N
,
Lq(Qn;w)
and
.
W1'p(Qn;S)
isomorphic to the classical (non-weighted) spaces
are isometrically Lq(Qn)
and
W1'p(Qn),
and the local imbeddings (17.20) and (17.17) can be derived using the classical Sobolev imbedding theorems provided some additional conditions
are fulfilled. For example, if
Qn C C
0'1
then these conditions read as
follows:
q
-
+ 1 = 0
for continuity,
+ 1 > 0
for compactness
p
(17.23)
q
-
p (for references, see the end of Subsection 12.13). We will indeed meet these conditions later in some particular cases.
18. IMBEDDINGS FOR THE CASE
1
18.1. Introduction. The domain conditions (17.21) and (17.18)
c p c q<
St
.
Our aim now is to reformulate the
which are expressed in terms of norms of the
spaces considered and which together with the local imbeddings guarantee the global imbedding] in terms of the weight functions
w,v0,v1,...,vN
.
Here
we will suppose that
vl = v2 = ... = VN and
consequently
(18.1)
we will deal with the spaces
W1'p(0;v0,v1)
and
Wl'p(S2;v0,v1)
(cf. Subsection 16.10).
249
We will consider domains 22
(18.2)
0
C RN
with the only restriction
0
Let us denote
d(x) = dist (x,252)
(18.3)
and let
n E N , be such that
52n C SZ ,
{x E 52; n1 < d(x) < n} C S2 nC {x C-:
(18.4)
n +1 1 < d(x) < n + 1}
52;
(draw a picture!), put
Stn = int (0 \ 0n
(18.5)
and suppose that CO'1
0n E
(18.6)
Obviously,
C S2
0 C 0
of the subsets
U 0n = Q
'
The sets
.
0n
will play the role
n=1
from Section 17.
Qn
18.2. An important auxiliary function. The weight functions.
(i)
In view
of the conditions (17.21) and (17.18) we need some estimates of the norm
in
n
Lq (Q;w)
i.e. of
hull
q,S2
there exist a number
n E N
r = r(x)
Stn
defined on 43x)
r(x)
(18.8)
cr1 < r(y) < cr
(ii)
n a 2
,
for
a.e.
0n
for
,
a positive measurable function
B(x,r(x)) x E S2 3n
c
>
1
r
such that
x E Stn
for a.e.
B(x,r) = {y E RN ;
ensures that the ball belongs to
w
and a constant
(18.7)
[recall that
For this purpose, we suppose that
n
x e 0n
ly - xl
y t B(x,r(x))
,
< r} ].
belongs to
Q
The condition (18.7) for
x E Q
and, moreover,
(see Lemma 18.5 below).
The weight functions v0 , v1 appearing in the weighted Sobolev
spaces (18.1) are supposed to be connected by the following condition: there exists a constant (18.9)
250
K0
such that
vI(x)r-p(x) < K0v0(x)
for
a.e.
x6; 52n
(iii)
Further, we suppose that there exist positive measurable
functions
b0
(18.10)
w(y) < b0(x)
(18.11)
b1(x) < v1(Y)
b1
,
defined on
and such that
0a
and
x E SZn
for a.e.
y E B(x,r(x))
w
These last conditions connect the weight functions
functions
w
x
a constant depending on
Let
18.3. Lemma.
with the B(x,r(x))
.
be a bounded domain in RN
A
and
a positive
p
A . Then there exists a sequence of points
function defined on
the
,
are bounded from above or below, respectively, by
v1
or
v1
and ensure that on the ball
r = r(x)
auxiliary function
,
xk f- A
such that (i)
'J Bk with Bk = B(xk,p(xk))
A =
k (ii)
there exists a number
depending only on the dimension
0
N
and such that
)(B (z) < 0 for every z E k
RN
k
Let
18.4. Lemma.
1
N- N+
< p,q <
1
p
q
>_ 0, r> 0,
xE
RN .
Then
the inequality 11/q < u(Y)lq dy]
(18.12) [
J
B(x,r)
K r
1/p
N/q-N/p+1(r-p
dy + J
J
B(x,r)
B(x,r)
holds for every
x
,
r and
u
Vu(Y)IdY
lu(Y)Ip
I
with the constant
u E wl'p(B(x,r))
K
independent of
.
[Note that here and in the sequel N
(18.13)
p
IaX (x)
Iou(x)Ip = i=1
.
i
251
Lemma 18.3 is part of the famous Besicovitch covering Zemma and its proof can be found, e.g., in M. DE GUZMAN [1]. Lemma 18.4 is in fact the Sobolev imbedding theorem for the ball the unit ball
B(x,r)
.
and then dilating it to
B(0,1)
Applying this theorem to B(r.,r)
the dependence of the imbedding constant on the radius
Let
18.5. Lemma.
n ? 3m
,
r = r(x)
r
.
be the function from Subsection 18.2 (1). Let
m > max (2, n)
mEN,
we express exactly
,
.
If B(x,r(x))n On x 0
,
then
2m
B(x,r(x)) C
Proof. Let us write*
n = S2 0 U Q
St
where
S20 _ {x e On; d(x) < n}
,
On =
f x E Stn; d(x) > n}
Obviously, it suffices to prove the following two implications:
(18.14)
B(x,r(x))n Stn z 0 > B(x,r(x)) C S2m
(18.15)
B(x,r(x)) n
(i)
Let
On
z E B(x,r(x)]
Iz - El
B(x,r(x)) C O
x 0
y C B(x,r(x)) n On
.
.
Then
Iz - yl + Iy - EI < 2r(x) + ly - El for
<
E E DS2
and from the definition of d(z) we have d(z) =
inf
Iz - EI
<_ 2r(x) + d(y)
EE31 Now,
d(y) < n
(18.16)
since
y E SZ
,
and by virtue of (18.7) we obtain
d(x) + n
d(z) < 3
Further,
Ix - Ej and thus
252
<
Ix - yl + ly - El < r(x) + ly - EI for EE DO
d(x) < r(x) + d(y) < 13 d(x) + which together with (18.16) yields d(z) <
_
2
2
n
3m 5 M + 1
,
(ii)
1 n
m ,
n
The last inequality implies that
.
and (18.14) is proved.
z E B(x,r(x))
Let
Iz - &I and consequently (18.17)
<
1
with regard to the choice of z E S20 m
23 d(x)
n
->
v e B(x,r(x)) () S2n
,
Ix - I -
Iz - xj
Then
.
Ix - El - r(x) for
>
E aS2
d(z) = d(x) - r(x) ? d(x) - 3 d(x) = 3 d(x)
Further,
Ix - EI ? IY-EI - Ix - yl and thus, since
implies
y E S2-
> Iy-&I - r(x) for EEasz
d(y) > n
d(x) = d(y) - r(x) > n -
3
d(x)
,
i.e.
d(x) > 2
3
which together with (18.17) yields
d(z) > 2 ? 2m > m + 1 m z E SZ
Consequently,
and (18.15) is proved.
The following theorem gives sufficient conditions for the continuity of the imbedding of weighted Sobolev spaces.
Let
18.6. Theorem.
1
p < q
<_
<
N ,
q
(18.18) Let
W1'p(SZn;v0,v1)
Lq(S2n;w)
-
N + 1 > 0
.
Let
p
for n E N
be the function from Subsection 18.2 (i) and suppose that
r = r(x)
the weight functions
v0
,
v1
, w
satisfy the conditions (18.9), (18.10),
(18.11). Denote (18.19)
sup
=
n
x C SZn
1/q (x) rN/q_N/p +1(x) b0
bi/ P(x)
If (18.20)
lim C
n-
n
= G <
253
then W1'p(1;v0,v1) Cj Lq(S2;w)
(18.21) Proof.
We will use Theorem 17.10 with
X = W1'p(S2;v0,v1)
and
Q. = 03n
According to this theorem, it suffices to verify that the condition (17.21) is satisfied.
R > 0 , we denote
Taking
QR = {x E Qn;
Lemma 18.3, used for sequence
(18.22)
{xk} C S;R
For
n ' n
,
lxl
< R}
ensures the existence of a
such that
0
Bk = B(xk,r(xk))
XB (z) < 0 k=1
0(x) = r(x)
and
A = S2R
On = {x E 52n;
,
and of a number
12R C U Bk , k
(18.23)
1x1 < R}
z E RN
,
k
denote
Kn,R = {k C N ; Bk n S23n x 0 }
U
According to Lemma 18.5,
.
Bk C Stn C Stn
,
and therefore
we can use
kEKn,R all estimates from Subsection 18.2. Further, by virtue of (18.22),
(18.24)
hull'
n
= Ilujlq
q,QR,w
3n
q'QR 'w
=
lu(y) lq w(y) dy <
1
3n QR
<
k E Kn, R J The inequality (18.10) and Lemma 18.4 imply (18.24*)
lu(Y)Iq w(y) dy
--< bo (xk)
I
J
Bk
254
Bk
lu(y)
q dy 5
[Kbl/q
rN/q-N/p +1 (xk)3 q [rP(xkJ
(xk)
u (Y) p dy +
Bk dY]q/p
J IVu(Y)Ip
+
k E K n,R
,
Bk
Using here the inequality (18.11). the condition (18.8), the estimate (18.9) gnd the definition of
13
(18.19)) we obtain
(cf.
n
1/q
x
b0/p( k) rN/q-N/p
rr I
J Iu(Y)Iq w(y) dy LK
Bk
bl
(xk)
v (Y)
r Iu(Y)Ip
J k
tL
1q/p dy + j IVu(Y)lp v1(Y) dy]
1
rp(Y)
B
B
k l1
rr
Kq G3 n[AAKU J Ju(y)Ip vO(y) dy + J JVu(y)Ip v1(y) dy r
q/p
l
LL
Bk
Bk
5 K1
q/p
q n
v0 (Y) dy + J
J Bk
where
Bk
K1 = Kq max
u
IVu(Y)Ip v1(Y) dY]
This inequality together with (18.24) yields
11q
q,Q
w
Iq p
K
J I``u(Y)Ip "0(Y) + IVu(Y)lp v1(Y)J dy
n
c
n,R
Bk dy]q/p
K1
n [
` pq/p K 1
kE
L KJ n,R
Vu(Y)I p v1(Y)J I
Bk
iq n1IuIIq1.p,S2,vo,v1
where we have used the fact that M
inclusion
[iuy)Ip v0(Y) +
q/p >_
1
,
the estimate (18.23) and the
i J B.C. St K
k=1 For
R -+ -
we immediately obtain the estimate 25
(18.25)
11u11 q,
Oq/P R1 , n Ilu11q,
1p'
Qn ,w
v0,v 1
Finally, (18.25) and (18.20) imply that the condition (17.21) is satisfied. 11
Analogously, we can formulate a sufficient condition for the compactness of the imbedding in question.
W1'p(Q;v0 v1)
(18.26)
Let
r
v0
,
,
Vi
for
Lq(S2n;w)
(
n E N
satisfy the assumptions of Theorem 18.6 and Zet
, w
63 n
be defined by (18.19).
If (18.27)
lim
',Qn = 0
nip then
W1'P(U;v0,v1)
(18.28)
Proof.
yY
Lq(S2;w)
.
Using the estimate (18.25) and the condition (18.27) we immediately
obtain that the condition (17.15) is satisfied, and (18.28) follows from Theorem 17.6. 11
18.8. Necessary conditions.
In order to derive necessary conditions for
the continuity or compactness of the above-mentioned imbeddings again in terms of the weight functions
v0
,
Vi
, w ,
let us change the assumptions
from Subsection 18.2 (ii), (iii). More precisely, let
r = r(x)
be the
function from Subsection 18.2 (i) and suppose that (i)
(18.29)
(ii) Stn
256
there exists a constant k0v0(x)
v1(x) r-p(x)
k0 > 0 for a.e.
such that x E Stn
there exist positive measurable functions
such that
;
b0
,
bI
defined on
(18.30)
w(Y) = b0(x)
for a.e.
(18.31)
x E Stn
and
y E B(x,r(x))
.
I(x) ' vi(Y)
[Compare with (18.9), (18.10) and (18.11)!]
Further, introduce the numbers
(18.32)
sup
b0/q(x)
xC- Qn
bl/p(x)
On =
18.9. Theorem.
Let
r
<- p,q < w
N/q-N/p +1
(x)
Let r = r(x) be the function from Subsection 18.2 (i) and suppose that the weight functions v0 v w 1 Satisfy the conditions (18.29), (18.30), (18.31). Let On be given by 1
.
,
,
(18.32).
If
(i)
(18.33)
W
1'p(Q;v0)
r
v1Lq(2;w)
then
(18.34)
1im
13
n
n+-
= I3
<
If
(ii)
W1.P(0;v0'v1)
(18.35)
Lq(Q;w)
then n
(18.36)
Proof.
lim na.
63
n
= 0
(i)
Suppose that (18.33) holds and that the condition (18.34) not fulfilled. Then there exist a sequence of natural numbers
k E IN
,
and a sequence
{xk}
nk > n
n
xk e S2 k
,
,
is f
such -that
1/q
(18.37)
b0(xk b
/p l
rN/q-N/p +1(xk) > k
k c- N
(xk
Put (18.38)
uk - Rr(xk)/8 X3Bk/4
'
k E N
257
where
is the mollifier with the radius
RE
ly - xkI
< ar(xk)}
uk e C0(Bk)
(18.40)
U
au (18.41)
on
1
0 < uk <
,
c 15
1
,
xE 1
r (xk)
with a suitable constant
(18.42)
;
2 Bk
k(x)
ax
aBk = iY E R
Then we have
.
(18.39)
k =
N
ff
and
a
i = 1,2,...,N
,
independent of
c
k
,
,
uk E WD'p(Sl;v0,v
[For details concerning mollifiers and their properties, see, e.g.,
R. A. ADAMS [1], Section 2.17, or A. KUFNER, 0. JOHN, S. FUNK [1], Sections 2.5 and 5.3; the property (18.42) is a consequence of our assumption
vol vl E Lloc(0)
,
cf. (16.19).]
Using (18.40) and (18.30) we obtain
qw(y)
If
'
ll1/q dYJ
2-N/q
JI
w(Y) dyJ
Bk/2
0
(18.43)
1/q (
[N(B(0,1))]l/q b01/q(xk) rN/q(xk)
while (18.39), (18.41), (18.29), (18.8) and (18.31) imply 1/p
{JIuk(y) Ip v0(Y) dy + J
0
<
0 r
(18.44)
Ip v1(Y) dyj
r
<
dy + Ncp If
l1/p r-p(xk) v1(y) dyJ <
J
Bk
Bk 1/p
(
0 Ik
v1(Y) r-p(Y) dy + Ncpr-p(xk
1 J
J
v1(Y) dyj
Bk
258
`B
1/p
r
rr
L(k-ICP + Nc) r-P(xk)
rN/p-1(xk) bi/P(xk)
= L
b (xk) dy] J
Bk
L =
where
L(k-0
1
1/P
cp + Nc) mN(B(0,1))]
r
Using the assumption (18.33), i.e. for
IuIIgVE2'w `- C IIuII1,p,SZ,v 0,v
u C W1,P(c;v0,v1)
1
we obtain from (18.43) and (18.44) that bl/q 0
1/
(18.45)
(xk) rN/q-N/p +1
(xk) < C
for every
k
b1 P(xk)
with a suitable constant
C
independent of
k
However, (18.45)
.
contradicts (18.37). (ii)
Suppose that (18.35) holds and that the condition (18.36) is not
fulfilled. Then there exist a positive number numbers
nk , nk ? n , k E N , and a sequence
e
,
a sequence of natural
3nk
f1
ixk}
xk C-
,
S2
such that
,
^1/q
b0/P(Xk) rN/q-N/p +1(xk)
(18.46)
>_
c
for
k E N
k
1
According to Lemma 18.5,
3nk
implies
xk E 12n
(18.47)
Bk = B(xk,r(xk)) C
uk
from (18.38), denote
For
U
k
k
= uk/IIuk1I1,p,O,v0,v1
Using (18.39), (18.43) and (18.44), in view of (18.47) we obtain that L PO 11
q,SZ
for
k E N
with
= Il uk q,R,w
nk' w
b1/q(x 0
k)
rN/q-N/p +1(x
1 bl/P(x 1
)
k k
L1 = 2-N/q L-1 [mN(B(0,1))]1/q
,
and consequently, due to
(18.46), we have
259
sup
fI
for every
u II1,p,0,v0,v1<<1
nk
hull
q'R
nk
AukII ,w
>_ L1 E ,w
q,Sl
k E N . However, then the condition (17.15) from Theorem 17.6
is not satisfied, which leads to a contradiction with (18.35). El
The sufficient as well as the necessary conditions derived
18.10. Remark.
in the foregoing theorems have concerned the imbedding of the space W1'P(St;v0,v1)
Obviously, the assumptions of Theorems 18.6 and 18.7
.
guarantee also the validity of the imbeddings W0'P(c;v0,v1)
y
WI'P(1;v0,v1)
c v Lq(S1;w)
Lq(Q;w)
and
is a closed subspace of
respectively, since
W1 'P(St;vO,vl
On the other hand, in the assumptions (18.33) and (18.35) of Theorem can be replaced by
Wl'P(S1;v0,v1)
18.9 the space
Wl'P(1;v0,v1)
This
.
follows from the fact that in the proof we have used only the properties uk
of the functions
by
from (18.38), which belong to
virtue of (18.42).
Combining Theorem 18.6 and part (i) of Theorem 18.9 or Theorem 18.7 and part (ii) of Theorem 18.9 we immediately obtain
18.11. Theorem (the continuous imbedding). Let 1 = p < q <
N
-
q
N+1 p
Let (18.48)
Let
W1'p(
r = r(x)
n;v0,v1) - Lq(1n;w)
n E N
be the function from Subsection 18.2 (i) and suppose that
the weight functions
v0
,
v1
,
satisfy the following conditions:
w
There exist positive constants positive measurable functions
b0
k0v0(x) --< v1(x) r-p(x)
260
for
,
k0 < KO b1
,
c0 < CO
defined on
K0v0(x)
,
Stn
,
c1 s C1
such that
and
-
O
c0b0(x) ` w(y) (18.49)
c1b1(x) `= v1(y)
for a.e.
x E 0n
and
C0b0(x) C1b1(x)
y E B(x,r(x))
.
Then
W1,p(S1;v0,v1) r Lq(0;w) [and
WO,P(o;v0,v1)c
Lq(S2;w)
4f and only if lim
(18.50)
n
n+m pahere
fjn
is defined by (18.19).
18.12. Theorem (the compact imbedding). Replace in Theorem 18.11 the assumption (18.48) by
(18.51)
Wl,p(S2n;v0,v1) Y (i Lq(0n;w)
for
n
.
Then
WI,P(Sl;v0)v1)
y
c Lq(S2;w)
[and W0P(1;v0,v1)
c c Lq(Q;w)
if and only if (18.52)
lim
f'jn = 0
n-18.13. Remarks.
(i)
For bounded domains the foregoing results have been
derived by P. GURKA, B. OPIC [2]
.
For certain special unbounded domains,
the problem was solved (by another method) in P. GURKA, B. OPIC B. OPIC, P. GURKA
[2],
[1]. The approach used here, i.e. the application of
the Besicovitch covering lemma to the bounded domain of Theorem 18.6) and then the limiting process
R --
S2nR
,
(see the proof
exploits some
ideas of R. C. BROWN, D. B. HINTON [1], [2]. Another possibility is the
application of an extension of the covering lemma to unbounded domains (see W. D. EVANS, J. RAKOSNIK [1]).
261
We have dealt here with the spaces
(ii)
the spaces
with
W1'P(S2;S)
W1'P(S2;v0,v1)
S = {v0,v1,...,v1}
.
Results for
can be easily derived if,
for instance, there exist two weight functions
v0
and a constant
vl
,
such that
c > 0
0(x)
<- cv0(x)
,
l(x) < cvi(x)
,
(18.53)
for a.e.
x E S2 , where
i = 1,2,...,N
v0
satisfy the conditions of Theorems 18.6,
vl
,
18.7. Indeed, from (18.53) we obviously have
W1'P(S2;S) C W1,P(Q;v0,v1)
18.14. The functions
r
b1_.
b
,
0--i-
.
The numbers
43 n
appearing in the
criteria of continuity and compactness of the imbeddings mentioned above are expressed in terms of the auxiliary functions
r
,
b0
,
b1
.
Since we
have supposed that such functions exist, it would be useful to know how to choose them. Thus, let us give some hints (i)
in this direction.
A trivial choice is provided by the formulas b0(x) = ess sup w(y) b1(x) = ess inf v1(y)
b0(x) = ess inf w(y)
,
,
,
b1(x) = ess sup v1(y)
,
the suprema and infima (here and in the following point) are taken over
y E B(x,r(x)) (ii)
.
Suppose that
there exist constants
w , v1 c
,
C
,
are defined for all 0 < c <
1
5 C < m
,
x E Stn
and that
such that
cw(x) S ess inf w(y) s_ ess sup w(y) = Cw(x)
,
cv1(x) < ess inf v1(y) < ess sup v1(y) < Cv1(x) for every
x E Stn
.
Then the inequalities (18.10), (18.11) and (18.30),
(18.31) are satisfied with b0(x) = Cw(x)
,
b1(x) = cv1(x)
,
x E S2 b0(x) = cw(x) If, in addition,
262
v1
w
,
b1(x) = Cv1(x)
,
have the special form
n
v1(x) = v1(d(x)) [recall that
w(x) - v(d(x))
,
d(x) = dist (x,20) I and if also the function
expressed in the form
r(x) = i(d(x))
can be
r
then the conditions (18.49) can be
,
-replaced by simpler but slightly more restrictive conditions cw(t) 5 w(T) < Cw(t) or all
--1
t E (O,n
cv1(t) < vl(T) < Cv1(t)
,
and for a.e.
)
T E (t - i(t), t +
In this case,
.Vhe conditions (18.50) and (18.52) can be expressed as follows: w1/p(t)
lim sup
(18.54)
t+0+
vl
For functions
(iii)
TN/q-N/p +1(t)
<
(or
= 0 ).
(t)
r(x) = r(d(x))
the assumptions (18.7), (18.8)
,
Can be replaced by r(t)
s_
for
3 t
t E (0,n
1)
,
(18.55) c rl
< r(t) < c r
18.15. Example.
Let
1ere we can take for
(i)
,
v0(x) = ds-p(x)
,
r(x) = d(x)/3
and
T e (t
E R , For
a,
,
and the number
x E Stn
(18.56)
n = 2
n t E (O,-1)
p < q <
1
w(x) = da(x)
for
,
and
x C 0
put
v1(x) = ds(x) b0(x) = w(x)
.
,
b1(x) = v1(x)
has the form Ld(x)]u/q-S/p+N/q-N/p +1
!;n = c sup n x E S2
The continuous imbedding W1,p(o;d8-p,ds)(
y
The condition N
Lq(0;da)
-
q
N + 1 > 0 p
guarantees the continuity of the local imbeddings (18.48). (i-1)
(18.57)
If
sup
0
is such that
d(x) < W
,
xC-0 263
then the condition (18.50) will be fulfilled if and only if a
a+
q
p
(i-2)
0
If
0
is such that
d(x) = m
sup
(18.58)
N+ p
q
,
xE 12
then the condition (18.50) will be fulfilled if and only if
(ii)
a
8+ N
q
p
N+ p
q
0.
The compact imbedding W1,P(c1;ds-P,dR) r
Ci
Lq(P;da)
y
The condition N - N
q
+ 1 > 0
p
guarantees the compactness of the local imbeddings (18.51). Suppose that lim
(18.59)
S2
is bounded or quasibounded (the latter term means that
d(x) = 0 ).
Ixl-xe2
Then the condition (18.52) will be fulfilled if and only if
a-
N+ 1> 0.
+ N q
p
q
p
The same conditions concern also the imbedding of
(iii)
W0'p(S2;ds-pd
into
Lq(SZ;da)
(see Remark 18.10 and Theorems 18.11,
18.12).
18.16. Example.
Let
w(x)
1
< p < q < m
,
in d(x)IY
,
= da(x)
v0(x) = ds-P(x) vl(x) = ds(x) for
x
such that
elsewhere in
264
12
.
a,
S, Y,
6 E R
.
For
x C 12
put
in d(x)I6
In d(x)16
d(x) < 2
or
d(x) > 2
Again we can take
,
w(x) = v0(x) = v1(x)
r(x) = d(x)/3
b0(x) = w(x)
1
b1(x) = v1(x)
r
(18.60)
(i)
and have = c
[d)]a/q-S/p+N/q-N/p+1
sup x E SZn
Iln
d(x)`Y/q-6/p
The continuous imbedding. The condition N
N + 1
q
p
0
>_
uarantees the continuity of the local imbeddings (18.48). If
d(x) < -
sup
then the condition (18.50) will be fulfilled if
,
xE0 *ild only if either
a- 8+ N- N+ 1> 0 q
.pr
(18.61) If
q
p
aq
p
p
+ NN+ 1= 0 and !-a <_ q q p
d(x) = m
sup
0
p
then the condition (18.50) will be fulfilled if and
,
xEO only if (18.61) holds. (ii)
The compact imbedding. The condition N - N + 1 > 0 q
p
"guarantees the compactness of the local imbeddings (18.51). If
0
is
bounded or quasibounded, then the condition (18.52) will be fulfilled if and only if either
aq
p
q
p
N
N
q
p
1> 0
or
18.17. Example.
+ N _ N+ 1 = 0 and Y- 5 < 0 q
Let
p
I
5 p 5 q < -
that (18.57) holds. For w(x) = ea/d (x)
We can take
p
q
r(x) = d2(x)
x E 0
,
,
,
a,
S C R . Suppose that
0
is such
put
v0(x) = d- 2p(x) e8/d(x)
b0(x) = w(x)
,
,
vI(x) = es/d(x)
b1(x) = v1(:;)
,
and have
265
(i)
e(a/q-s/P)/d(x) [d(x)]2(N/q-N/P +1)
sup
=
n
xE0n
The continuous imbedding. If
N/q - N/p + 1 ? 0
,
then the
condition (18.50) will be fulfilled if and only if < 0
(18.62)
q - P
(ii)
N - P + 1 > 0
If
The compact imbedding.
,
and
0
is bounded
or quasibounded, then the condition (18.52) will again be fulfilled if and only if (18.62) holds.
In the foregoing examples, we have apriori supposed that
18.18. Remark.
(k)
N q
+1 ?0 - N p
(
>0
)
when deriving conditions for the corresponding continuous (compact)
imbeddings. As will be shown later (cf. Lemma 19.14) the continuity (compactness) of the imbeddings mentioned in Examples 18.15, 18.16, 18.17 implies the condition (*) and, consequently, it is a necessary condition.
Similarly it can be shown that the condition of the quasiboundedness of
0
(cf. (18.59)) is necessary for the compactness of the imbeddings
appearing in these examples. This follows from B. OPIC, J. RAKOSNIK [1].
18.19. Weakening the conditions on auxiliary function
r
The condition (18.8) on the
.
is restrictive, but it was used substantially in
r
the proofs of the foregoing theorems. If we suppose that 0
is bounded and
v0 = vl
(
= v )
then (18.8) can be omitted. More precisely, the following analogues of Theorems 18.6, 18.7 and 18.9 hold.
18.20. Theorem.
Let
p < q <
1
9 - P + 1 ? 0
,
let
0 C RN
be a
bounded domain. Let
W1,P(0n;v,v)
j Lq(S2n;w)
[W1'P(0n;v,v)
Let there exist a number b1
266
defined on
0n
ry (r
for
nE N
Lq(Qn;w)
n E N and positive measurable functions
such that
r
,
b0
r(x) < d(x)/3
,
w(y) 5 b0(x)
y E B(x,r(x))
and
x E Stn
b1(x) ` v(Y)
,
.
Denote
1/q
(x) rN/q-N/p (x) bi/P(x) b0
su xE Stn
W1'P(St;v,v) (
Lq(Q;w)
W1'P(o;v,v) Y (,, Lq(St;w)] if
lim 9 n
=
<
n+
lim j1n = 01. nom
The proof is a slight modification of the proofs of Theorems 18.6 and 18.7.
Instead of the inequality (18.24*) we derive the estimate rN/q-N/p(xk)]q
Iu(Y)Iq w(y) dy <_ [KbI/q(xk) B
Iu(Y)Ip dy + LI
J
k
Bk
+ r'(xk)
I
IDu(Y)
Bk
and since the boundedness of r(x) < d(x)/3
together with the inequality
St
implies
r'(xk) 5 (diam S2/6)P
,
we finally obtain the following analogue of (18.25): 11
ullq
q,Qn,w 18.21. Theorem.
Let
there exist a number defined on
Stn
` Oq/P K 1 Qq n Iu,jq1,p,St,v,v
1 < p,q <
,
let
St
be a bounded domain in RN . Let
n E N and positive measurable functions
r
,
b0
b1
such that
r(x) < d(x)/3
,
267
w(y) = b0(x) for a.e.
b1(x) _> v(Y)
,
x E 0n and y E B (x, r (x)) W1'P(S2;v,v)
be defined by (18.32). If
43
y
(-,, Lq(S2;w)
n
lim rn _ [lim n--
Let
Lq(S2;w)
[W1'P(S2;v,v) then
.
<
=0
n
The proof is again a modification of that of Theorem 18.9. Using the fact
we again derive the formula (18.45) for the function uk
that
v0 = v1 = v
from
(18.38), but now in a little different way. We have
,
,IukII1,P.Q'V,v
t/
v(Y) dy +
r
Ncp
(( I
1/p
Bk
Bk <
r-p(xk) v(y) dyj
I
11/p 11/p ( I1 + Ncpr-p(xk)I v(y) dyl
<
Bk (1
11/p bl(xk) dyl
111/p
((
1 + Ncpr-p(xk)I
<
lJ
Bk < L rN/p -1(xk) bi/P(xk)
for
k E N such that
since for these
k
,
nk 3 1/(3cN1/p) Ncpr-p(xk)
1
.
and
S2nk
= {x E
,nk; d(x)
> nk} = 0
Then we complete the proof as in the
case of Theorem 18.9.
18.22. Remarks.
(i)
Obviously, the space
18.21 can be replaced by the space (ii)
W1'p(St;v,v)
W0'p(Si;v,v)
in Theorems 18.20,
(cf. Remark 18.10).
Note that in the case of the sufficient conditions we have now
replaced the numbers
0n
from Theorems 18.6, 18.7 by
n
from (18.63),
while in the case of the necessary conditions we have used the same number as in Theorem 18.9. Consequently, we cannot combine Theorems 18.20 and 18.21 and have no analogue of Theorem 18.11 and Theorem 18.12. 268
19. POWER TYPE WEIGHTS
In this section we will deal with imbeddings of special
19.1. Introduction.
weighted Sobolev spaces
W"p(Q;ds,ds) into weighted Lebesgue spaces Lq(Q;da)
.
ecall that d(x) = dist (x,aQ)
;
mere we will suppose that the domain For the case
< p < q < -
1
is bounded.
0
we will use the results from Section 18;
moreover, we will also consider the case
= q < p <
1
The results of
this section are due to P. GURKA, B. OPIC [2]. Imbedding theorems of the type mentioned above have been investigated for the case p = q
by A. KUFNER [2] under certain additional assumptions about the domain
f0
Let us start with the definition of a special class of domains.
A bounded domain
CO,K
19.2. Domains of the class
S2 L RN
is said to
belong to the class (19.1)
CO'K
0 < K <
,
1
if the following conditions are satisfied: (i)
(19.2)
There exist a finite number (yi,yiN)
,
yi = (y
and the same number of functions
of Cartesian coordinate systems
m
i1,yi2,...,yiN-1)
a. = a.(y!) 1
1
1
:_
defined on the closure of
the (N-1)-dimensional cubes (19.3)
A. = {yi;
lyijl <
d
for
j = 1,2,...,N-1}
= 1,2,...,m ) such that for each point 1 E {1,2,...,m}
x E a0
there is at least one
such that
269
(19.4)
x = (Y1,yiN)
The functions
(ii)
exponent
yiN
'
K
,
ai
ai(yi)
satisfy on
i.e. there exists a constant
(19.5)
lai(yi) - ai(zi)I < AJyi - z'IK
for every
yi, zi E
(iii)
the Holder condition with the
Ai
There exists a positive number
A > 0
such that
A <
such that the sets
1
Q1
defined by (19.6)
Qi = {(Yi,YiN); Yi E L'i
ai(Yi) - A < YiN < ai(Yi) + Al
,
satisfy (19.7)
Ui = Qi n 52 = {(Y1,YiN
(19.8)
Ti = Qi 0 30 = {(Y"
(i = 1,2,...,m)
n
yiN
Yi E Li
'
Suppose
0 E CO'K
,
be a domain from CO'1
yiN = ai(Y,)}
0 < K < 1
.
For
n E N let
such that
{x E 0; d(x) >
(19.9)
ai(Y,) - A < yiN < ai(Y,)}
,
.
19.3. Partition of unity. 0
yi E Ai
}Con C {x E 0; d(x) > n + 1 }' n
and denote
SZn = int (0 \ 0n)
(19.10)
Obviously
n
C 0
C n+l x
[Compare these sets
cf. (18.4). For
n
the boundedness of
0
L)
S2
n=1S2n n
with the analogous sets defined in Subsection 18.1,
sufficiently large the two definitions coincide due to 0 .1
There exists a number {Q1,Q2,...,Qm}
(19.11)
with
from (19.6) forms a covering of the closure of the set
Q i
270
n E N such that the system
Stn
.
Let
{a1' 2,...,Om}
be the partition of unity corresponding to the covering (19.11), i.e.
Oi E C'(RN) (19.12)
supp i
,
for x E an
Oi (x) = 1
WW_
19.4. The distance.
L2th ri (19.14)
1
m
i=1
(19.13)
0`i6
Qf ,
Denote
di(x) = dist (x,r.) from (19.8). For d(x) = di(x)
A
> 0
sufficiently small we obviously have
x E U
for
Vhere Ut i
= S2 n supp of
U* C L'
,
Moreover, the following estimate holds:
(19.15)
ai(yi) - yiNl 1/K 1
1 + A
for
<_ di(x) < ai(yi)
- yiN
J
x = (yi'yiN) E Ui
i = 1,2,...,m
,
(see e.g. A. KUFNER [21, Lemma 4.6).
The following two theorems have been proved in A. KUFNER 121 using the local coordinates
and the one-dimensional Hardy inequality
(yi'yiN)
(0.2) with respect to the variable
y,
1N
19.5. Theorem.
Let
1
< p <
E > K(p - 1)
.
Wl'p(Q;dE,dE)
y
12 C- CO,K
,
0 < K
1
and
1
and
Then
L'(Sl;dn)
where J
E/K - p Kp
19.6. Theorem. Let
1
for
E > Kp
for
K (p - 1)
,
12 `
,
<
CO,K
E < Kp. ,
0
271
E x K(p - 1)
.
WO'P(Q;dK,dK)
c LP(1 ;dn)
Then
where
for for for
E/K - p n = 1 E - Kp 1
l K (E - p)
E > Kp
0<E c
-<
,
Kp
E x K(p - 1)
,
0 .
If we use the inequality (6.20) (for
p = q) instead of the classical
Hardy inequality (0.2), then we can improve the foregoing theorems. Moreover,
using the methods from Section 17, we obtain assertions about the compactnes of the imbeddings mentioned:
19.7. Theorem.
(19.16)
Let
1
p <
Q E CO'K
,
0 < K <
,
1
.
Then
Wl'P(o;dK,d6) C, LP(S0;dn)
E/K - p for E > Kp , n > E - Kp for K(p - 1) < E < Kp n>-K for E K (p - 1) . n
-5
(19.18)
Wl'P(Q;dK,dK) C
if the inequalities for
19.8. Theorem.
(19.19)
Let
1
n
y
LP(c;dn)
in (19.17) are strict.
< p < m
O E CO'K
,
,
0 < K <
W10,P(o;dK,dt)(; LP(0;dn)
where
n > E/K - p E - Kp n n>= K(E - p) n>-K
for > Kp , for 0 < E Kp , for E 0 for E = K(p - 1) E
<_
E Y K(p
1
.
Then
(19.21)
W0'P(c;d5,dE) (;
if the inequalities for
LP(c;d")
n
in (19.20) are strict.
Now, we extend the above imbeddings to the case
p < q
.
In the proof
we will use the following result from Example 18.15.
Suppose
1
,
N/q - N/p + 1
>_
0
[or
N/q - N/p +
1
> 0
Then
119.22)
W1'P(S2;du-P,d1)
y
Lq(R;da)
k,or
r
W1,P(R;dY-p,dY)
Lq(0;da)
tf and only if (19.24)
a - Y + N _ N p
q
q
a _ Y + N p
q
19.9. Theorem. +a,
+ 1 ? 0
p
q
N+
1> 0 ].
p
1 < p < q <
Let
a E R . Then
(19.26)
N
N
q
p
,
+ 1 ? 0
,
QE CO,K
0
<_
Lq(f;da)
if (19.27)
QKp+q-p+1>0
> Kp
or (19.28)
K (p - 1)
<
a
p
+ N - N # - _K
0
p
q
or
(19.29)
a _
S < K(p - 1)
q
Proof. Using Theorem 19.7 for
(19.30)
hull
p,Q,dY-P
ac
K(P - 1)
+ N
p
q
-
N
p
u E W1'P(c;d8,d8)
+K>0 we obtain
l ull
1,P,c,d8,d8
2i
1
where
(19.31)
Y
( 8/K
for
it
for K(p - 1)
S - Kp + p
In both cases we have
(19.32)
N
au
B
Y
<=
,
i=I axi P,S1,dY
<
and consequently,
,
N
(diam 111Y-S
<
p
> KP
S
l
2
au p axi
i=1
J
P,Q,d8
The inequalities (19.30), (19.32) imply (19.33)
WI'P(Q;d6,d6) c WI'P(0;du-p,d1) .
If (19.24) is satisfied, then we have (19.22) which together with (19.33)
immediately yields (19.26). The conditions (19.27), (19.28) follow from (19.31) in view of (19.24).
Now let (19.29) be satisfied. Then there exists a number
w E (0,K]
such that a -
K(p - 1) + N
q
Denoting holds with (19.34)
6(w)
p
p
6(w) = K(p - 1) + w
,
instead of
6
W1,P(1;d8(w),d6(w))
Since the inequality (19.35)
N + K ? W > 0
-
q
p
6(w) >
we obtain from this inequality that (19.28) ,
and consequently,
G Lq(Q;da) implies
8
Wl'p(Q;ds,ds)C3, Wl,p(Q;ds(w),d0(w))
the imbedding (19.26) follows from (19.35) and (19.34). Similarly we can prove
19.10. Theorem.
Let
1
4-P+1 >0,
a, 6 E R . Then
(19.36)
WI'P(S1;ds,d6)
if
6> Kp
274
,
Lq(SZ;da)
a- 8+ q
KP
q
p
+ 1> 0
D E C 0' K,
0
1
,
or
0<8 or
S
aq -
S X K(p - 1)
Kp ,
p
-
+ N q
N
p
+K
>_ 0
6 <0 , a-KS+N_N+K>0 q
p
q
p
or (19.37)
S = K(p -
1)
a - K(P - 1) + N - N + K > 0 p
,
q
p
q
The proof of the following two theorems concerning the compact imbedding is again similar to that of Theorem 19.9; we only use the imbedding (19.23) instead of (19.22)
and, of course, the condition
(19.25)
instead of (19.24)].
19.11. Theorem. Let a, S E R . Then (19.38)
1
N
-
N
+ 1 > 0 , 0 E CO,K
,
0 < K < 1,
p
q
W1,P(Q;ds,ds) y C Lq(0;da)
if
>Kp, qa-
+N-N+1>0 q p
KP
or
K(p _ 1) < a < Kp
a- pS+N-N+K p
,
q
>0
q
or (19.39)
B < K(p - 1)
Let
19.12. Theorem.
1
K(p - 1)
a _
,
q
p
+
q - pN + K > 0
N
N - N +
< p < q <
1
0ECO'K, 0 < K <
0,
>
1
p
q
a, 8 e R . Then (19.40)
Lq(0;da)
W0'P(Q;ds,d8)
if
> Kp
aq
,
KP
+N-N+1> 0 p q
or
0<
B
<-
KP
,
S X K(p - 1)
,
q
- S+N p
q
N
p
+K>0
or 0
q
PS +
q
p+
> 0
275
,
or
-K(P p
6=K(p-1)
(19.41)
+4-P+K>0
Q
Theorems 19.9 - 19.12 give only sufficient conditions for
19.13. Remark.
the corresponding imbeddings. We will show that for K =
1
these conditions are also necessary except for the conditions (19.29), (19.37), (19.39) and (19.41) for the imbeddings (19.26), (19.36), (19.38) and (19.40), respectively. First, let us prove some auxiliary assertions.
Let
19.14. Lemma.
1
< p,q <
°°
.
Let
G
be a bounded domain in RN and
suppose that
WO'P(G) r Lq(G)
(19.42)
[WO'P(G) r Cj Lq(G)
Then
P+ 1 q- p+ 1 = 0 [ Nq
(19.43) Proof.
>0
In (19.42) we consider the weighted spaces with weights identically
equal to one. In Theorem 18.21 we can take b1(x) =
1
,
r(x) = d(x)/3
,
b0(x) =
which then yields (together with Remark 18.22 (i))
n
c
r(x)]N/q-N/p+1
supP [d xEGn
n
and the necessary condition
n
lim 3n =
<
n-
[ lim On = 0 ]
implies
n4 °
(19.43).
19.15. Lemma. a,
Let
1
< p,q <
let
a E R . Suppose
(19.44)
WO,P(O;d,ds) ( Lq0;da)
[or (19.45)
Then
77A
1
Wl'P(Q;ds,d)
C,
C3 Lq(0;dn)
0
be a bounded domain in RN
,
q - P + 1 > 0
N q
Proof.
N
-
Let
+ 1 > 0
Let
u E
q
p
q
P
+ N -
0 < p < d(x) < 1 D and define
W1 'p(G)
+ 1 > 0]
G C G C 0
,
.
for
N p
q
D = diam 0 <
,
p
q
be a domain in RN
G
p - dist (G,30) > 0
(19.48)
a - 6 + N - N + 1= 0
,
and denote
Then
x C- G
u(x) = 0
x E 0 \ G
for
.
Then (19.48)
immediately implies that
W1'p(G) Cj W0'p(S2;ds,ds) and
Lq(S0;da) g Lq(C)
.
The imbedding (19.44) [or (19.45)] implies that (19.42) holds and Lemma 19.14 yields the first inequality in (19.46)
[or in (19.47)].
Now, we use Theorem 18.21 and Remark 18.22 (i) where we take r(x) = d(x)/3
,
b0(x) = da(x)
,
b1(x) = dd(x)
,
and we obtain
n
n
[d(x)]a/q-B/p+N/q-N/p +1
sup
= c
xE52n 1
The necessary condition
n
A
lim F)n = Z <
or
implies the
n+-
n->m
second inequality in (19.46)
lim Bn = 0]
[or in (19.47)].
D
Comparing Lemma 19.15 with Theorems 19.9 - 19.12 (where we take K= 1) we immediately see that the conditions (19.46), (L9-.47) are necessary and sufficient. More precisely:
19.16. Theorem.
[ S x p - 1]
.
Let
1 < p <_ q <
0E C
0,1
a, P E R ,
B> p- 1
Then
W1,P(0;dg,ds)(
Lq(R;da)
Lq(0;da)]
if and only if
277
N _
N+ 1> 0, a- s+ N- N+
q
p
Let
19.17. Theorem.
Bzp-
p
1
q
P
_<
q <
1
__>
0
p
q
S2c C 0'1
,
a,
C- R ,
S
> p -
1
Then
1
W1,P(Q;dPd
Ci Lq(Q;da)
[WD'p(c;ds,ds) c c Lq(0;da) ] if and only if N - N
+ 1> 0,
19.18. The case
p
q
1
N
+ N -
p
q
< q < p <
+ 1> 0.
p
q
As was mentioned before Theorem 19.5 and
Theorem 19.7, an important role was played by the one-dimensional Hardy inequalities (0.2) and (6.20) (the latter for
p = q ). In the case
p > q
we will again substantially use the inequality (6.20). To this end, let us summarize the results derived in Examples 6.8 and 8.21 (ii):
Let
1
0
(b1/q q tE dt
(19.49)
6, n E R .
,
ll 1/p
br
Then the inequality
Jlu'(t)Ip to dtj
I
I
0
0
holds
for
(i)
u E ACL(0,b)
n< p
(19.50) (ii)
(19.51)
for
(19.52)
for
u E ACR(O,b)
,
e
>
g-
- 1;
P
p
if and only if
nP-P-
u E ACLR(O,b)
1
or n` p - 1,
if and only if
e> q P- P, -
nER,
19.19. Remark.
E> n
1
n > p - 1
(iii)
if and only if
1
Checking the proof of the necessity of the conditions
(19.50) - (19.52) we can
see that these conditions are necessary
for the inequality (19.49) to hold on the (smaller) classes TR(O,b)
278
,
TLR(O,b)
,
respectively, where
TL(O,b)
TL(O,b) _ {u E C-([0,b]); supp TR(O,b)
{u E C-([0,b]); supp u 0 {b} = 0}
TLR(O,b) = TL(O,b)' TR(O,b)
Let
19.20. Theorem.
u n {0} = 0}
p
1
<=
.
S t E C 0' K
,
0< K` 1,
,
a, 8 E R
Then
W11P(St;dd8)
(19.53)
Lq(0;da)
Lj
if 8 > K(p - 1) + K P
(19.54)
a
,
q
q
-
S Kp
+1-1+1 >0 q
p
or (19.55)
K(p - 1) < 8 < K(p - 1) + K P q
,
a q
-
S
p
+ K(1 - 1 + 1) > 0 q
p
or
8 < K(p - 1)
(19.56) Proof.
(i)
(19.57)
,
a>-K
First we show that under the above assumptions W1,P(a;d8,d6)
c Lq(0;da)
According to Theorem 17.10, it suffices to verify that
(19.58)
lim
sup
nom
where we take
hull
q,S,n,da
=
<
ljullX<=1
X = W1'P(0;d8,d8)
(for the set
Stn
,
cf. (19.10)). Let us
denote (19.59)
V = {u E C-(0);
llu
1,p,0,d8,d8 Then
V
Take
is a dense subset of
u E V
.
W1'P(U;d8,d8)
Using the local coordinates
(cf. V. I. BURENKOV [1]). (y',y i
corresponding partition of unity
have for x E 0n
and the
from Subsections 19.2, 19.3, we
n>n m
(19.60)
{4i,}
)
iN
u(x) = u(x)
m
fi(x) = i=1
S u(x) i=1
279
.
ui = UOi
with
.
from (19.14), we
di(x)
Consequently, using the weights
obtain m
Ilu
a q,52n ,d
L
=<
i=1
11uill
m
n a= 4,52 d
G11uill
i=1
q,
S2
n
(19.61)
a
i1 suPP $i' d i
m`
Iluill
E
i=1
a q,Ui,di
Let us now estimate the norm (19.62)
II U.I,q
a =
*
q,Udi
Iu(x)Iq da(x) dx = J
i
ai(Y1) =
l
J
Iui(YI'YiN)Iq di(YiYiN) dY.N
J
dyi
Ai ai(Y,)-A If
a ? 0
ai(Yi)
then the second inequality in (19.15) and the substitution
,
t
- YiN
t E (O,A)
,
yield
A
i[J ui(y ,ai(Yi) q,Ui,d
(19.63)
J
.
i
If
a < 0
l
(rq
- t)
to dt] dyi
0
then we use the first inequality in (19.15) and obtain similarly
,
as before A
(19.64)
Iluillq
I:
a
` (1 + A)
-a/K
[JIui(y'a i i(yi) - t)
J
q,Ui,di
q I
to/K dt dyi I
A. 0 1
The inner integrals on the right-hand sides in (19.63), (19.64) can be estimated by the Hardy inequality (19.49) on the class b = A
,
u(t) = u.(y',a.(y') - t]
,
E = a
or
yields the estimate A l
1ai(Yi) - t)
(19.65) J
Ai
[JI-i
tE dtl dyi 1
ACR(O,b)
(with
E = a/K ), which finally
q/p
P
`Cq.1
u(Yi,ai(Y1) -
dt
to dtJ
t))
dyi
A1 0
If
n < 0
then Holder's inequality and the estimates (19.15) together
,
with (19.14) obviously lead to the inequality A
jj A
P
d
lq/P dyr to dtj
ui(yi'ai(yi) - t)
Id t
i
J
0
1
(19.66)
1(Ai)]
(p-q)/P
If
t
A. 0 i
t) I
P
iN
lq/P to dtJ dy'1 i ll
1
1,p,R,dn,dn
ayiN p,0,dn 11
0
n
ui(yi,ai(yi) -
q
au.
Similarly, for
I
a
ay
1
we obtain
A
d
(19.67) IA
lJ
i0
ui(yi,ai(Yi)
Idt
- t) p to dtq/p dyi <
1,p,Q,dK n,d Kn
1
We have to distinguish four cases: Let
(i-1)
a z 0
6 > 0
,
.
K = a
Then we put
and by
n = 6/K
(19.63), (19.65) and (19.67) we have (19.68)
< Kqjiujjq
lulIq
1 q,U*,da
1,p,Q,d6,d6
K = K1 = Cc2/q ) provided (see (19.51))
(with
(19.69)
6 = K(p - 1)
(i-2)
Let
a < 0
or
6 > K(p - 1)
6 ? 0
,
.
,
a>
Then we put
pI
Kp
E = a/K
(19.64), (19.65) and (19.67) we arrive at (19.68) (with C(1 + A)-o'/(Kq) c2/q
(19.70) (i-3)
6
a ? 0
and by
K = K2 =
provided (see (19.51))
<= K(p - 1) Let
n = 6/K
,
,
,
a > - K or 5 < 0
.
6 > K(p - 1)
Then we put
c = a
,
,
a>
p
n = 6
- K+ 1) p
and the
28
inequalities (19.63), (19.65) and (19.66) immediately imply (19.68) (with K = K3 = Cci/q a < 0
Let
(i-4)
B < 0
,
E = a/K
Then we put
.
,
n
inequalities (19.64), (19.65), (19.66) yield (19.68) (with
and the
B
K = K4 =
= C(1 + A)-a/(Kq) ci/q ) provided (see (19.51)) (19.71)
a > - K
.
From (i-1) - (i-4) we conclude that (19.68) holds with provided
K = max (K1,K2'K3,K4)
a
satisfy the conditions (19.54) -
B
,
(19.56), and then we obtain from (19.61) that
lull
< mKI,ll
n ,d a q,SZ u E V
holds for every
1,p,Q,dB ,d B
The same estimate holds for every
.
V
due to the density of
u E W1'p(0;d9,d9)
< mK
consequently, (19.58) is fulfilled with
.
,
and
Thus, (19.57) holds.
The compactness of the imbedding in question will be proved
(ii)
only for
W1'p(Q;d9,d9)
in
a
,
satisfying (19.54) (the proof for the cases (19.55),
a
(19.56) is analogous). Since the second inequality in (19.54) is strict,
a- S + 1- 1+ 1> 0, q
there exists
Kp
q
p
such that the numbers
e > 0
and
B
a = a - c
satisfy
(19.54), too. Then it follows from part (i) of the proof that
W1'p(Q;d9,d9)(
Lq(S2;da)
,
i.e. there exists a positive constant
(19.72)
IIuII
q,St,d
a
Using the fact that
< KIIu1I
such that
K
1,p,R,dB ,d B
d(x) < 1 n
for
x E
SZn
,
we derive from (19.72) the
estimate
qQ da
I
Iu(x)Iq da(x) dx = J
.(.)Iq da(x) dt(x) dx
J
pn
On jIuIIq,0n,da
nE
7A?
` n4 IIujI1,p,Q,dB,dB
which holds for every lim
This estimate implies that
u E
sup
= 0
ul
,
q,0n,da
n;= 11-Ix
and (19.53) follows by Theorem 17.6.
1< q < p <
Let
19.21. Theorem.
0
I E C 0'K
,
,
0< K 5
1
R
a,
Then (19.73)
WO'p(Q;d',ds) (
LQ(S;da)
y
if
> K(p - 1) + K
,
q
q
-
Kp
+
q
- p1 + 1 > 0
or
0 < R < K(p - 1) + K P q
q
p
1 + 1) > 0
+
p
q
or
q
0
P + 1) > 0
+KQ
-4
The proof is analogous to that of Theorem 19.20.
The conditions in Theorems 19.20, 19.21 have been only sufficient. Similarly as in the case provided
K = 1
19.22. Theorem. (19.73)
p ? q
,
these conditions are also necessary
.
Let
1
<_ q < p <
St C C 0'1
,
a,
E R . Then
Lq(Q;da)
W1'p(Sl;ds,ds)
if and only if
(19.74) Proof.
SER ,
a
q
- S+1 q
p
1 +1 >0 p
If (19.74) holds, then we have the compact imbedding (19.73)
according to Theorem 19.21.
Conversely, let us suppose that (19.74) is violated for some
a
i.e. assume that (19.75)
a
p
- gp -- 1
.
283
In view of (19.52) and Remark 19.19, the condition (19.75) implies that the Hardy inequality (19.49) C
(with a finite constant
[with
E = a
on the class
)
there exists a sequence of functions
,
is not fulfilled
= B ]
r1
TLR(O,b) = C-(O,b). Consequently,
un E C-(O,b)
such that
b
is dt = j I u'(t)Ip n
(19.76)
1
n E IN
,
0
b Jun(t)Iq to dt
(19.77)
n
for
0
Now, we will use the first coordinate system Let
6
be the constants from Subsection 19.2 [cf. (19.3) and (19.6)]
A
,
from (19.2).
(yl'Y1N)
the corresponding function from the partition of unity described
and
in Subsection 19.3. Without loss of generality, we can suppose that the set {(yl'YlN); Iyi"
is contained in 0 < a* < a
,
a1(Y1)
'1N < al(y )}
- '* <
for sufficiently small numbers
supp ¢1
0 < 6* <
6*,
<
6
a*
,
,
COrRN-11
E
Further, we introduce a function
.
6'`
such that 0 <
(P (z')
for
1
@(z') = 1
and for
n E N
(19.79)
and
for
Iz'
x E S2
RN-1
z' E
(19.78)
d'`
< 2 ,
@(z') = 0
u
n
if
x = (Y 1' 1
is the function from (19.76), (19.77) with
and consequently,
a*
0
Obviously, the support of
v
n
is contained in the set
346
l('1''1N)' Y1 `
,
and, moreover, there exists a domain (19.80)
36 * 4
IN ) E U*1
u e C-(O,A*) n
>
if x e 0 \ U 1
0
where
Iz'I
we define
(YP) u n (a (Y') - Y 1 1 1 1N
vn(x) _
for
supp v C Gn C Gn C U n
1
al(Y1) -
G C RN
IN
(0,A )}
such that
instead of
b
The function
(cf. (19.5)) and consequently, it
O1
is Lipschitzian on
a 1
follows from (19.79) that
vn E Wp'P(G
n
Moreover, in view of (19.80), vn E Further,
W10,P(SI;d6,ds
d1(x) = d(x) for x E Ui (cf. (19.14)) and we obtain
IIvnil,Q,d p
The estimates
IDI
IIvnIIp
s - IIvnilp,U
<_
S
$
.
= ;Iv nil
p,U l, d
1
(19.78)) and (19.15) (with
(cf.
1
i, d
K =
) yield
1
=
al((Y1)
(19.81)
=
un(a1(y1) -
)I
I
a1(yi)-A
Al
lun(t)IP is dtI dy1 = c2
< c1 J 1 Q
lJ0
n
axi
for
(P
J
un(t) P is dt
0
111
The properties of the functions av
dy1NI dyl
J
l
and
a1
imply that
P (x)
<= c3llunlal(Y1) - Y1NJ IP +
x = (Y1'y1N) E UL
lu°(al(Y1) - ylN)Ip1
and we obtain similarly as in (19.81) that
avn p
avn
ax-.
i P,Q,d
u(t)
= c4
axi P,Ul,d
is dt + J
Iun(t)IP J dt]
lJ
0
0
hence
285
SC
II vnIIp
1,p,Q,Ads
(t)Ip t8 dt +
5
lun(t)lp is dtI
I
0
.
11
0
Since the first integral on the right-hand side can be estimated by the a = 8 ) we finally p = q second (cf. Example 8.21 (ii) with b = a b = A*
have from (19.76) (with (19.82)
with
c6
= c6
llvnllp
l,p,sl,d8,d 8 n
independent of
0(yi) =
Using the fact that
a llq n q,R,d
,IV
that
)
= llvnll
1
*
for
< 6*/2
y1'l
,
we obtain
a
p,U1'd1
(19.83)
[jiu(t)l
c 7
I
dyi = c8
q to dt
I
111
0
0
I
[cf.(19.81)] with
to dt
I
IY'
n
independent of
c8 > 0
From (19.82), (19.83) and (19.77) it follows that the sequence Lq(St;da).
is unbounded in
which is bounded in the imbedding of
W0'p(S2;d8,d8)
the less so, compact.
19.23. Remark.
into
Lq(R;da)
cannot be continuous, and
In the proof of Theorem 19.22 we in fact have shown that
is not even continuous. The same is
Lq(SZ;da)
into
W0'p(S2;d8,d8)
true for
is compact or it
the imbedding of
W1'p(Q;dd
(see the following theorem).
(19.84)
Theorem.
Let
S2EG'0'1
< q < p <
1
W1,p(52;dd8)
c(
L4(Q;da)
if and only if either a
(19.85)
8
> p - 1
+
6 + 1 -
1
q
286
-
,
p
q
p
> 0
a,
,
Consequently,
E
either the imbedding of
19.24.
{v n}
8 E R . Then
or
- 1<
(19.86)
1, a>- 1
p
or (19.87)
S-<-
1
q-p+q-p+1 > 0
,
In the cases (19.85) and (19.86) the proof is analogous to that of
Proof.
Theorem 19.22. For
6 < -
we have
1
W1,P(Q;d6,d6) = W01,P(O;d6,d6)
(cf. A. KUFNER [2], Remark 11.12 (ii)), and the result follows from Theorem 19.22.
E
19.25. Remarks.
The necessity of the condition (19.87) cannot be
(i)
proved in the same way as in the case of necessity of the conditions (19.85), (19.86): If we used functions
vn
defined analogously as in (19.79), we W1'p(0;d6,d6)
would not be able to guarantee that they belong to 6 < -
1
since for
the inclusion
C(0) C W1'p(Q;d6,d6) does not hold.
We have derived necessary and sufficient conditions only for
(ii)
0 C
0,1
,
i.e. for
K =
1
.
In the case
0 < K <
1
it is possible to find
necessary conditions for the validity of the corresponding imbeddings (by the same methods as in the proof of Theorem 19.22; moreover, also for the case
p < q
)
but the conditions are different from the sufficient ones.
20. UNBOUNDED DOMAINS
20.1. Introduction.
In Section 17 we derived general criteria for the
continuity and compactness of imbeddings of weighted Sobolev spaces into weighted Lebesgue spaces (Theorems 17.8 and 17.10). In Section 18 we reformulated these general results in terms of the weight functions; an important role was played by the distance
d(x) = dist (x,aO)
.
287
Here we will deal with a special type of unbounded domains and the role of
will be played by the function
d(x)
20.2. The domain for some
0
n E N,
is such a domain that
St C RN
2,
n
{x E S2;
(20.1)
Let us suppose that
.
x E RN
,
IxI
> n} = {x E RN;
IxI
> n}
IxI
.
This class of domains will be denoted by
(20.2)
;b
in fact,
;
if there exists a compact set
12 E .0
(20.3)
S2 = RN \ K
K C RN
.
We will mainly deal with the following special cases: K = G
where
K = 0
,
K = {0}
is a bounded domain. Then
G
Q = RN
or St = RN \ {01
or
0 E , . The role of the set
Let
such that
SZ
=
RN \ G
.
from Subsection 17.2 will be
Qn
played by
Stn = {x E S2; IxI < ni
(20.4)
and we denote (20.5)
Stn
int
=
(S2
\
52
Again we have 0n
S2
n+1 z
moreover, according to (20.1), for complement of the closed ball
20.3. The function a function
c r s'
1
r = r(x)
r
.
the set
Stn
coincides with the
B(0,n)
S2 E
defined on
2)
Stn
.
We will suppose that there exists see (20.1)] and a constant
[for
such that
(20.6)
r(x) < 3 IxI
(20.7)
c-1 < r(y) <
288
Let
n > n
r
for a.e.
r(x) ` c r
x E Stn
for a.e.
,
x E Stn
and
y E B(x,r(x))
20.4. Remark.
If we compare the assumptions about
and
0
with those
r
of Section 18, we see that there are certain differences - in the classes of domains considered,
- in the definition of the sets - in one property of
r(x)
that the ball
xl/3
[see (18.4) and (20.4),
[see (18.7) and (20.6)].
The important auxiliary function by the function
0n
r = r(x)
is now 'controlled' from above
which together with the
B(x,r(x))
belongs to
n
relation
12 E 0 ensures
provided
B(x,r(x)) n 03n x 0 This is the situation which occurred in Section 18 due to the condition (18.7) [see Lemma 18.5], and we may introduce the following and
n = max (n,2)
.
convention:
20.5. Convention. All assertions formulated in Subsections 18.6 to 18.12 remain true if we suppose that 1 E 2 [instead of (18.2), define 0 by n (20.4) [instead of (18.4)] and assume that r = r(x) satisfies (20.6)
[instead of (18.7). All other assumptions (about the weight functions v1 , w
,
numbers
about the auxiliary functions d^)
n
63n ,
r
,
b0
,
b1
,
b0
b1
,
,
v0
about the
) remain unchanged [compare also the identical conditions
(20.7) and (18.8)].
The proofs of these 'new' theorems are literally the same as those of the 'old' ones, and therefore, the formulation as well as the proofs are left to the reader.
Now, we will give some examples in which we will use the following notation: For (20.8)
x E '0
,
put
a* = inf {IxI; x E c}
and denote by (20.9)
¶
0,1
the set of all
0 E 2 such that
12 = RN \-G
with
G E C
0,1
.
Theorems 18.11, 18.12 together with Convention 20.5 imply the following results.
20.6. Example.
Let
1 < p < q < -
,
a, B E R ,
12 C 0
0,1 ,
aX > 0
.
Then
289
W1,P(c2;
s-P,
x
G Lq(52; Ix!a)
,
1x13-P, lxl
W1'P('R;
C
XIm)
if and only if N
- P + 1? 0,
Q -
- N+1 >0
a
P+
qN
N
p+
1
S 0
- N+
1
< 0 ].
-
q N
q
[Here we set
r(x) = lxl/3
20.7. Example.
For
,
p
Let
1
q
,
-
S
+ N q
p
p
b0(x) = lxlm
< p = q <
,
b1(x) = lxls .1
x E 0 put w(x) = lxlm lnYlxl
,
v0(x) =
lx
B-P lnaxl
v1(x) = lxls lndlxl Then
(i)
W1'P(Q;v0,v1)
Lq(Sd;w)
if and only if N-N+1>_0 q
p
and either
a a-a+N-N+1<0 q
p
q
p
q
p
or
B+ (ii)
N+ 1= 0 q
p
W1,P(s;v0,v1)
c Lq(Q;w)
N-N+1>0 q
and either
290
p
a-s+N-N+1<0 q
p
q
P
a` 0; q
if and only if
or
S2 E
m, S, Y, 6 E R ,
p
,
011
+
q
p
[Here we set
Let
q
xl/3
p
b0(x) = w(x)
,
W1,P(sl. e8lx
eSlxl)
,
b1(x) = v1(x)
.]
Then
.
Lq(Q; ealxl)
eslxl, eslxl) ry
L
,
Q E `D O' 1
< p < q < -
1
< 0
, ,
p
r(x) =
20.8. Example.
Y-
+ i = 0
_ q
Lq(12; ealxl) ]
if and only if
a-s<0
N-N+1>>0 p
q
q
p
a-S<0 LN-N+1>0, q q p p [Here we set
r(x) =
20.9. Remark.
1
b0(x) = ealxl
,
,
b1(x) = eSIxI
]
Let us go back to Example 20.6. The condition
a*
H E 0
(together with the assumption
0,1
) has guaranteed the validity of
the 'local' imbeddings (18.18), (18.26) since the corresponding weighted spaces
and
W1'P(12n;v0,v1)
Lq(S2n;w)
are for
a;;
> 0
isometrically
isomorphic to the corresponding non-weighted spaces and we can use the classical Sobolev (Kondrashev) imbedding theorems. This approach fails if
a
= 0
.
Nevertheless, for
can use the results from Section 18 since
d(x) = dist (x,M) = IxI obtain for
:
S2 = RN \ {0}
W1,P(Q;
Q = RN \ {0}
we
(see (18.2)) and
iU = {0} x 0
according to Example 18.15 (and Lemma 19.14) we that
xlR-p,
Ixl8)(; Lq(12;
xla)
if and only if
N-N+1 q p
0, qa-s+N-N+1=0. p p q
The same result obviously holds if we take a certain difference: while the spaces W1'P(RN \ (0};
Ixls p,
lxls)
Q = RN
.
However, there is
W1'P(RN \ {0}; IxI8-p,
IxI8)
and
are well-defined since the conditions (16.18)
291
and (16.19) are satisfied for every =
lxls -p
vl(x) =
,
xls
S E R , in the case
the conditions (16.18) are satisfied for S
,
and the conditions (16.19) are satisfied for W1,p(RN;
dealing with the spaces
20.10. The case
lxls-p,
S E (p - N, Np - N)
we will consider
p > q
S > p - N
lxls)
,
.
Therefore, when
W0'p(R';
lxls-p,
lxls)
.
Radial weights.
.
v0(x) =
S2 = RN,
Now we will consider imbeddings
of the type
Wl,p(Q;v0,vl) L Lq(S;w) for
< q < p <
1
co
We assume that
.
v = v(x)
(20.10)
v(x) = v(lxl)
x E S2
,
,
see (20.8)]. Such weight functions are called
a*
[for
v E W(a.t,co)
and restrict ourselves to weight
of the type
functions
with
c E
radial weights.
Moreover, we introduce two special subclasses of the class an unbounded interval WB(I)
(20.11)
W(I)
for
I C R : WC(I)
or
denotes the class of all
v E W(I)
which are bounded from above and from
below by positive constants on each bounded or each compact interval
J C I,
respectively.
We will make use of the following two auxiliary assertions:
20.11. Lemma. 1
2 2
}
(i)
(ii)
(iii)
292
Let
R > 0
.
Then there exists a partition of unity
R
=
with the following properties: R,
2 E CW(RN)
supp 0R C B(0, R + 4)
,
RN \ B(0,R)
,
supp 42 C R
on RN
(iv)
0 < 0i < 1
(v)
01(x) + 02(x) = 1
(vi)
there exists a constant
,
i = 1,2
for
x E RN K > 0
independent of
R
such that
i (x)
ax
for
s K
i = 1,2
j
,
j
The proof is standard and is left to the reader.
Let
20.12. Theorem.
1
< p < -
that there exist a constant (20.12)
0 (t)
2 E `0
,
and a number
k > 0
to > a*
t > t0
for a.e.
->- k v1(t) t -P
v0, vl E WC(a*,-)
,
.
Suppose
such that
.
Then the set
Cbs (2) = {g E
(20.13)
is a dense subset of Proof.
W1'P(SZ;v0,v1)
u
where
and fix
u E W1 'P(52;v0,v1)
Let
function
supp g ( S2
is bounded} vi(x) = vi(IxI)
c > 0
,
i = 0,1
Then there exists a
.
,
E
uEE Cm(S2) (-" W1,P(S2;v0)v1) = V
(20.14) such that
(20.15)
lu - uEI1 1,p,S2,v0'v1
<
E
2
(cf. V. I. BURENKOV L1]).
Let
be such that
f E C (R)
0-s f(t)
tER,
for
1
<_
(20.16)
f(t) =
Choose (20.17)
R > n
t < 5/4
for
1
n
(for
,
f(t) = 0
h > 0
see (20.1)) and for
lx
for
t
>_
7/4
denote
R}
Fh(x) = fh
J
,
x E RN
I
Further, for
s > 0
denote
Sts = Ix E 0; The function
Fh
xl < s}
,
Sts = int
from (20.17) belongs to
(S2
\ Sts)
C'(RN)
.
and satisfies
293
0 i Fh(x) < Fh(x) =
for x E
1
for
1
RN ,
x E 12R+5h/4 U a0
'
(20.18)
x E RN
for
< cf h
supp Fh C B(0, R + 2h)
j
,
= 1,2,...,N
.
If we define (20.19)
with
u
(20.20)
ue,h(x) = uE(x) Fh(x)
E C- (R)
ue,h
,
from (20.17), then obviously
Fh
from (20.14) and
x E S1
,
supp uc,h C B(0, R + 2h)
,
1R+h ) C C 12
supp (uE -
.
These properties together with (20.18) and (20.12) imply that for
h > max {R, t0-R} 1/p
I1/p
(20.21)
<
dxJ IJIuc - uE,hIP v0
IuEIP v0 dx]
[
I
OR+h
0
and P
a
(ue - uE,h)
ax
l1/p
vl dxJ
J
auE p
11/P
v1 dx J
1R+h
J
auE p (20.22)
Ivclp lax,(1 - Fh)
12R+h
J
ax.
f
l
;
,
-
ax.
J
I"EIP h-P vl dxJ 1
12R+h\12R+2h
P
[I
vI (x)
1/p
+ 3cf I
dxJ
au E J
12R+h
294
ax
J
p
I
S+h
I uE (x) I P
1/P
r
v1 dx +
Iuc J
R+h
(IxI)
v
12R+h
< K
<
v1 dxJ
1/p
l1/p vl dxJ + cf[
12R+h
i
1/P
P
a
+
v0
dxll J
1
Ix
P
11/P
dxJ
21/p,
with
K =
max {1,3cfk-1 P}
] = 1,2,...,N
,
.
Since
u
C W1'p(0;v0,v1)
according to (20.14), the estimates (20.21), (20.22) and the Eproperties of imply that there exists a number
SIR+h
h > 0
such that
E
uE -
2 ,
1,P,0,v0,v1
which together with (20.15) yields u - u
E,h 1,P,R,VOIv1
< C
Thus our theorem is proved since
.
uC,h E Cbs(Q)
according to (20.20). 0
Now we are able to prove some imbedding theorems.
20.13. Theorem.
Let
< q < p < w
1
Suppose that there exists a number
where
such that
v(t)tN-1,
q, p)
W1'P(Q;v,v) (j (.
(20.24)
w(x) = w(Ixl)
Proof.
w, v e WB(a*,o)
<
is given by (1.19). Then
AL
with
H > n
w(t)tN-1,
AL(H,
(20.23)
S2 E D 0'1
,
,
Lq(S2;w)
v(x) = v(Ixl)
The set
V = {u e- C'(0); W1'P(Q;v,v)
is dense in
Ilu
1,p,R,v,v
(see V. I. BURENKOV [17). Due to this density and
to Theorem 17.6, it suffices to verify that (20.25)
lim sup {IluIIq,Qn,w; u E V,
W
where we put
Qn =
12n+5
IuI1,p,52,v,v < 1} = 0
[Note that (20.25) is the condition (17.15); the w, v E WB(a,,-)
condition (17.14) is satisfied due to the assumption Let
u e V ,
1
u1,p,S2,v,v unity from Lemma 20.11 with a fixed
.J
and let
{,i,¢2}
be the partition of
n E N ,
n > H
H
,
from (20.23).
Then
295
u=u1+u2
where
ui = uOi ,
supp ul C B(0,n+4)
,
supp u C RN \ B(O,n) 2
and we have uI
q
u(x)lq w(x) dx = q,Qn,w
RN\B(0,n+5) J
lu2(x) Iq w(x) dx
(20.26)
RN\B(0,n)
RN\B(0,n+5) tN-1 dt dO
nJlu(t,0)lq W(t)
1
1U2(x)Iq w(x) dx
S1
with
S1 = {x E RN;
E C (n,-)
IxI = 1}
and
According to the definition of
.
for every fixed
u2(n,0) = 0
0
u2
we have
,
and-the one-
,
dimensional inequality (cf. Theorem 5.10) implies
(lu2(t,0)q w(t) tN-1 dt < Aq
[J(to)
q/p
P
v(t) tN-1 dt j
n
n with
An = q1/q
w(t)tN-1,
(P,)1/q'
v(t)tN-1,
AL(n,
q, p) .
Using this estimate in (20.26) we obtain by Holder's inequality and Lemma 20.11 (iv), (vi) that Du
llullqn
q,q ,w
<
An
J
f
l
Si
p
at ?(t
,
o)
11q/P
v' (t) tN-1 dtj
dO _<
n
W (S)](P_q)/p
An ((
JU (t .0)
v(t) t
N-1
dt dOJ
JS1
n <
1-1(S1)](P-q)/q
An
Nq
296
JJJIlu(x)lp v(x) dx +
INP Kp l
0
q/p
<
N J}aX) P
+
j =1
S2
Nq [mN-1(S1(p-q)/q max {Nq Kq, 1}
cq = =
where
v(x) dx Iq/p < cq An
From (20.23) it follows that c o ndition
n-
(20 25) is satisfied
20.14. Theorem.
Let
1
lim An = 0
< q < p <
.
and consequently, the
,
0
w, v0' v1 E WB(a*,m)
S2 E `,D 0'1
,
such that (20.12) holds. Suppose that there exists a number
H > n
such
that
AR(H, -,
AR
where
w(t)tN-1,
q, p)
<
,
is given by (6.7). Then W1,P(1;v0,v1)
with
V1(t)tN-1
v0(x) = v0(IXI)
c y Lq(Q;w) v1(x) = vv1OxI)
,
w(x) = w(IxI)
,
The proof is similar to that of Theorem 20.13, only instead of Lemma 20.11 we use Theorem 20.12. Then we can work with functions vanish for
IxI
g(t) = u(t,O) = 0
u E Cbs(S2)
which
sufficiently large (see (20.13)), and consequently, for
Hardy inequality for
t
near infinity. Thus we can use the one-dimensional
g E ACR(n,W)
.
The following two theorems will deal with general unbounded domains
in RN
.
We again define
a*
by (20.8) and
by (20.5) and consider
Stn
radial weight functions.
20.15. Theorem.
Let
w, v0, v1 E WB(a*,.) (20.27)
where
< q < p <
Let
S2 C RN
be unbounded,
and
t(a*,
w(t)tN-1,
1(t)tN-1, q,
p)
<
.4 is given by (8.98). Then
(20.28) with
1
W1,P(S,v0,v1) c Lq(Z,w)
v0(x) = v0(IxI)
,
v1(x) = V1(IxI)
,
w(x) = w(Ixl) 297
Moreover, let
A E WB(a*,.)
be decreasing in
for some
(H,")
H > a*
and let lim a(t) = 0
(20.29)
t-*
Then
(20.30) with
Lq(S2;wA)
c
a(x) = A(Ixl)
.
Proof. Using the density argument we can consider
by zero to the whole R
N
.
Extending
and introducing the spherical coordinates
g(t) = u(t,O) E C0(a -)
we have that
u E C0(0)
for every fixed
0
.
u
(t,0),
Due to
(20.27), we can estimate the inner integral in
tN-1
J lu(t,0)Iq w(t) q,0,w = J S1 a*
llullq
dt dO
by the one-dimensional Hardy inequality according to Theorem 8.17 and arrive finally at the estimate (20.31)
uj
au
q,0,w < c 1
c2IIuII1,p,0,vCv1
at p,O,v1
Consequently, we have proved (20.28). In order to obtain (20.30) it suffices to show proof of Theorem 20.13 (20.32)
Take
lim
ueX
hull
IjullX`1
n>H
,
I Ullq
q,51n ,wa
= 0 , where
q,S2n,w.1
X = WO'p(c2'v0,v1)
Then
.
=
J l_(x) lq w(x) A(lxl)
dx
2n
,1(n) J lu(x) lq w(x) dx =`
pn
298
similarly as in the
that
-
sup
n,
-
a(n) c2
jujjq
in view of the monotonicity of
and of (20.31). The condition (20.32)
A
now follows by (20.29).
20.16. Theorem.
w, v E WB(a*,-)
Let
< q < p < -
1
0 C RN
be unbounded,
Suppose that there exists a number
.
w(t)tN-1,
.4 = 4 (H,
(20.33)
Let
.
v(t)tN-1,
q, p)
<
H > a*
such that
°°
Then
(20.34) with
W0'p(c2;v,v) S, Lq(0;w)
v(x) = v(Ixl)
Moreover, let
w(x) = w(lxl)
,
A E WB(a*,-)
satisfy the assumptions of Theorem 20.15.
Then
(20.35)
W0'p(E2;v,v) C C> Lq(0;wa)
a(x) = A(lxl)
with
First we will prove (20.34). According to Theorem 17.10 it suffices
Proof.
to verify that lim
(20.36)
sup
nQn = Qn+5
where
Let
q,Q
Ilull<1 ,
< m
n
hull
nEN,
,
w
X = W0'p(S2;v,v)
be the number from our assumptions and let
H
partition of unity from Lemma 2 0 . 1 1 . Take
u E CO(0)
be the <
, l l u l l
n > H
.
X
1
,
n E N,
Then we have
llullq
n
J
q,Q ,w
flu(t,0)lq w(t)tN-1 dt dO 1
S1 H
with
u2 =
(cf. (20.26)), and since
q
w(t)tN-1
dt <
J
299
(lout
5 c y4q
p
P
v(t) tN-1 dtI
l J lat H
due to Theorem 8.17 with A from (20.33), we obtain analogously as in the proof of Theorem 20.13 the estimate
fl
with
independent of
c1
C0(2)
q,Qn,w
in
n
.
This estimate together with the density of
implies (20.36) and thus, (20.34) is proved.
X
The step from (20.34) to (20.35) is the same as in the proof of Theorem 20.15. 11
foregoing theorems we have derived sufficient conditions for
In the
the corresponding imbeddings. Now, let us give a necessary condition.
v0E W(a*,-), w, v1EWC(a*,-). n such that Let
20.17. Theorem.
0(t)tN-1,
(20.37)
where
1
1(t)tN-1,
.)
<
P, P)
(R,
is given by (8.69).
If (20.38)
with
W0I,P(I;v0,v1) C Lq(Q:w)
v0(x) =
,
4(R, -,
(20.39)
Proof.
o(IxI)
I(xl)
vl(x) =
w(t)tN-1,
w(x) = w(Ixl)
,
,
then
1(t)tN-1, q, p)
<
Suppose that (20.39) is not satisfied. Due to Theorem 8.17 and the
condition
w, v1E WC(a*, m), the corresponding one-dimensional Hardy inequality M
ljg(t)Iq w(t) tN-1 dtl
(
R
1/q
)
does not hold (with a finite constant
M
[Jg?(t)IP vl(t) `C
tN-1
dt 1/p
R
C ) on the class
C0(R,-)
(cf.
Remark 19.19), and consequently, there exists a sequence of functions W
gn E C0(R,-) 300
such that
=
tN-1 dt
nEN,
1
gn(t)IP v1(t) R
tN-1 dt
I g(t) Iq w(t)
--. m
for
n -
.
R
For
Then
n e N put gn (Ixl)
for
0
for x E 51 (l B(0,R)
un E CORN \ B(0,R))
(20.40)
x E RN \ B(0,R) .
and
fg(t)Iq
I Iu(x) Iq w(x) dx =
tN-1
w(t)
dt dO --.
S1R
S2
n-- -
for while au
P
n(x)
(20.41)
tN-1
<
ax i
v1(x) dx
Jlgn(t)IP v1(t) R
J
dt dO = mN-1(Sl)
S1
for every
n E IN
.
On the other hand, it follows from (20.37) that p
l g(t)l
0(t) t
N-1
dt
C
R
with
C
R
independent of
n
,
N (20.42)
fg?(t)IP vl(t) t N-1 dt
P
nllp,S2,v0
c0i=1
and consequently, au
n p
ax.
= c0 N mN-1(S1)
P,c,v1
due to (20.41).
By virtue of the estimates (20.42) and (20.41), the sequence
is bounded in Wp'P(11;v0,v1)
fun} C C0-(S2) C unbounded in
Lq(Q ;w)
,
but it is
due to (20.40). Consequently, the imbedding (20.38)
cannot hold.
20.18. Remark.
The reader can easily see that Theorem 20.17 remains true 301
for
< p < q < -
1
provided we replace (20.39) by
,
w(t)tN-1,
(20.43)
J.) (R,
v1(t)tN-1, q, Pj
<
On the other hand, in Section 18 we have derived necessary conditions for (20.38) to hold without the (restrictive) assumption (20.37) (see, e.g., Theorem 18.9).
Now we will apply the foregoing theorems to some special weight functions.
Let
20.19. Exam le. Let
(1)
12 E
_ q < p < m
1
a* > 0
,
a, BER .
,
B ji p - N
,
.
Then the following three
conditions are equivalent: WO1'P(S2;
IxIB-P,
W0,P(S2;
IxIB-P,
r IxIB r 6 Lq(Q; Ixla) IxIB) S Lq(12;
Ixla)
a- B+ N- N+ 1< 0. (ii)
12 E e
Let
p
q
p
q
0,1 ,
a* > 0
,
B > p - N
.
Then the following three
conditions are equivalent-
W1'P(0; IxIB-P, IxIB) W1,P(S2;
yy
1.1B-P, IxIB)
Lq(S2;
Ixla)
L' (Q; I xla)
,
a- B+ N- N+ 1< 0. p
q
(iii) WO'P(S2;
Let
IxIB-P,
p
q
or
S2 = Rn \ {01 IxIB)
S2 = RN ,
5;e p - N
Then the space
.
Lq(S2; lxM
is continuously imbedded into
for no
aER 20.20. Example.
Let
1
_ q < p <
,
a*
1
>
1
and put IxIB-p
w(x) = Ixla 1nYIxI
,
v0(x) =
lndlxl
vI(x) = IxIB lnalxl
302
.
,
a, B, y,
d C- R
U)
If
s / p - N
then the following three conditions are equivalent:
,
( Lq(Q;w)
W0'P(Q;v0,v1) W0i'P(0;v0,v1)
i
Lq(Q;w)
either (I
q
S+ N- N+ 1< q
p
0
p
or
(ii)
a
S + N - N + 1 = 0
q
p
q
Y
,
p
6
If, moreover,
p
and
S2 E 5)0'1
+
-
q
1
q
1< 0. p
S > p - N
then the following
,
three conditions are equivalent:
W1'p(0;v0,v1) j ( Lq(E2;w) W1,P(Q;v0,v1)
i Lq(Q;w)
,
either Ot
+ N- N+ 1< 0
-
p
q
q
p
or -
+
q
20.21. Example.
Let
,
.
P
q
P
-6+I-1<0
+ 1 = 0
-
1 q < p <
1
0 E
0'1
,
a,
B E R,
0
Then the following five conditions are equivalent:
(i)
W1'p(0; eslxl, eslxl Cj C, Lq(Q; ealxl
(ii)
W1'P(0; eslxl, eslxl) r ( Lq(Q; ealxl)
(iii)
WO,P(S2; eslx
(iv)
W1,P(o; eslxl, eslxl) C Lq(O; ealxl)
(v)
a
q
-
$
p
<0
eslx
Lq(11; ealx ,
.
If we weaken the conditions on conditions (i), (iii),
)
(v)
0
and suppose only
0 E
then the
are equivalent.
303
20.22. Remarks.
The results of this section are due to B. OPIC,
(1)
P. GURKA C1]. (ii) Let us go back to the estimate (20.42). Its first part is in fact
the N-dimensional Hardy inequality for our special functions
un
implies that for these functions, the norms
and
Ilu
n 1,p,5Z,v0,v1
and
II
(see (16.24)) are equivalent. As was mentioned in Section
I``
IIIunIll1,P,S2,v 1
16 (see Lemma 16.12), the problem of equivalent norms is closely connected with the validity of the N-dimensional Hardy inequality. We will deal with this question in the next section.
21. THE N-DIMENSIONAL HARDY INEQUALITY
21.1. Introduction.
At the very beginning of this book we have stated our
intention to describe conditions which guarantee the validity of the Ndimensional Hardy inequality 1/q (2 1.1)
[1lu(x)l
w(x )
C
dxj
N
r
1=1
J
au (x) axi
p
}}
1/p
v.(x) dxJ
f
Some partial answers have been given in Chapter 2; moreover, Lemma 16.12 enables us to use also the results derived in Chapter 3. Since we investigated in Sections 18 - 20 mainly the case
vl = v2 = ... = vN = v
,
we can write (21.1) in the form
(21.2)
[JIU(x)l q w(x) dx]
l/q
I [10u(x) lp v(x) dxJ
C
l/P
J
where N (21.3)
p
IDu(x)lP = i=1
ax (x) i
We will establish simple necessary and sufficient conditions for the validity of (21.2) on some classes following cases:
304
K
containing
C0(0)
mainly in the
(A)
bounded (in most cases
S2
v(x) = dB (x) (B)
w
and
with
v
d(x) = dist (x,2Q)
E O
unbounded (
S2
S2 E C
or
0'1
and
)
w(x) = da(x)
;
from some special subclasses of ) ),
S2
radial weights.
In both cases we will make use of Lemma 16.12, and therefore,
we will
C
start with some assertions about equivalent norms. First, let us consider the case (A). Here the question of equivalent norms is solved by the following lemma.
21.2. Lemma (A. KUFNER [11, Proposition 9.2).
<- 1,
K
Let
1
Let Q e
< p <
S E R . Then the norms and
111-111
1,P,Q,d
where (21.4)
1,p,S2,d8
(21.5)
=
[Jvu(xP d(x) dx]ll1/p 1/p
[Ju(x) IP ds(x) dx + I1IuIIIP,P,Q,dsl
lu
are equivalent on the space
Wp'P(Q;ds,ds)
provided
-1KPIC<s
K
we put °°
- Kp/(1 - K) _ - m
<3
1
so that the last condition reads
.J
Lemma 21.2 together with Lemma 16.12 and the results of Section 19 (see Theorems 19.10, 19.21) imply
21.3. Theorem. Let 1 < p < q < a, S E R , and either 0 < S < K (P - 1) or
1K <
0
a -
,
N
,
S
q
p
a
KS
q
p
-
N
+1 >0
+N-N+K q
+N q
>_
,
ci E C 0' K
,
0
1
0
P
N+ K>0 p
305
Then there exists a finite constant 1Jq
(
(
IJJu(x)Iq do'(x) dxj
(21.6)
such that the Hardy inequality
C
111p
(
Jjvu(x)IP d6(x) dx]
C
S2
S2
holds for every function
21.4. Theorem.
Let
W0'P(S2;ds,ds)
u c-
q
1
a E CO'K
0
,
1
,
a, RER
and either
q - P + K(q - P + 1) > 0
0 < S < K(p - 1) or
1= < S
`=
0
a
,
q
-
KS + p
Then there exists a finite constant (21.6) holds for every function If
Let
1
< p,q < m
exists a finite constant
for every function
q
.
p
such that the Hardy inequality
u E W0'P(1;d6,ds)
C
S2 E C 0'1
,
,
S < p -
1
.
Then there
such that the Hardy inequality (21.6) holds
u E W1'P(S2;d6,d6)
if and only if either
p=q<°°, N-N+1>0 q P
1
- 1 + 1) > 0
(1
then we can give necessary and sufficient conditions:
S2 E CO'1
21.5. Theorem.
C
K
a-B+N-N+1>0 q p p q
,
or 1
<= q< p <
21.6. Remark.
°°
,
q-
P+
4-
P+ 1 > 0
As was mentioned in Subsection 16.1, the Hardy inequality
provides a useful tool for deriving estimates for capacities. Using the foregoing results, we can specify the isoperimetric inequality (16.5) and obtain that for
1
s p < q <
12E C
0'1 ,
6 < p -
1
and
1P
(x,C) _
CiIP ds(x))
i1 there exists a finite constant
306
B > 0
such that the inequality
r d°`(x) dxj 1/q = B[(p,4) -cap (K,H)] 1/P K
holds
for every compact set
N_ N+1=0 q
a
,
p
if and only if
K C 12
q
- S+N p
q
N
p
+ 1>_0.
Here we have used Theorems 16.3 and 21.5. If we use Theorem 16.5, we can derive analogous results for the case Now we will consider the case bounded domains.
1
(B)
< q < p < -
.
from Subsection 21.1, i.e. un-
The following two theorems form a counterpart of Lemma
21.2.
21.7. Theorem. domain,
Let
r(a*,
0(t)t
N-1
-
, v1Wt
Then there exists a constant
(21.8)
lullp,Q,,, < C 0
for every
0 C RN \ {0}
be a non-empty unbounded
Suppose that
Vol v1 E
(21.7)
Let
< p <
1
C > 0
N-1
p, p)
<
such that
11 Vullp12,v 1
u E W0'P(1;v0,v1)
with
v0(x) =
0(Ixl)
,
v1(x) = v1(Ixl)
[and
consequently, the norms dxlll 1/p
(r
(21.9)
IIIuI111,p,12,v1
=
IJ vu(x)IP vl(x) J111
and
1/p
If u(x) 1P v0 (x) dx + 111-11Ip,
(21.10)
l
P,0,v1J
12
are equivalent on the space
Proof.
W1'P(i2;v0,v1) I.
Using spherical coordinates, the condition (21.7) and the one-
dimensional Hardy inequality, we can derive the inequality (21.8) by the same method as we have derived the first inequality in (20.42) in the proof of Theorem 20.17 from the condition (20.37). 307
0,1]
In the next theorem, we will deal with domains
o E `,D
which satisfy the condition
xE0
(21-11)
,
t>
=-j t x E 0
1
.
This class of domains will be denoted by [E,D 0, 1I
(21.12)
Let
21.8. Theorem.
1
k > 0
that there exist numbers 0 (t)
0 E Zt ,
< p <
>_ k v1(t)
t-P
Vol vl E WC(a},-) such that
t0 ? a*
and
for a.e.
t
and suppose
> t0
.
Assume that v0(t)t
BR(a*,
(21.13)
N-1
Then there exists a constant for every
-
,
v1(t)t
C > 0
with
u E W1'P(0;v0,v1)
N-1 ,
p, p)
< m
such that the inequality (21.8) holds v0(x) = v0(Ixl)
,
vl(x) = vl(IxI)
and
consequently, the norms (21.9) and (21.10) are equivalent on the space W1,P(Q;v0.v1)]
Proof.
Due to our assumptions, it suffices to prove the inequality (21.8)
only for functions
from the dense subset
u u
,
21.9. Remark.
Cbs(Q)
(cf. Theorem 20.12). For such
we proceed analogously as in the proof of Theorem 21.7.
In Theorems 21.7, 21.8 we have shown that the conditions
(21.7) and (21.13) are sufficient for the equivalence of the norms (21.9), (21.10) on
W1'P(0;v0,v1)
and
W1'P(0;v0,v1)
,
respectively. Obviously,
these conditions are also necessary if we suppose that the domain has the special form
St ={ x E RN ; with some
R = RN
r
0
.
Ixl
> r}
In the case of the space
W1'P(Q;v0,v1)
,
it can be even
.
Using Theorems 21.7, 21.8, Lemma 16.12 and the results from Section 20 (see Examples 20.6 - 20.8, Remark 20.9, Examples 20.19 - 20.21) we immediately obtain necessary and sufficient conditions for the Hardy inequality 308
1/q (21.14)
w(x) dxj
u(x)Iq
K =
to be valid on the class
lll 1/p
C[J
s
X(0)
V u(x)Ip v1(x) dx
, which will be specified in the
following examples.
21.10. Example.
Let
w(x) = Let
(i)
< p,q < W
I
a
x
S E R
xls-P
v0(x) =
0E£
a,
,
v1(x) = Ixls
,
[o C ,V0'11
a* >
,
and put ,
0 / p - N [g > p - N]
0 ,
Then the Hardy inequality (21.14) holds with a finite constant
class
K(0) = W01'p(0;v0,v1) 1
<m
p = q
[ K(0) = W1'P(Q;v0,v1) N
N+1
q
p
if and only if either
p
q
on the
C
- B+N - N+1
a
0
.
q
0
p
or
H = RN \ {0}
Let
(ii)
p - N
S
,
L 0 > p - N [. Then the Hardy in-
equality (21.14) holds with a finite constant W0'P(Q;v0,v1)
(21.15)
[ K(Q) = W1'P(H;v0,v1)
`= p
1
<=
Let
(iii)
q < -
0 = RN
9 - P + 1
,
,
K(Q) = W
1,p
(S2;v0,v1)
21.11. Example. Y,
6ER
Let ,
w(x) =
1
>-
0
4 - P + Q
,
.
p + 1 = 0
Then the Hardy inequality
on the class
C
-
K(Q) = WO'P(H;v0,v1)
if and only if the condiiton (21.15) is satisfied.
< p,q < W
S/p-N
,
H t .)
[ Q C
0'1 ],
a* >
1
,
[ S > p - N [ and put
Ixla
lnYlixl,
v0(x) =
lix
-P In Ixl,
v1(x) =
Then the Hardy inequality (21.14) holds with a finite constant
class
K(Q) _
if and only if
p - N < 0 < Np - N
(21.14) holds with a finite constant or
1
on the class
C
K(H) = WO'P(H;v0,v1)
[ K(52) = W1'P(H;v0,v1) ]
x10 lndlxI. C
on the
if and only if one of
the following two conditions is satisfied:
(i)
1
`p=q<
N_ q
v
p
+1 ?0
309
and either -
°-
q
S
+
p
N
N + 1< 0
q
p
or
+
-
(11)
+ 1 = 0
-
,
p
q
p
q
0
1
and either
a- B+ N- N+ 1< 0 q
p
q
p
or
+
-
21.12. Example.
+ 1 = 0
-
Let
p,q <
1
w(x) = ealxl
q-
, ,
,
P+
p< 0
q-
S2 E 0 ,
S E R . Put
a,
v0(x) = v1(x) = eSIxI
,
Suppose that one of the following two conditions is satisfied:
a,, > 0 (21.16)
a., = 0
0 ,
B>0
;
or
a* = 0
,
B
<0
p>N
,
Then the Hardy inequality (21.14) holds with a finite constant class
K(Q) = W01'p(Q;v0,v1)
1
=` p< q <
C
on the
if and only if either
N- N+ 1? 0,
°-
q
q
p
-
S
0
p
or
l
q
`=
°' ,
a q
B
p
<0
.
The same conditions are necessary and sufficient also if we suppose that
Q E
0,1
,
replace (21.16) by
0
and consider (21.14) on the
class
21.13. Remark.
v1(x) _
xls
The inequality (21.14) for the special case
w(x) = Ixla
appears very frequently in the literature, mostly in
connection with estimates used in the theory of partial differential equations. From the numerous results let us mention at least the paper by A. E.
GATTO, C. E. GUTIERREZ, R. L. WHEEDEN [1] who derived exactly the necessary and sufficient conditions (21.15) for
310
2 = RN \ i0}
and
K(Q) = CD(S2)
by
another method, using the theory if fractional integral operators in weighted Hardy spaces.
For the convenience of the reader, we
21.14. Concerning equivalent norms.
will summarize the conditions which guarantee the equivalence of the norms (21.9) and (21.10) for the particular weights appearing in Examples 21.10, 21.11 and 21.12. Some of these conditions can be derived directly from the Hardy inequality, since (21.8) is nothing else than a special case of (21.14) where we put B
( y
,
d
w = v0
and
q = p
.
However, the set of admissible values
a
is in fact bigger, including also values excluded apriori in the
)
assumptions of Examples 21.10 - 21.12. Suppose 1
(i)
< p <
For the weight functions v0(x) = Ixla
v1(x) = IxIs
,
a, 8 ER ,
'
the norms (21.9) and (21.10) are equivalent
(i-1)
on
if
W1'P(52;v0,v1)
02 E V ,
a* > 0
S2 E IZ
a* > 0
p- N
,
a<
S=p-N
,
a<-N
-p
or ,
,
or
2 = RN \ {0}
,
S
ie p - N
,
a=S-p
or
S E (p - N, Np - N)
t = RN ,
(i-2)
,
on
W1'P(Q;v0,v1)
c2 E 0*
,
a* > 0
a=S-p
,
if ,
S>p-N
,
a<8-p
or
02 = RN \ {0)
,
B
>p-N
,
or
Q = RN ,
S E (p - N, Np - N)
,
a=
-p
.
311
(ii)
For the weight functions v0(x) = Ixla lnylxl
v1(x) = lxls lnalxl
,
a,
,
Y,
the norms (21.9) and (21.10) are equivalent Wl,Op(St;v0,v1)
on
(ii-1)
fp
N
a < S- p
p
N
a= S- p
a* >
S2 E :0
if
1
and either
or Y
`-
6
or
p- N, a<- N or
p-
p- N, a=- N,
p
or
p - N
a = - N
W1'p(0;v0,v1)
on
(ii-2)
,
> p- N, a<
d = p
,
< - 1
0 E
if
a* >
1
;
and either
p
or > p
a=
N
p,
< d
or
p - N
,
a < - N
,
p - N
,
a = - N
,
> p- 1
or d
> p -
,
y K d- p.
For the weight functions
(iii)
v0(x) = ealx
,
vl(x) =
the norms (21.9) and (21.10) are equivalent (iii-1)
312
1
on
if
a,
S E R ,
d E R
SZ E 9)
a* > 0
,
a=S
,
(a;S) g, (0;0)
or
=
RN \ {0}
,
a<0
,
aSS
>0
,
a<S
,
p>N
or
0 = RN \ {01 or
O = RN \ {0} (iii-2)
on
0 E 43* >0
W1'p(S2;v0,v1)
a
1
1< p < N if
N>
1
if
a* > 0 or
,
p=
a<0
S=0
,
52 = RN \ {0}
or
0 = RN
and either
B
or 0
a< 0{ p1< p < N
21.15. Some extensions.
(i)
if
N>
1
In this section we have been in fact concerned
with two special types of weights depending on d(x) = disc (x,252)
or on Ix
= disc (x,{0})
.
It is possible to extend many of the foregoing results to the more general case of weights of the type (21.17)
where
v(x) = v(dM(x)) v E W(O,m)
and
dM(x) = dirt (x,M)
M C M C Q and
,
mN(M) = 0
.
(See also Example 12.10 where
M was its edge, i.e.
M C 20
but
0
was a polyhedron
M x 2Q .)
One can expect that some of the general theorems from Section 18 can be used with an auxiliary function
r = r(x)
of the type
r(x) <_ 3 dM(x)
313
or
more precisely,
r(x) < 3 min {d(x), dM(x)}
(18.7), (20.6). The dimension
m
[compare with formulas
of the manifold
m
will play some role.
Some results concerning the continuity and compactness of the imbedding W1,p(D;v0,v1) C Lq(Q;w)
with weight functions of the type (21.17) are mentioned in A. KUFNER, B. OPIC, I. V. SKRYPNIK, V. P. STECYUK [1]; the case
p = q
,
M C a0
is
dealt with in A. KUFNER [2], J. RAKOSNIK [1] and E. D. EDMUNDS, A. KUFNER, J. RAKOSNIK [1]. (ii)
In Section 10 we have investigated the Hardy inequality for higher
order derivatives in the one-dimensional case. Obviously, imbedding theorems and Hardy-type inequalities involving derivatives of higher orders (and even fractional derivatives) can again be derived for N-dimensional domains N >
1
.
Some results concerning the case
p = q
can be found in
314
,
A. KUFNER
[2]; as concerns the approach described in Sections 17, 18, cf. B. OPIC, J. RAKOSNIK [1], where also further references can be found.
0
Appendix
22. LEVEL INTERVALS AND LEVEL FUNCTIONS In this additional section, we will give the proof of HALPERIN's Theorem 9.2 which is a fundamental tool for the proof of the Hardy inequality with
0 < q <
I
.
The proof will be divided into several auxiliary
assertions. Let us start with some notation.
1
22.1. Level intervals.
For
and for
let us denote
(a,B) C (a,b)
p E W(a,b) fl L (a,b)
B(
f(a,B) =
(22.1)
B(
f(t) dt
J
,
p(a,B) =
to
)
p
J
p(t) dt
,
R(a,B) = f(a' P(a,B)
a
a
The interval
f E M+(a,b)"')L1(a,b)
and
is called a level interval (of
(a,B) C (a,b)
with respect
f
if
(22.2)
R(a,x) < R(a,s)
If the level interval then it
for every
x E (a,B)
is not contained in any larger level interval
(a,B)
is called a maximal level interval.
By
(22.3)
L=
L (a,b,f,p)
LM =
,
LM(a,b,f,p)
we denote the system of all level intervals and of all maximal level intervals
(a,B) C (a,b)
22.2. Remark.
,
respectively.
A natural question arises whether the systems
L
and
L M
can be empty or not. The answer is given by the following example.
22.3. Example. (22.4)
f(t)
Let us take -
1
,
(a,b) = (0,1)
p(t) = t
for
and
t E (0,1)
According to Subsection 22.1, the interval
(a,B)
with
0 s a < B <
1
is
315
a level interval if and only if
x
f (t) dt <
a x
(22.5)
f (t) dt
J
Jr
for every x c_ (a, B)
1 p(t) dt
p(t) dt
a
Using (22.4) we obtain after a simple calculation that (22.5) is equivalent to the inequality B < x
x E (a,B)
for every
,
which obviously cannot hold. Consequently, the system is empty for
LM
system
f
,
L
as well as the
from (22.4).
p
Moreover, the reader can easily verify that the systems are empty if the function
Let
22.4. Lemma.
f/p
(a, B) C (a,b)
is decreasing on
,
x - (a,B)
.
(a,b)
L
and
LM
.
Then the following three
conditions are equivalent: (i)
R(a,x) < R(a,B)
(ii)
R(a,x) < R(x,B)
(iii)
R(a,B) < R(x,B)
Proof.
After some elementary calculations we succesively obtain from the
definition of
R(a,B)
that the following inequalities are equivalent: R(a,x) < R(a,B)
,
p(a,B) f(a,x) < f(a,B) p(a,x) [p(a,B) - p(a,x)] f(a,x)
[f(a,B) - f(a,x)] p(a,x)
p(x,B) f(a,x) < f(x,B) p(a,x) R(a,x) < R(x,B)
Thus we have obtained that (i) 4
(ii)
.
.
Similarly we can prove that
(ii) " (iii).
22.5. Remark. 316
Obviously, Lemma 22.4 remains true if we replace the
<
sign
by the sign
<
_
,
or
,
conditions (i),
(ii) and (iii).
22.6. Theorem.
(i)
simultaneously in all three
>
Every level interval is contained in a maximal level
interval.
If
(ii)
(al,bl)
(a2,b2)
,
a < a1 < a2 < bl < b2
b
The system
(iii)
then
,
LM =
are level intervals with (al,b2)
is a level interval.
LM(a,b,f,p)
is either empty or it is a denumerable system of non-overlapping intervals. Proof.
Let
(i)
level interval
be the system of all level intervals containing the I = (a0,b0) Introduce in S a partial ordering -< by S
.
the rule 11
I2 <' I1 C 1 2
We have to show that
.
contains a maximal element. By virtue of the Zorn
S
lemma, it suffices to verify that every ordered subset
S C S
is bounded
from above.
Therefore, let
S = {I
;
IY = (aY,b ) E S
define
,
y E r} be ordered and
Y
-Y
Irf = (aM,bM) = U I Y YEr
Obviously
aM = inf
YEr {yn}, {yn} C F
a
,
M
n-)-m S
sup
b
YEr
.
Then there are two sequences
Y
such that
am = lim a Since
b m
Y
,
Yn
b
m
= lim b-
n- Yn
is ordered, we have also bM = lim b
n- Yn
If we show that
IM = (aM,bM)
is a level interval, i.e. that the impli-
cation (22.6)
X E IM -->-
R(aM,x) <= R(aM,bM)
holds, then the proof of the assertion (i) will be finished. But for
x E IM
317
x F Iyn
such that
there exists a yn E F
and since
,
is a level
Iyn
interval, we conclude that ,x) < R(a
R a ( yn
,b
)
yn
.
yn n
Now, (22.6) follows by passing to the limit
The definition of level intervals and Lemma 22.4 imply that
(ii)
R(a2,b1) = R(a2,b2) = R(bi,b2)
R(a1,a2) < R(al,bl)
.
Consequently,
R(al,a2) < R(a2,b2)
R(ai,b1) < R(bi,b2)
,
and Lemma 22.4 implies R(a1,b2) < R(a2,b2)
(22.7)
The inclusion
R(al,b1) < R(al,b2)
,
together with the second inequality in (22.7)
(al,b1) E L
yields
R(al,x) < R(al,b1)
(22.8)
Analogously, the inclusion
if
<< R(al,b2) L
(a2,b2)
E (al,b1)
Lemma 22.4 and the first in-
,
equality in (22.7) yield R(x,b2) > R(a2) b2) > R(al,b2)
if
x E (a2,b2)
,
and consequently, again by Lemma 22.4 we have R(a1,x) <_ R(al,b2)
(22.9)
if
x E (a2)b1)
The inequalities (22.8) and (22.9) immediately imply that (iii)
If
LM
is nonempty (cf. Example 22.3) then the assertion
follows from point (ii) above.
22.7. Level functions.
where
I. (1 I. = 0 1
318
If
11
LM .4 0
,
then
LM = {In = (an,bn); n = 1,2,...}
(22.10)
(ai,b2) E L
if
i # j
.
Denote
if
U In (22.11)
I =
LM x 0
n
if LM=O
0
and define the level function to
,
f0
f E M+(a,b)f L1(a,b)
[of
with respect
p e W(a,b)(1 L1(a,b) ] by the formula
fn(x) =
(22.12)
For
R(an,bn) p(x)
for
x (7- In = (an,bn)
f( x )
f or
x
{
E
(a, b)
I
\
denote
(a,R) C (a,b)
f0(a, R0(a,R) =
(22.13)
p(a,s)
The following lemma is an easy consequence of the definitions.
Let
22.8. Lemma.
of
f
and
I
n
LM = LM(a,b,f,p) it 0
= (a n,b
n
)
Let
.
f0
be the level function
a maximal level interval. Then
f0(an,x)
for
x E In
(i)
f(an,x)
(ii)
f(an,bn) = f0(an,bn)
(iii)
f(a,x)
< f0(a,x)
for
a E (a,b)
(iv)
f(a,x)
= f0(a,x)
for
a, x E (a,b) \
,
\ I
x E (a,b)
,
I
,
x > a
Proof. It follows from Subsection 22.1 that R(an,x) = R(an,bn)
for
x E I n
and consequently, by (22.12) we have
x R(an,bn) p(t) dt =
f(an,x) _` R(an,bn) p(an,x) = 1
a
n
x =
f0(t) dt = fo(an,x)
I
a
.
n
Thus (i) is proved. The proofs of assertions (ii) - (iv) are similar. 11
319
22.9. Theorem.
Let
be the level function of
f0
every level interval of
(i)
f
.
Then
is a level interval of
f
f0
,
i.e.
L(a,b,f,p) C L(a,b,f0,p)
(22.14)
every maximal level interval of
(ii)
is a level interval of
f0
f
i.e.
LM(a,b,fQ,P) C L(a,b,f,p)
(22.15)
the functions
(iii)
f
and
have the same maximal level intervals,
f0
i. e. LM(a,b,f,P) = LM(a,b,fO,P)
(22.16)
for each level interval
(iv)
f0
there exists a constant
,
such that
k = k(J)
f0(x) = kp(x)
(22.17)
(f0)0 = f0
(v)
Proof.
of
J
(i)
x e J
for
.
J = (a,s) E L(a,b,f,p)
Let
.
According to Theorem 22.6 (i)
In = (an,bn) E LM(a,b,f,p)
there is an interval
the definition of the level function
f0
such that
R(an,bn) I p(t) dt
f0(t) dt
a =
= R(an,bn) x
x
p(t) dt
I
n
]
,
and consequently,
R0(a,x) = R0(a,6)
which implies that (ii)
Let
p(t) dt
a
a x E (a,b
for every
J E L(a,b,f0,p)
x e (a,B)
.
J = (a,s) E LM(a,b,f0,p)
.
and the above proof of point (i), neither points of some level intervals of
f
.
According to Theorem 22.6 (ii) a
nor
f(a,6) = f0(a,B)
320
f0(a,x)
for every
3
can be interior
Consequently, from Lemma 22.8 (iii),
(iv) we obtain
f((X ,x)
By
x
1
for every
.
(22.12)] we have
[cf.
x
R0(a,x) =
J C In
x E (a,8)
This implies R(a,B) = R0(a,B) for every
R(a,x) < R0(a,x) J E LM(a,b,f0,p)
and since
we obtain
,
R(a,x) < R0(a,x) i.e.
L(a,b,f,p)
J
x e (a,B)
R0(a,B) = R(a,B)
for
x r J
.
(iii)
The assertion follows from (22.14), (22.15).
(iv)
Let
J E L(a,b,f0,p)
According to Theorem 22.6 (i) and to
.
In = (an,bn) C- LM(a,b,f,p)
(22.16) there is an interval
such that
J C I
n
and by (22.12)
f0(x) = R(an,bn) p(x)
This implies (22.17) with
for every
k = R(an,bn)
x E In
.
Lemma 22.8 (ii) implies that
(v)
R(an,bn) = R0(an,bn)
,
which immediately yields the assertion. LJ
22.10. Theorem.
Let
(a,B) C (a,b)
,
x e (a,8)
.
Then
R0(x,B) < R0(a,B)
(22.18)
and R0(a,B) < R0(a,x)
(22.19)
Proof.
Suppose that
(i)
a
,
a
are finite, i.e.
a,
B E R . In order to
prove (22.18), suppose on the contrary that (22.20)
R0(a,B) < R0(x,B)
The function
R0(y,B)
exists a point
x0
.
is continuous on
[a,B)
,
and consequently, there
such that
x0 = max {y e [a,x]; R0(y,B) =
min
R0(s,B)}
s E[a,x] If
x0 = x
,
then (22.20) would imply that
contradicts the definition of
x0
.
R0(a,B) < R0(x0,B)
Consequently,
x0 < x
,
which
which imme-
diately yields 321
for every
R0(x0,6) < R0(s,6)
(22.21)
s c- (x0,x]
.
Using Lemma 22.4 (and Remark 22.5) we obtain from (22.21) that R0(x0,s) < R0(x0,6)
(22.22)
R0(x0,x1) =
(22.23)
Putting
s = x
(22.23) implies
and consequently,
in (22.22) we obtain in view of (22.23) that
since the assumption
x < x1
Now
,
R0(x0,s)
max
sE[x,]
R0(x0,x) < R0(x0,6) ` R0(x0,x1)
(22.24)
[x,6]
.
such that
x1 E [x,6]
there exists a point
s E (x0,x]
is continuous on
R0(x0,s)
The function
for every
x = x1
leads to a contradiction:
while (22.22) (for
R0(x0,x) ? R0(xO,6)
R0(x0,x) < RO(x0,6)
.
implies
s = x )
.
Thus, we constructed an interval
(x0,x1)
containing the point
x
and such that (22.25)
(x0,x1) E L(a,b,f0,p)
Indeed, if
then (22.22) and (22.23) yield
s E (x0,x]
R0(x0,s) < R0(x0,6) <= R0(x0,x1)
while for
,
we have from (22.23)
s e (x,x1)
R0(x0,s) < R0(x0,x1)
.
According to (22.25), the formula (22.17) together with the definition RO
of
yields f0(x0,s)
R(x0 O,s)
=
p(x
In particular, we have (ii)
Now, let
0,
s) = k
for every
R0(x0,x) = R0(x0,x1)
(a,6) C (a,b)
s E (x0,x1]
which contradicts (22.24).
be a general interval (i.e.
possibly infinite). Then there exist two sequences that
an 1 a
,
$n f
6
x E (an,n)
and that
(i) of the proof we have RO(x,6n) < RO(an,6n) 322
,
n e N
{a
for every
n},
a
and/or
i6n } C R
n E IN
.
such
By part
and (22.18) follows by passing to the limit (iii)
n -+ m
.
The inequality (22.19) follows from (22.18) by Lemma 22.4 and
Remark 22.5.
Let
22.11. Lemma.
al < a2
((x 2'62) C (a,b)
Bl < 62
,
.
Then
R0(a2,62) < R0(a1,81)
(22.26)
Proof.
(a1,61),
Suppose that
(22.18) with
al < a2
a = al
and
6 = 62
.
x = a2
,
and
82
(22.19) with a = al
61 < 62
and
x = Bl
,
Using first the inequality and then the inequality we obtain
R0(a2,62) <= R0(a1,62)
R0(a1,62) <= R0(a1,61)
which implies (22.26). al = a2
The proof for the case
or
is similar and is left
B1 = 62
to the reader.
22.12. Theorem.
Let
be the ZeveZ function of
f0
Then there exists a non-increasing function
f
D = D(x)
with respect to on
(a,b)
p
f0(x) (22.27)
Proof.
p(x) = D(x)
For
(22.28)
x E (a,b)
for a.e.
x e (a,b)
t > 0
and
define
H(x,t) = R0(x,t1)
where
tl = min {x + t
Thus for small (22.29)
t
L 0 < t
,
<
2(x + b)}
2(b - x)
we have
H(x,t) = R0(x, x + t)
Assume that (i)
for every fixed
decreasing for (ii)
t
1 0
x E (a,b)
the function
H(x,t)
is non-
;
for every fixed
t > 0
the function
H(x,t)
.
such that
is non-increasing 323
in
x
.
According to (i), the limit of (22.30)
D(x) = lim H(x,t)
H(x,t)
for
4
t
0
exists. If we define
,
t40
for
is non-increasing on
D(x)
then the function
a < x1 < x2 < b
due to (ii). Indeed,
(a,b)
we have
H(x1,t) ? H(x2,t) and consequently,
D(xl) = lim H(x1,t) ? lim H(x2,t) = D(x2) t10
t10
On the other hand, we have D(x) = lim H(x,t) = lim R0(x,x+t) _ t10
t10
x+t
x+rt lim t10
f0 (s) ds
= Jim t40
f
x
=
x+t p(s)
lim 1
d+s
t10
x
x E (a,b)
f0(s) ds J
x x+t I
for a.e.
1
f
(x)
0 p(x)
p(s) ds
t
.
Thus, we have arrive at the formula (22.27). In order to complete the proof it remains to show that the assumptions (i), (ii) are fulfilled. (i)
(22.31)
It suffices to verify that for
H(x,t2) = H(x,t
0 < t2 < t1 < 2(b - x)
From Theorem 22.10, formula (22.19), we have R0(a,y)
Putting here
a = x
,
R0(a,B)
for
B = x
t1
R0(x, x + t2)
y E (a,8) and
y = x + t2
R0(x, x + t1)
which implies (22.31) according to (22.29). (ii)
324
We have to verify that
,
we obtain
(22,32)
H(x2,t) < H(x1t)
(ii-1)
x1 < x2
2(b - x2)
0 < t <
If
for
then
,
H(xi,t) = R0(xi, xi + t)
i = 1,2
for
and (22.32) follows from Lemma 22.11 where we put
i = 1,2
xi
a.
i
(ii-2)
t = 2(b - x1)
If
i = 1,2
,
and (22.32) follows again from Lemma 22.11 where we put
1(xi + b)
If
2(b - x2) < t
2(b - x1)
<
H(x1,t) = R0(x1, x1 + t)
,
then
< 2(x2 + b)
(22.32) follows again from Lemma 22.11 where we put
S2 = 2(x2 + b)
of
f0
p
.
,
the inequality
a1 = x1
,
F1 = x1 + t
.
22.13. Proof of Theorem 9.2. respect to
Si =
H(x2,t) = R0(x2, 2(x2 + b))
,
x1 + t < x1 + I(b - x1) = 2(x1 + b)
a2 = x2
ai = xi
i = 1,2
,
(ii-3)
Since
,
then
,
H(xi,t) = R0(xi, z(xi + b))
=
Bi = xi + t
,
.
Let
be the level function of
f0
f
with
Then the property (9.4) is a consequence of the definition
and of Lemma 22.8.
The property (9.5) is a consequence of Theorem 22.12.
Thus, it remains to prove that (9.6) holds, i.e. that b
(f 0(x)
p
b
P(x) dx `
J[ P(x),
p P(x) dx
.
a
a If
f(x
x E (a,b)
\
I
with
I
from (22.11), then
f0(x) = f(x)
.
Consequently,
it suffices to show that b
n
(22.33)
fP(x)J
Jr
a where
n
(an,bn)
b
p
n
p
p(x) dx `
P(x) dx
an
lP(x))
are the intervals from (22.10). 325
If
p =
1
then (22.33) follows from Lemma 22.8 (ii)
,
(even with the
equality sign).
Thus, suppose that we have that for
p >
1
.
From the definition of
[cf. (22.12)]
x E (an,bn) b
(22.34)
f0
cp() with
f0(x)
cn =
b
If(t) dt/ a
a
n
p(t) dt
J
n
If b
Jn -
n
P(x) dx = 0
R-(-x) )
J
a
p
(x)
n
then (22.33) holds trivially. If
Jn x 0
22.8 (ii) and Holder's inequality yield b
b
(n
Jn =
P-1
a b
n
that
cn-1 f(x) dx
a
n
then the formula (22.34), Lemma
n
f0(x) dx
J
,
n
p -1
p1/P'(x) f(x) p1/P (x) dx fP(x), a
n
b
n
1/p' ( ( n
lJ
1/P
f (x)) P p (x) dxJ
(P
(x)
a n However, this inequality implies (22.33) since b
jn
= cpn
P (X) dx < J
a
326
n
n
[1
J
n
z 0
and
References
ADAMS, D. R.: [1]
A trace inequality for generalized potentials. Studia Math. 48 (1973), 99 - 105. MR 49 # 1091
[2]
Weighted nonlinear potential theory. Trans. Amer. Math. Soc. 297 (1986), no. 1, 73 - 94. MR 88m:31011
ADAMS, R. A.: [1]
Sobolev spaces. Academic Press, New York-San Francisco-London, 1975.
MR 56 # 9247 ANDERSEN, K. F.; HEINIG, H. G.: [1]
Weighted norm inequalities for certain integral operators. SIAM J. Math. Anal. 14 (1983), no. 4, 833 - 844. MR 85f:26012
ANDERSEN, K. F.; SAWYER, E. T.: [1]
Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators. Trans. Amer. Math. Soc. 308 (1988), no. 2, 547 - 558.
BEESACK, P. R.: [1]
Hardy's inequality and its extensions. Pacific J. Math. 11 (1961), 39 - 61. MR 22 # 12187
[2]
Integral inequalities involving a function and its derivatives. Amer. Math. Monthly 78 (1971), 705 - 741. MR 48 # 4235
BENSON, D. C. [1]
Inequalities involving integrals of functions and their derivatives. J. Math. Anal. Appl. 17 (1967), 293 - 308. MR 34 # 2809
BERGH, J.; LOFSTROM, J.: [1]
Interpolation spaces. An introduction. Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 58 # 2349
BLISS, G. A.: [1]
An integral inequality. J. Math. Soc. 5 (1930), 40 - 46.
BRADLEY, J. S.: [1]
Hardy inequalities with mixed norms. Canad. Math. Bull. 21 (1978), no. 4, 405 - 408. MR 80a:26005
327
BROWN, R. C.; HINTON, D. B.: [1]
Weighted interpolation inequalities of sum and product form in Proc. London Math. Soc. (3) 56 (1988), 261 - 280.
[2]
Weighted interpolation inequalities and embeddings in in Canad. J. Math.).
Rn
Rn
(to appear
BURENKOV, V. I.: [1]
Mollifying operators with variable step and their application to approximation by infinitely differentiable functions. Nonlinear analysis, function spaces and applications, Vol. 2 (Pisek 1982), 5 - 37, TeubnerTexte zur Math., 49, Teubner, Leipzig, 1982. MR 84a:46068
CHIARENZA, F.; FRASCA, M.: [1]
A note on a weighted Sobolev inequality. Proc. Amer. Math. Soc. 93 (1985), no. 4, 703 - 704. MR 86h:46052
DUNFORD, N.; SCHWARTZ, J. T.: [1]
Linear operators. I. General theory. Interscience Publishers, Inc., New York-London, 1958. MR 22 # 8302
DZHALILOV, K. A.: [1]
On an imbedding theorem with weight (Russian). Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 7 (1986), no. 4, 25 - 31. MR 88f:46071
EDMUNDS, D. E.; KUFNER, A.; RAKOSNrK, J.: [1]
Embeddings of Sobolev spaces with weights of power type. Z. Anal. Anwendungen 4 (1985), no. 1, 25 - 34. MR 86m:46032a
EVANS, W. D.; RAKOSNTK, J.: [1]
Anisotropic Sobolev spaces and quasidistance functions. Preprint no. 55, Math. Inst. Czech. Acad. Sci., Prague, 1989.
FABES, E. B.; KENIG, C. E.; SERAPIONI, R. P.: [1]
The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), no. 1, 77 - 116. MR 84g:35067
GARCrA-CUERVA, J.; RUBIO DE FRANCIA, J. L.: [1]
Weighted norm inequalities and related topics. North-Holland Math. Studies, 116, North-Holland Publishing Co., Amsterdam-New York 1985. MR 87d:42023
GATTO, A. E.; GUTI$RREZ, C. E.; WHEEDEN, R. L.: [1]
328
Fractional integrals on weighted Hp spaces. Trans. Amer. Math. Soc. 289 (1985), no. 2, 575 - 589. MR 86k:42037
GRISVARD, P.: [1]
Espaces intermediaires entre espaces de Sobolev avec poids. Ann. Scuola Norm. Sup. Pisa 17 (1963), 255 - 296.
GURKA, P.: [1]
Generalized Hardy's inequality. Casopis Pest. Mat. 109 (1984), no. 2, 194 - 203. MR 85m:26019
[2]
Generalized Hardy's inequality for functions vanishing on both ends of the interval (to appear in Analysis).
GURKA, P.; KUFNER, A.: [1]
A note on a two-weighted Sobolev inequality (to appear in Banach Center Publ.).
GURKA, P.; OPIC, B.: [1]
Ar-condition for two weight functions and compact imbeddings of weighted Sobolev spaces. Czechoslovak Math. J. 38 (133) (1988), 611 - 617.
[2]
Continuous and compact imbeddings of weighted Sobolev spaces I, II, III. Czechoslovak Math. J. 38 (133) (1988), no. 4, 730 - 744; 39 (134) (1989) 78 - 94; to appear. MR 89j:46034
DE GUZMAN, M.: [1]
Differentiation of integrals in Rn Lecture Notes in Math., Vol. 481, Springer-Verlag, Berlin-New York, 1975. MR 56 # 15866 .
HALPERIN, I.: [1]
Function spaces. Canad. J. Math. 5 (1953), 273 - 288.
HARDY, G. H.: [1]
Note on a theorem of Hilbert. Math. Z. 6 (1920), 314 - 317.
HARDY, G. H.; LITTLEWOOD, J. E.; POLYA, G.: [1]
Inequalities. University Press, Cambridge, 1952. MR 13 # 727
HEINIG, H. P.: [1]
Weighted norm inequalities for certain integral operators II. Proc.Amer. Math. Soc. 95 (1985), no. 3, 387 - 395. MR 87h:26027
HORMANDER, L.: [1]
Linear partial differential operators. Springer-Verlag, HeidelbergBerlin-New York, 1963.
KADLEC, J.; KUFNER, A.: [1]
Characterization of functions with zero traces by integrals with weight functions I, II. asopis Pest. Mat. 91 (1966), 463 - 471; 92 (1967), 3430, 5924 16 - 28. MR 35 1/
329
KOKILASHVILI, V. M.: [1]
On Hardy's inequalities in weighted spaces (Russian). Soobsc. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 37 - 40. MR 811:26014
KUDRYAVTSEV, L. D.: [1]
Direct and inverse imbedding theorems. Applications to the solution of elliptic equations by variational methods. Translations of Mathematical Monographs, Vol. 42, Amer. Math. Soc., Providence, R. I., 1974. MR 49 # 9618, 33 # 7838
KUFNER, A.: [1]
Imbedding theorems for general Sobolev weight spaces. Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 373 - 386. MR 40 # 6253
[2]
Weighted Sobolev spaces. Teubner-Texte zur Math., 31, Teubner, Leipzig, 1980 (first edition); J. Wiley and Sons, Chichester-New York-BrisbaneToronto-Singapore, 1985 (second edition). MR 84e:46029, 86m:46033
KUFNER, A.; HEINIG, H. P.: [1]
Hardy's inequality for higher order derivatives (Russian) (to appear in Trudy Mat. Inst. Steklov).
KUFNER, A.; JOHN, 0.; FUZ`IK, S.: [1]
Function spaces. Academia, Prague, Noordhoff International Publishing, Leyden, 1977. MR 58 # 2189
KUFNER, A.; OPIC, B.: [1]
Some imbeddings for weighted Sobolev spaces. Constructive function theory '81 (Varna, 1981), 400 - 407, Bulgar. Acad. Sci., Sofia, 1983. MR 85d:46042
[2]
How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25 (1984), no. 3, 537 - 554. MR 86i:46036
KUFNER, A.; OPIC, B.; SKRYPNIK, I. V.; STECYUK, V. P.: [1]
Sharp embedding theorems for a class of weighted Sobolev spaces (Russian). Dokl. Akad. Nauk Ukrain. SSR Ser. A (1988), no. 1, 22 - 26. MR 89e:46038
KUFNER, A.; TRIEBEL, H.: [1]
Generalizations of Hardy's inequality. Confer. Sem. Mat. Univ. Bari 156 (1978), 1 - 21. MR 81a:26014
LANDAU, E.: [1]
A note on a theorem concerning series of positive terms. J. London Math. Soc. 1 (1926), 38 - 39.
LEE, K. C.; YANG, G. S.: [1]
330
On generalization of Hardy's inequality. Tamkang J. Math. no. 4, L09 - 119. MR 881:26047
17 (1986),
LEVINSON, N.: [1]
Generalizations of an inequality of Hardy. Duke Math. J. 31 (1964), 389 - 394. MR 30 # 2111
LEWIS, R. T.: [1]
Singular elliptic operators of second order with purely discrete spectra. Trans. Amer. Math. Soc. 271 (1982), no. 2, 653 - 666. MR 84a:35215
[2]
A Friedrichs inequality and an application. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 185 - 191. MR 86i:35111
LIZORKIN, P. I.; OTELBAEV, M.: [1]
Imbedding and compactness theorems for Sobolev type spaces with weight I, II (Russian). Mat. Sb. (N. S.) 108 (150) (1979), no. 3, 358 - 377; 112 (154) (1980), no. 1 (5), 56 - 85. MR 80j:46054, 82i:46051
MAZ'JA, V. G.: [1]
Sobolev spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1985. MR 87g:46056. (An extended and improved version of the books Einbettungssatze fur Sobolewsche Raume 1, 2, Zur Theorie Sobolewscher Raume, Teubner-Texte zur Math., Vols. 21, 28, 38, Teubner, Leipzig, 1979, 1980, 1981
MUCKENHOUPT, B.: [1]
Hardy's inequality with weights. Studia Math. 44 (1972), 31 - 38.
MR 4741418 [2]
Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207 - 226. MR 45 # 2461
MYNBAEV, K. T.; OTELBAEV, M. 0.: [1]
Weighted function spaces and the spectrum of differential operators (Russian). "Nauka", Moscow, 1988. MR 89h:46036
NECAS, J.: [1]
Les methodes directes en theorie des equations elliptiques. Academia, Prague and Masson, Paris, 1967. MR 37 # 3168
NIKOL'SKII, S. M.: [1]
Approximation of functions of several variables and embedding theorems (Russian). Second edition, "Nauka", Moscow, 1977. MR 81f:46046
OINAROV, R.: [1]
Density of smooth functions in weighted spaces and weighted inequaliti (Russian). Dokl. Akad. Nauk SSSR 303 (1988), no. 3, 559 - 563.
OPIC, B.: [1]
N-dimensional Hardy inequality. Theory of approximation of functions, Proceedings (Kiev 1983), 321 - 324, "Nauka", Moscow, 1987.
331
[2]
Necessary and sufficient conditions for imbeddings in weighted Sobolev spaces. asopis Past. Mat. 114 (1989), no. 4, 165 - 175.
OPIC, B.; GURKA, P.: [1]
N-dimensional Hardy inequality and imbedding theorems for weighted Sobolev spaces on unbounded domains. Function spaces, differential operators and nonlinear analysis, 108 - 124. Pitman research notes in mathematics series, 211, Longman Scientific and Technical, 1989.
OPIC, B.; KUFNER, A.: []]
Weighted Sobolev spaces and the N-dimensional Hardy inequality (Russian). Imbedding theorems and their application to problems of mathematical physics, 108 - 117, Trudy Sem. S. L. Soboleva, No. 1, 1983, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1983. MR 85j:46053
[2]
Some inequalities in weighted Sobolev spaces. Constructive theory of functions '84 (Varna, 1984), 644 - 648, Bulgar. Acad. Sci., Sofia, 1984.
[3]
Remark on compactness of imbeddings in weighted spaces. Math. Nachr. 133 (1987), 63 - 70. MR 88m:46041
OPIC, B.; RAKOSNIK, J.: [1]
Estimates for mixed derivatives of functions from anisotropic SobolevSlobodeckil spaces with weights. Preprint no. 54, Math. Inst. Czech. Acad. Sci., Prague, 1989.
PACHPATTE, B. G.: [1]
On some extensions of Levinson's generalizations of Hardy's inequality. Soochow J. Math. 13 (1987), no. 2, 203 - 210. MR 89i:26018
PORTNOV, V. R.: [1]
L(1)(S2 x R+) Two imbeddings for the spaces and their applications (Russian). Dokl. Akad. Nauk SSSRp'b155 (1964), 761 - 764. MR 28 4243 1
RAKOSNIK, J.: [1]
On imbeddings of Sobolev spaces with power-type weights. Theory of approximation of functions, Proceedings (Kiev, 1983), 505 - 507, "Nauka", Moscow, 1987
SAWYER, E. T.: [1]
Weighted inequalities for the two-dimensional Hardy operator. Studia Math. 82 (1985), no. 1, 1 - 16. MR 87f:42052
[2]
A weighted inequality and eigenvalue estimates for Schrodinger operators. Indiana Univ. Math. J. 35 (1986), no. 1, 1 - 28. MR 87m:35164
SEDOV, V. N.: [1]
332
Weighted spaces. The imbedding theorem (Russian). Differentsial'nye Uravneniya 8 (1972), 1452 - 1462. MR 46 # 9715
SHUM, D. T.: [1]
On a class of new inequalities. Trans. Amer. Math. Soc. 204 (1975), 299 - 341. MR 50 # 10183
SINNAMON, G.: [1]
A weighted gradient inequality. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 329 - 335. (See also: Weighted Hardy and Opial type inequalities to appear.)
[2]
Operators on Lebesgue spaces with general measures. Thesis, McMaster University, 1987, 149 pp.
STECYUK, V. P.: [1]
Thesis, Inst. of Appl. Math. and Mech., Akad. Nauk. Ukrain. SSR, Donetsk, 1986.
STEPANOV, V. D.: [1]
Two-weighted estimates for Riemann-Liouville integrals. Preprint no. 39, Math. Inst. Czech. Acad. Sci., Prague, 1988.
[2]
Two-weighted estimates for Riemann-Liouville integrals 1, 2 (Russian). Preprints, Akad. Nauk SSSR, Far East Branch, Comput. Center, Vladivostok, 1988.
SYSOEVA, F. A.: [1]
Generalizations of a certain Hardy inequality. Izv. Vyss. UEeb. Zaved. 240 Matematika 6 (49) (1965), 140 - 143. MR 33 1
TALENTI, G.: [1]
Una diseguaglianza integrale. Boll. Un. Mat. Ital. (3) 21 (1966), 25 - 34. MR 33 # 4220
[2]
Sopra una diseguaglianza integrale. Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 167 - 188. MR 36 # 1600
TOMASELLI, G.: [1]
A class of inequalities. Boll. Un. Mat. Ital. (4)
2 (1969), 622 - 631.
MR 41 # 411 TREVES, F.: [1]
Relations de domination entre operateurs differentiels. Acta Math. (101 (1959), 1 - 139.
TRIEBEL, H.: [1]
Interpolation theory, function spaces, differential operators. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 801:46023
ZYGMUND, A.: [1]
Trigonometric series. Vols. I, II. Second edition, Cambridge University Press, London-New York, 1968. MR 38 # 4882
333
