Weighted inequalities of Hardy type

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UNIVERSIDAD COMPLUTENSE

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WEIGHTED INEQUALITIES OF HARDY TYPE

Alois Kufner Academy of Sciences, Czech Republic

Lars-Erik Persson Lulel University of Technology, Sweden

R,- 5? I Dtl

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This book is dedicated to ZLATA and LENA for their understanding, patience and long-lasting support

Contents

Preface

xi

Conventions and Notation

xv 1

Introduction

Chapter

1

Hardy's Inequality and Related Topics

1.1 Weighted Lebesgue Spaces and the Hardy Operator 1.2 Ha^rdy's Inequality With Derivatives 1.3 Some Notation and Modifications L.4 Hardy's Inequality for Some Special Classes 1.5 1.6 1.7 1.8 1.9 1..1.0

11 11

22 25

of Fbnctions The Role of the Interval Compactness of the Hardy Operator Some Limiting Inequalities - Preliminary Results Limiting Inequalities - General Results

27

Miscellanea

51

Comments and Remarks

59

31

35 42 45

viii

Contents.

Chapter

2.I

2

Contents ix

Some Weighted Norm Inequalities

Preliminaries

65

2.2 A Special Operator 2.3 General Hardy-type Operators. The F\rndamental Lemma

2.4 General Hardy-type Operators. The Case p ( 2.5 General Hardy-type Operators. The Case p ) 2.6 Some Modifications and Extensions 2.7 Comments and Remarks Chapter

3

4.I

4

g g

The Hardy-Steklov Operator

Higher Order Hardy Inequalities

Preliminaries

4.2 Some Special Cases I 4.3 The General Case 4.4 Some Special Cases II 4.5 Reducing the Conditions 4.6 Overdetermined Classes (k: 1) 4.7 Overdetermined Classes (k > 1) 4.8 Overdetermined Classes (Another Approach) 4.9 Again the Interrial (0, *) 4.10 Comments and Remarks

Chapter

5.1

5

Flactional Order Hardy Inequalities

Introduction

67 77

3.1 Introduction 3.2 Some Auxiliary Results 3.3 The Case p < q 3.4 The Case p ) q 3.5 Some Applications 3.6 Some Generalizations and Extensions 3.7 Comments and Remarks Chapter

65

89 99

tt4 116

119 119 L27

5.2 An Elementary Approach. The Unweighted Case 5.3 The General Weighted Case 5.4 Hardy-type Inequalities and Interpolation Theory 5.5 F\rrther Results 5.6 Comments and Remarks Chapter 6 Integral Operators on the Cone of Monotone F\-rnctions

6.1 6.2 6.3 6.4 6.5

Introduction The Duality Principle of Sawyer Applications of the Duality Principle More General Integral Operators Comments and Remarks

248 254 275

284 295

301 301

303 313

322 331

L29

References

335

136

Index

355

t47 155

162

165 165

t67 L74 191

196 199 207

221 237

242

246 245

Preface

oAll analysts spend half their time bunting through the litera' ture for inequalities which they want to use a,nd cannot prove'" G. H. Hardy in his Presidential Address at the meeting of the London Mathematical Society, November 8, 1928, quoting Harald Bohr.

Inequalities are an essential part of virtually all areas of mathematics and there is no doubt about their importance and usefulness in various applications. This is of course reflected in the vaste literature that exists on the subject. To many readers, inequalities exhibit a certain elegance and beauty and since they are also of great independent interest, they may be viewed as the eaergreen of mathematics. This book is devoted to a particular field in the big picture of inequalities: of integral inequalities in weighted Lebesgue spaces. These weighted integral inequalities are generalizations of those given in the fundamental work of G. H. Hardy and his contemporaries in the early 1920's. During the last three decades, the study of Hardy or Hardytype operators focussed on the characterizations of weights, for which such operators are bounded on weighted Lebesgue spaces. These re sults are of interest and importance - not only because the mappings

xii

Prcfou n\i

Weightd Inequalities of Hardy fupe

are optimal in the sense that the size of the weight classes cannot be improved - but also because the weight conditions themselves are of intrinsic interest. This intensively investigated area of mathematical analysis resulted in the publication of numerous research papers and also some books. We mention the monograph of B. Opic and A. Kufner, Hard,y-type inequalities (quoted here frequently as [OK] in the bibliography) because its theme is precisely the weight characterizations of such operators and their relationship to the study of weighted Sobolev spaces. Hence it may be considered to be the precursor of the present book - or perhaps the present book may be seen as a continuation of that work on Hardy's inequality. The classical Hardy inequality is also the starting point in our considerations, and the results, namely the characterization of weights for which the Hardy operator together with its conjugate is bounded on weighted Lebesgue spaces, are referred frequently. Consequently we start in Introduction with a brief survey of classical and "modern" forms of Hardy's inequality including also some historical remarks. In particular, some special cases (with power weights) of special interest for many applications are pointed out. In Chapter 1 some complementary results and information are glven, e.9., some alternative

criteria for Hardy's inequality to hold, the inequality in difiercntiol form, the role of the interval of integration and limiting (CarlemanKnopp type) inequalities are included. The natural extension of the Hardy operator - called here Hardy-type operator - is investigated in Chapter 2. Special cases of such operators are the Riemann-Liouville fractional integral operator of order greater or equal to one, and its conjugate, the Weyl fractional operator. The weight cha,racterization for which these operators are bounded on weighted Lebesgue spaces is deduced from the general results. Chapter 3 considers the Hardy-Steklov operator, a variant of the Hardy operator. Here various applications and weight characterizations are given, and in particular it is observed that the weight classes for which this operator is bounded are strictly larger than those for which the Hardy operator is bounded. Chapter 4 deals with the weighted Hardy inequality in differential form. Here the boundary conditions are of importance. In fact they determine the weight classes in a fundamental way.

In Chapter 5 the fractional order Hardy inequality is considered in a weighted setting. The results given here relate to those given in Chap ters 3 and 4, and in addition exhibit some interesting connections with the theory of interpolation in Banach function spaces. In order to study mapping properties of classical operators on Lorentz spaceg is necessary to consider Hardy-type operators defined on decreasing functions. Chapter 6 is concerned with weight cha.racterizations, where the operator is defined on monotone functions. Here the key result is a duality theorem from which the characterizations follow. This book is not intended to provide a comprehensive treatment of the mapping properties of Hardy-type operators. We do not consider systematically higher dimensional generalizations, the corresponding discrete results, nor weight characterizations for which this operator is bounded on weighted amalgams, Orlicz, modular or ideal spaces. Our intension is to provide the readers with a basic overview on the subject and to introduce them to the complexity of the arguments which generate the results. Those readers who wish to pursue the subject further may consult the Comments and Remarks at the end of each chapter which contain some applications and suggestions for further study. Although care has been taken to include all relevant material, certain aspects, such as questions of sharp constants, limiting cases of the parameters involved, have usually been omitted to avoid very cumbersome technical arguments. How to read the book: Each chapter is divided into sections which run in sequence through the particular chapter, similarly as the numbers of formulas. Hence, 4.5 is the fifth section of Chapter 4 and (4.20) is the twentieth formula of Chapter 4. For the convenience of the reader, we added a list of symbols, notation a.nd conventions, although, not all notions - in particular for the operators - are used systematically and consequently throughout the book. As indicated above, the aim of the authors has been to collect and present certoin of the new results in this fascinating area without any attempt to give an exhaustive picture (which, in fact, seerrur to be even impossible under our limiting capacity). Naturally, many results of other authors have been included and thus, concerning

it

xiv

Weighted Inequalities ol Hordy

I\pe

certain parts of the book, the authors feel to be more eilitors tha,rr authors. Therefore, it is our duty to thank our colleagues who contributed in the preparation a^nd finalization of this toct. First of all, it is unevitable to mention the contribution of Prof. Hans P. Heinig (McMaster university, Hamilton, canada) which wasi very substaniiut. tti" big influence at this text is particularly obvious in Chapter 3 and in parts of ChaPter 6. It is also a pleasant duty to tha,nk our colleagues who have read some preliminary versions of the ma'nuscript carefully and made comments and suggestions which have improved the mate rial and presentation of this work. we want to express our gratitude to Dr. Lubos Pick and Dr. Petr Tomiczek, to Prof. vladimir D. Stepanov, Prof. viktor Burenkov and Prof. Vakhtang Kokilashvili' and to Dr. Anna Wedestig, Dr. Sorina Barza, Dr' Maria Na'syrova, Dr. Pankaj Jain and Dr. Alexandra cizmesija. The expertize patience and good cheer of Ms Eva Ritterov6 (Mathematical Institute, Czech Academy of Sciences, Prague) who typed the manuscript was invaluable and we are endebted to her. Finally we thank the czech Academy of sciences, the swedish Royal Academy of sciences a^nd the Lule& university of Technolory for their (not only financial) support.

Conventions and Notation

Conventions a

Prague,

Lulei

A. Kufner, L.-8. Persson

All functions

are assumed to be reol-uolued. functions are functions measurable and positive almost everywhere (a.e.).

a Weight

CrCo,Ctr.. . denote positiue constants whose exact value is not important. For two positive (non-negative) quantities A, B, A S B means that A ( cB with some c ) 0, AZ B means that A ) cB with some c > 0, A x B means that simultaneously A S B a.nd A ) B. Expressions of the form

0.oo, 0/0,

m/x

are taken uqoul to ,"ro. 1s is the characteristic function of the set E, i.e.

Xa(t)=t if x€.8, xe(r):0 if s4E.

xvi o

Weighted Inequolities of Hordy fupe

Conuentiorts ond

Decreasing means ttnon-increasing"l Increosing means tt non-decreasingtt

r

(otherwise, we use the expression stri,ctly decrcosing and strictly increosing).

T ; X -+ Y

4: A:

A(o,b;u,u) A(a,b;u,u)

A'rA* orbrcrdrArBrCrD PrQrTr "'

is continuous (bounded), i.e., that

pt:*

llull.76

for every u € X.

xrtid'

Constants and Parameters

means that the operator ? maps the set (space) X into the set (space) I/. X .-+ Y means that the imbedding of the normed linear space X into the normed linear space (i.e., the identity operator I : X -+ Y)

ll"lly 5

Nototion

d:h

for0
some special integers some parameters from (0,

-)

forp:1

k(p,q)

II

Notation

o tr\rnctions and Measures

o Sets

E

the real line (-oo, oo) the half-line (0, -) the set of positive integers the lc-tuple (0, 1, . .. ,k - 1), tc € N the set of integers the open interval the closed interval half-open intervals the set of functions with absolutely continuous derivatives of order rt - I : ACo functions from ACk-r satisfying some special boundary conditions the incidence matrix

E(l,t)

:{r€(0,*):f(r)>t}

Mo,Mt N

subsets of

R

R+ N N1 //,

(o,b) [o, b]

[a, b), (o,

D]

ACK-I AC

ACk-r(Mo,Mi

Nj

the set of functions with zero mean nalue

ft f+ fo t'

adecreasingfunction an increasing function level function of / decreasing rearrangement of

f**

),y aoi@ 6,

/

distribution function of / a positive measure on (0, oo) the Dirac function

o tr\.rnction Spaces

Cl(0,oo) Cff(0, oo)

the space of functions with continuous derivative the space of infinitely differentiable functions with compact support in (0, oo) a weighted Lebesgue space the dual space of L"(.) the norm in Z"(u.') :,"(1) the "usual" Lebesgue space : L"{w") the Lorentz space a weighted Lorentz space

L"(.) (L"(r)). ll. ll",,, L" L"(-) U,8 Ito(-) Wl'o@) f some special

Sobolev spaces

w'd'@) [ with the norm llgll = lly'llp"

xviii

Weighted Inequolities ol Hordy Tlpe

o Operators G

the geometric mean operator

H H

the Hardy operator (fffXt) f G)at the conjugate Hardy operator @ il@) = 1| 767at the Hardy averaging operator (H"f)(r): I f $)dt a modified Hardy operator a modified Hardy operator a generalized Hardy averaging operator the conjugate operator to H. the generalized Hardy-type operator

(G/X")

Ho

.?r -?3

H-

fr. K

k K"

k" M

sl '9"

: exp(t Jf tn11t;at1

ff

:

[d k(r,t)f (t)dt the conjugate operator to K : [d k'(r,t)f (t)dt the conjugate operator to K, (x

f)(r)

the manimal function the moving averaging operator the Steklov operator

Classical forms of the Hardy inequality In a note published in 1920, G. H. HARDY [2] stated (without proof) that if o)0, f(")>-0,p) l and Ji f@)ds isconvergent, then

s3T

lntroduction

: [i

the Hardy-Steklov operator

Qf)@): #,9i f Q)dt

Duality

(g,fl:tyltysi

I"* (:

l,' rc,,)'*=(h)'

I"-

(0.1)

!p(r)dx

1925 [3] he wrote: "I did not give a proof, being occupied primarily with the corresponding theorem for infinite series." His main aim was to find a new, more elementary proof of Hilbert's inequality for double series, and he showed in [3] that in fact this inequality follows from the discrete version of (0.1):

In a note published in )at

(g,fl,,: tf@g(t)w(t)dt

duality another duality

oo /., n {J

€

/ _ rp t Io* I s (4 )' I"o* 7=t\n7:, / \P-rl A' \P

(o,, > 0,p

>

1). (0.2)

In fact, G. H. HARDY had dealt with inequality (0.2) earlier (see [1]) after a communication with M. RIESZ, but without the best constant (in his estimate, the constant

If

l@

-

1)]p appears).

2

Weighted Ineqnlities of

Hody fupe

Introductian

FinallS in his lg2snote [g] G. H. HARDY wrote: ,,In a letter

dated 21 June 1921, Prof. LANDAU communicated to me a direct proof of the theorem [i.e., of (0.1)] which gives the correct value lp/@ - 1)]p. H" also pointed out that, if the integral theorem were extended to the case o : 0, then the theorem for series, with the correct constant, may be deduced at once by taking (r): or (0 < f

f(r): oz (L <, <2),...." Hence G. H. HARDYstated and proved the following inequality: Suppose that t@) ) 0, p > l; that f is integroble ouer any finite interval (0,X) and fn is integrable ouer (0,o). Then :r < 1),

I,- (: l,' rav,)' *

=

(h)' I,* rp(z)da.

I,- (: lo' rav,)'x€dn. (-=)' valid

-

with

During the last decades, inequality (0.a) wa.s extended to the form

(I' ([ -

"

(1"'

ran@)tu)

(0 6)

c, b real numbers satisfying

oo(o(b(cre, -

u,a weight functions, i.e. measurable functions positive a.e. in the interval (c,D), p,q real parameters satisfying

0 0 il and only i!

able functions .f

A
(0.7)

where

fo*

rotd*'*

(0.4)

p ) I and € < p - 1 - for all measurable non_ / (see [HLp, Theorem 330]), where the constant

tp/b - I - e)ln is the best possible- Let us also mention the following ducl inequality (which can be easily derived from (0. )):

(-+-)" /- rp(r)r"dr; (0.b) ,'d, s-\e*t_p1 l^* (: [_* ravr\P Jo \c"/c / Jo - with p > 1 and e > p-l- for all measurable non-negative functions / and the constant fp/(e+t-dlp is sharp (see again iHle,

, ,:.::lo for the

(1,' "toor)"'

(I'

i-p,(t)dt)

,

f =#

(0.8)

ca^se

llp<{(m, and

o,:

r/r

(1"' (1,'

'av,)'" (I"

,,-,'r,ror)''o' ,t-n'6)dx)

for the case

holds

Theorem 330]).

r(qd,) u@)d-),o .

with

-

negative functions

it

Modern forms of the Hardy inequality

(o.s)

This is the original form of Hardy's integrolinequality which later on has been extensively studied and used as a model example for the investigation of more general inequalities. Very shortly after G. H. HARDy's proof of (0.3), the frrst ueighted, modification appeared, namely the famous inequality

3

with|:*-i.

0
(0.e)

4

Weighted Inequalities oJ Hordy fupe

Inhodvction

5

The dual inequality which is an extension of (0.5) has the form

(1"'

{1,'

p1o')o

'@)*)"0

'

c (!"'

it

is known that (0.10) holds for all measurable functions orul only if

Acm

f > 0 ,f

(see [OK, Section 1])

with s : qlp

r,: for the case

(1"' ,toor)''' (l.u *-e'$)dt) .::?u

I 1p 1 q (

#)'" (t*,U)''"'

or, for p 1q,

r(:) rg-'-f P'Q): -.1"'o frg- 1-yr61

(0.11)

1 (see V. M. MANAKOV [1]). These estimates remain true also for the constant C in (0.10) with

where

u,:

k(p,d: (r*

row(')d') " , 1o''o;

/ (0.12)

-

.4 replaced by

,f.

Remark 0.1. In this book, we primarily consider m, 0 ( g ( oo, omitting the limiting values p

oo, and

ur-o'14dr)"",0'", (1"' (1"' 'too')''o (l,o "-e'@dt)

for the case 0 < q
the cases l. < p

: 1,F, e :

(

oo.

The reader can easily reformulate the corresponding results, using appropriately the definition of the corresponding Lebesgue spaces (and their duals). We are not considering systematically the problem of the 6est constants in these inequalities. All constants C are assumed positive and finite and the only information given is that C is comparable to A (C A), where A is given by (0.8) or (0.9), or that C is comparable to A = (C t A) where A is given by (0.12) or (0.13).

Best constants

Shortly about the history

The best constant C in (0.6) satisfies

A
(0.14)

and

ot" (+)''o'

ot c sqrh@)r/q' A for q
(0.15)

The constant k@,q) in (0.14) appears in various forms.'For example,

k(p,q)

:

pr/c(rl)r/P'

or

k(p,q)

= orhlot'f/r'

Since the Hardy inequality was proved by G. H. HARDY, a vast amount of literature exists on the inequality, its nariations a,nd gen-

eralizations. Therefore a complete list on the subject can not be grven. We refer to the books [HLP], [MPF] and [OK] and mention only some results. The characterization of the weights u, u for which the Hardy inequalities (0.6) and (0.10) hold in the range

p-q>L goes back to 1969 (G. TOMASELLI [rJ a,nd c. TALENTI [U) and to 1972 (8. MUCKENHOUPT [1] who gave a nice and simple

6

Weighted Inequalities ol Hadg fupe

direct proof);

**y

Inbvdtction

authors also refer to the "untitled and unpub

lished manuscript" of A.

ARIOLA. The

was investigated independently

in 1g78 by J. s. BRADLEY [1] and V. S. KOKILASHVILI [1] a,nd (maybe e.,ur, *"liur) by V. G. MAZ'JA and A. L. ROZIN (ro V. G. MAZ'JA [t]). These last 1979 by

two authors also characterized the

Necessary and sufficient conditions on the weights

in the

case

(n2) for 1 < q

[1J.

Remark 0.2. In applications, most frequently power weights (i.e., weights of the form r), ) e R) appeax. For the convenience of the

Example 0.3. (i) The inequality

I


(0.16)


holds for 1 < p

roat)q
r"o,)"0

oo

([

(1,"@o')

/ rb rr/P scll fn@)xqdx) \J0 ) "d')

g>P-l'c2Bt-1-t pp

-t.

(0.17)

="(I*

y61,aa,)'/'

9-1;

(0.24)

(0.25)

(ii-2) tor 1 < g < p < oo if and only if (0.r8)

if and only if

g>p-1,0:P;-f,-t.

1o.zay

or

The inequality

(1,* (1,*

The inequaliry

(0.22)

with0

(ii)

yo1,1*aa,)'/o

(if)

(0.21)

if

g>p-1,crBl-#-r.

reader, we include now some examples. The given necessary a,nd suf_ ficient conditions of the validity of these inequalities can be easily verileJ by checking whether the corresponding numbers ,4 from (0.g) and (0.9) or .A from (0.12) and (0.13) are finire.

(I*

(0.20)

with0<6<mholds (i-1) for I

0
go back to the l98Z thesis by G. SINNAMON

holds for

hold,.

/ fb/ f" \q \l/c / rb rr/p yo(t)xlaa) (/, (/, f(t)dt) rdr) =r(/,

ocq<1,p>1

="

> q the inequalities (0.16) and (0.1g) do not

Example 0.4. (i) The inequality

ca.se

1
(1,* (1," r(qat)q r'0,)"0

For p

(See also [OK, Example 6.fl. Compare the results for p = g with the inequalities (0.4) a,nd (0.b).)

1
(iii)

case

7

fl>P-1, c 2P!-\-t pp

(0.26)

or (0.1e)

A-1.

(0.27)

8

Weighted Inequolities ol

Hady fupe

Introdtction

Let us emphasize the influence of the finiteness of the right endpoint b: in comparison with Example 0.4, the set of admissible values of. a, p is substantially richer.) (See also [OK, Example 6.8].

Example 0.5. (i) The inequality

(1"* (1"" r@at)q ro,)''o

,"

(1"* vo1,'1,0a,)t/e

=Bl_f,_t

(0.2e)

(i-2) for L < p 19 < oo if and only

(I' /e(c)oe-rll nsl'dr)Lte =" holds for L < p 3q < oo if a^nd only if g>p_1,

if

([

(0.31)

0>p-1, a<-1.

I\Jc\Jc//\Ja I ( I f1)dtl

holds for

rd,rl


g>p-r, dsP!-\-t, pp 6

(0.34)

9>p-1, a( Bt-4-t. pp

(0.35)


6

(0.38)

if and only if

9
In the case p g the corresponding numbers a,nd inequalities (0.36) and (0.37) do not hold. See also

if and only if

(See also [OK, Example 6.9].)

I

/ t@

with0
(ii-2) tor I < g < p <

(0.37)

/ i \r/p t (/. /P(s)'e-rllnxf ar) =

(0.32)

(ii) The inequality \l/c

c:pl_f,_r,

(0.36)

(1,' r(qdt) ]rr','r'a')'/o

or

\s

1 is

(ii) the inequality

9
/ f@ / f@

: p-

(1,' (1,' rodr) |trn't'a')'/o

or

92p-1,0<-1;

e

excluded. In this case, we can improve the inequality by adding some logarithmic terms on both sides and considering the interval (0,1). More precisely, (i) the inequality

(0.28)

with0
g
Example O.6. In inequalities (0.a) a,nd (0.5), the value

9

(0.3e)

A and A are infinite

[OK, Example 6.10 and Remark 6.11].

Remark O.7. The

progress

in the theory of Hardy type inequali-

ties and their more general extensions has continued also after 1990 (when the book [OK] appeared). Here, we will report on some of these developments, and we will deal mainly with the following topics:

(a) Some limiting cases of the Hardy inequality for

p +

(see Sec. 1.2).

(b) Some weighted norm inequalities (see mainly Chapter 2).

oo

l0

Weighted Ineryalities of Hardg

Ilpe

(c) Inequalities with the Ha^rdy-Steklov operator (Chap. B). (d) Higher order Hardy inequalities (Chap. ). (e) Flactional order Hardy inequalities and their connection with

(/)

interpolation theory (Chap. S). Integral inequalities on the cone (Chap.6).

of

1

monotone functions

Hardy's Inequality and Related Topics

1.1.

Weighted Lebesgue Spaces and the Hardy Operator

Weighted norm i nequalities The weighted Lebesgue space

:

L"(a,b;w)

L"(w),

(

1.1)

where 0 < s ( oo and ur is a weight function on (o, D), consists of all measurable functions f : (t) on (a,D) such

f

ll/11",.

ll/ll-,t., If

:= : :

.F/ denotes the

that

(1"' vax"w@)dr)"' €ss supo
. *,

l/(r)l < o".

0 < s < oo, (1.2)

Hordy operotor, (u il@) ,=

I"

f g)dt,

(1.3)

Hordg's Ineqtolitg orul Reloted ToVics 13

L2

Weighted Inequalities of

Hoilg Tlpe

Dualrty

then the Hardy inequality (0'6) can be rewritten as (1.4)

llH/llo," S cllJllp," and

it

holds for functions

t

U0

if

a'nd only

Define the duality on the weighted Lebesgue space .Lt(to) I < oo by the inner product

if

Acm

:

f)

b, (1.5)

Then one easily where A is given by (0.8) or (0'9), respectively' define Together with the Hardy operator H,we can immediately the conjugote HordY oPerator H: (H

f)(r) ,:

I,o

(1.6)

f (t)dt.

sees

then holds for functions .f > 0 if and only

if

A
s'- -L, s-l :

Specifically, ll9ll",,.,'-.,

6

suPll4l..-=r

-

ur-"'

lk'/)l

.

so that

- 7s'1.r-s'y.

(1.11)

Indeed: Htjlder's inequality shows that

fi

lk,.f)l (1.8)

(/'

respectively'

(1.5) to obviously, the step from I[ to ,F and from the condition the condition (1.8) can be made by a simple substitution' Whatismoreimportantisthefactthattheinequalities(1'a)and the form (1.?) are prototypestf a general weighted norm inequality of

-

Inthesubsequentchapterswewillobtaininequalitiesoftheform

f'

since

s/s'

: s - 1 : ll!' .f

:

(l

le(')

l"'.-

Sometimes' these operators functions of / such as e.g. mon(> are defined on certain special classes tone functions.

"'

r' p1 a,)'

/

"u

1), and

if

l9l"'-t, gn g-r-"' /llolli,/.j,_,,,

:

where?isageneralintegraloperator.Thustheinequality(1.9) asserts that T maps f,P(u) (continuously) into bt(u): T z U(u) -> Lq(u)

" u t'r t" - @) d*)"

ll/11",",' llgll,,,.,'-,'

(1.e)

ll"/lln," S cll/llpp

(1.9) for several particular operators

(1.10)

-

that the dual space to z"(u) can be identified

(I"(r)). (1.7)

llfifllo,"s cll/llpp

s@)!@)ar, I e L"(1n)

with the space tr"'(r?) where

The corresponding (conjugate) Hardy inequality

where

l"

with 1 <

then ff /11",,,, I and (g,f) :11911",,',,-,'. _ Moreover, the Hardy operators I/ and H arc mutually conjugate; more precisely,

if H : U(u\

-t Lq(u'), t 1p,{ (

then(I/).:Fand

fi

, 7o' 1ur-d y

+

Lr' 1ar-n'1.

oo,

14

Weighted Inequolities

of Hotdy fupc

Hardy's Inequality ond Related Topics

Indeed: By F\rbini's theorem,

fbrtfiP b,Hfl = Jo | s@) JoI

fft)dtax

Moreoaer, the constant

: I f(t) I Ja Jt

s@)dxdt

:

lr(u)).

Some other criteria As we know, the Hardy inequality (0.6) holds if and only if A < m where A is given by (0.8) for p S g and by (0.9) for p > g. Now we will give some alternative criteria. But first, let us introduce the notation

V(r)

:: [" o'-o'1t'1at. Jo

(1.12)

\l/c o:.::?u / rb"@o') ,r/n'1r\.

\J"

l.l.

Let

|

< p < q < oo. Then the HwdV inequolity

(1"' (1"' r@dr)o u@)a,)"' holds

=,

(1"'

Prool (i)

Necessity of

(1'13) holds for all the function

(1.16)

condition (1.14). suppose that the inequality

/ > 0 with a consta"nt c < oo and choose for / f,(r) :

x6,t1@)ur-P'

(r)

with t e (a,b) arbitrary but fixed. Then (1.13) implies

(1"' ([,r-r'1s)ds)o u{,ya,)'/o

. c (1"' ,,-o,t*)0.)

l.e.

/ rt tr/c
Then we can rewrite the number A as

Theorem

satisfies

B
U,fic)

where the first (last) brackets denote the duality in trq(u) (in

C in (I.IB)

15

rr,w6ya,)'/o

(ii) Sufficiency of condition (1.14). Due (1.13) is equivalent to the inequality

(1.r8)

to duality inequality

(1"' (1,' ,(t)dt)e 'r-o'1c)dx)

for oll functions .f > 0 if ond only if

B
s"

(1.14)

( fu so'1,yur-o'1,1a*\lo \Ja /

(r.17)

for g 2 0 with the same constant C. For with support in (a, 9 b) we

B:: "::?,

(l'

x (1"' Le.

"

:

aFp'(qdt)

^,

-rrn@)

":l?un

have by integration by parts and by Hiilder's inequality

(1,'

\ l/q

,'-nav,)' or) ,

(1"' ,Urr'(qdt)

r (r.15)

(1.15*)

::

:

I"' (1,' nror) ;-n,1r)dr o'

f"u (!,0

o(ildt)e

s@)

(1"' ,,-,'1,1ar) o,

= / l,' o{due-c)/q'(x) (!,0 nOor)''-'

Hordg's Ineryalitg ond Relateil Toyics L7

16

Weighted Inequolities ol Hardg

Tgpe

ar-p' (t)dt) u{o' -r) " (l "'

/ o'

s,

Estimating the integral on the right hand side by Minkowski's inte gral inequality, we obtain

(x)dr

(l:so'(r)u'-t'(dQ''o' iro

Jr

(1.18)

: ,, (1"' yro/o1r1ut-n'',)tu)

where

tr,:

f"b (l' ,avr)c@'-t) (1"' "-o'ttat)q

u@)h

'

:

o

:

l4rru ( "'(\Jol" \Jcl"

(1,'

(!'

,avr)@'-')P o'-'' pva*)

Denote

h(r) :=

(!,'

-t) navr)qb'

:

,'-o'l!dt)q u@)d'' l"' o (1"' n

nt lro{r)u(r)d,n: l) Irn-(t)latvt(r)u(r)dr l"'

: t' f' vn'tt)vq(r)u(r)ld'tda : l l Fn' tt)vq (r)u(x)ldrdt "o

:

"'

l"uVn'lr)l l"'

u(')vq(r)dad't

and due to (1.15-) we have

a,3

Bq

e,tall(1"' !"0

1o/r

/

/

) - BtYln '

This together with (1.18) implies that

Applying F\rbini's theorem, we obtain

rt :

\P'

sQ)dt) ,'-o'1";a"

( fu sn, 1r1ur-0, 1*10r\"0 , p,B ' / \"/o= i.e. inequality (1.17), and thus (1.13)' holds and' moreover' Jtlp,

c


Theorern L.2. Let0 < g
<6,P)

L,Lfr:L/q-l/p'

Let v be giuen by (1.12). Then the Harily inequolity (1.13) holds for aII functions f >- 0 if orul onlY iJ

(1.le)

B
u.: (1,' (1"' 'ur''{')a")"/o u-'/o1t)dvt") Moreouer, the constont

u'-o'{4a*)o/o at.

o

(1'20)

C in (1.13) sotisfes

Or-r/r 6tl/o' r-r/t' 2-r/c B S C

<

qr/cpl

B

.

(1.21)

18

Wei.ghted Inequaliiies of

Hardy's Inequolity and Relotd Topics 19

Hordy T\pe

in (1.21). Let us define

Proof. We write

r: : :

:

:

l"' (1"' r(qdr)'

u@)d'r

"0,:

u@)vq(r)v -e @)d,r l"' (1"' r(qdt)q

, t

We will show that

C

v-o-r{")dv(s)dx u@)vq(r) l,o I"' (1"' r(qd')'

f"u

v-o-'1",(1" (1"' raw) u@)vq(t)!")oru,

I ll f' u(r)vq (r)tu \ v-' (s)ldv (s)

)

here, we have used Fubini's theorem and the fact

c(

",,: ;

that

qr/t14)uq

(1.22)

7Bo.

(l"o (!,u ,,olat)'/n v;ro'p)dv,('))

with

ff < !]

fot

v1(u)

s. Applying now Hijlder's inequality with the parameters f; and ($)' : we obtain

!,

If

we choose for

, tn(|"' (1"" ,av,)'ffi)''' "'It

>

Fbr this purpose, denote u(n) :: uL-p'(r) and let u1 and u1 be such that 0 ( u1 ( u and 0 ( u ( u1. Suppose that the functions u1 and w1, w1(r) ,: rl-'' (r), are integrable. F\rrthermore, denote

s ' I"'l(1"" ropr) v-'(")] " L(/.

(l"u (!,' ,oor1'/o v'/o'(*)d,r@))

r(x)::

is easy to see that bY Theorem 1.1,

/

: [' ,1-o'gar: [' Jo Ja

u1(t)dt.

the function

(l,o

,rtyor)''* (1"' *t1t1at)'/oo' -r(,),

then we have

(1"' (1"' r@0,)'

#S)''' 'o'(1"' r'avr")a")'/o

Thus, we have shown that inequality (1.13) holds with C S qrlqy'B ( m provided (1.19) holds. To show that the condition (1.19) is necessary' let us suppose that inequality (1.13) holds with C < m and derive the lower bound

1""

f{r).,

(1,',,o0,)"* I" (l"' .uoo,)''oo' w1$)d,t

=+

/o'

(1,' u,(qdt)

o

(1,",1,;a")"/o

1W

Weightert Inequalities of Hodg

20

Hardy's Inequolity ond Related Topics

Integration by parts, the last estimate and

(l'f3)

lead to relations

If

that

we show

B

<

1zq1t/o

(f,)'"

then the lower bound

in

(1.21)

will follon'. For this purpose, write

o: ( s\"0

r \f /

= (1,'

":,0

(+)' (|"'r'to')"'

(1""

"61at)'/'

u,@)a,)'/o

:

", I"

(1"'

,,uro(-

uo,

I,',(")r"))''q

(1.23)

v-'/olsyav@) (t.24)

and

rt:

, (1"' (1"" ,ttor)n u,pva*)'/o =

t

(1"' (1"'

:

\

:

1""

vnpla(-

"

n

)

" "

(1"' vo1r1u@)ar)/

(1"' (l,u

I,',1,p")

: I,' (|,",(s)as) Yt-t(t)dv(t) : !r' vc-t+c/2p (t)]v -ono 1t7dv (t)

l/q

n

:, =

,av,)'

u(r)do

l(!,",(")as)

Hiilder's inequality with parameters

"avr)'"

(1"' ',1t1at)'/o

ofr*)'t-'@')

rr s

!

""td

(;)' :

and from (1.24) and F\rbini's theorem we then have

(!",oa')'/o v;^'6d,'("))''o : ""'''

'

B' *yr/zn-r/q 1r\dv (x)

C

> qr/s@t)r/4!&.

of integrable we de Theorem convergence functions and applying the Monotone rive the foregoing estimate with Br replaced by Bo, i'e' the esti-

By approximating u and ur by increasing

mate (1.22).

sequences

=

o,/t2,/o

*

$

leads to

(1"" (l"uro")''o''o-'*ct2p)('/q) $)d' G))

n

u-r/z1fldv+) " (lr'

(1"' (1,' ,,(Dat)''' (1"" -,(qdt)'/q' ',p1a)'/o

(1,'

2l

f

,b

I"' (lr',r{";ar)'/o

v, trr-,/

o

1s1av

vrlc'+r/zp(t)

llav g)

.

22

Weighted

Inquolities of Hordy

: {lryyl'U,' : s"r

Hardy's Inequolity ond Reloted Topics 23

Type

is finite:

u{,1a,)'lo y,/t'1t1aVg),

llg'llp,, <

Then the sets Wl'p(u) ana Wftp(u) are normed linear spaces with the inequality (1.25) is equivalent to the Hardy

same norm llg'llr," and the

inequality (0.6) (for g e Wl'p@)) and/or to the Hardy inequality (0.10) (for g eWrdp@)). Indeed, it suffices to take

and (1.23) follows. The proof is complete.

1.2.

*.

g(o):

Hardy's Inequality With Derivatives

l"' f {r)0,

(: (r//)(c))

and/or

The differential form of Hardy's inequality

Let us denote by

g(r): ACk-r(a,b): ACk-r

with k € N the set of all absolutely continuous functions on (4, b) with absolutely continuous derivatives up to the order lc - 1, and

(see also

l,o

f tDor(: (fr/X"))

[OK, Lemma 1.10]).

Consequently, the Hardy inequality (1.25), which can be rewritten also in the form

consider the inequality

(!"'

'aw,(")*)'/o

. c (1"' v't x'op1a*)'/'

llgllc," S cllc'llp,", (1.2b)

expresses the continuous imbedding

for functions g e ACo which satisfy either 9(a)

:

(1.26)

o

or (r.27)

e(b):0' F\rrther, denote by

wl'o@) and VIzIP(u) the sets of all functions g € .ACo which satisfy respectively lhe initial cond,ition (1.26) or (1.27)L and for which the right hand side of (1'25) rTbe subscript L lRl indicates that the function vanishes at the left [right] endpoint of the interval (a'b). Of course' for c : -oo, g(a) = 0 means that

lim"*--

g(c) = 0, and analogously for

6:

oo.

(1.28)

wl,o@)

4

Lq(")

(1.2e)

who@)

4

Lq(")

(1.30)

and/or

and the condition (1.5) is necessary and sufficient for (1.29) while the condition (1.8) is necessary and sufficient for (1.30). In Chaps. 4 and 5, generalizations and modifications of the inequality (1.28) are studied. Specifically in Chap. 4 we investigate, for k € N, & ) l, the higher order (/c-th order) HardV inequolity llgllq," S C llg(o) llo,,

(1.31)

for functions g e ACk-l subject to certain additional boundary conditions which extend the conditions (1.26) or (1.27). More precisely, we look for (necessary and sufficient) conditions on the parameters

24

Hordy's Ineryality arul Reloted Topics 25

Weighted Ineqrolities ol HodU Tbpe

ptq a\d on the weight functions u'u' under which inequality (1'31) t oid" for some particular subsets of. ACk-t. F\rrthermore, in Chap. 5 we will investigate also lroctionol order Hord,y inequalities, i.e. inequalities of the form llgllc,"

Scllg(^)llo,,,

o

<

I

< 1,

and

lls(^)llo,"

Scllg'llp,,,

o

<.\ < 1

where the expression llg(r);;',r, in accordance with the commonly used notion of fractional order derivatives, is interpreted as

(l"' I' _'ffi'

6'

/'

1.3.

Some Notation and Modilications

validity either in the form (0.6) or (0.10) or in its of Hardy's inequality (1.25) are expressed in terms of the constants A form ilifferentiol and .4 (cf. (0.8), (0.9) and (0.12), (0.13), respectively). In order to emphasize the role of the weight functions u, u, and also the interval (o,b), we will sometimes use the notation

Remark L.4. (i) Necessary

A: A(a,b;u,u), A: A(o,b;u,a). (ii) Several authors have used the mdifieil

Y1a'aY)'

L'(a,b;u):

can be also considered for functions

(1.35)

/

such that

fweLs' i.e., normed (for 1 S s < oo) by

s ewl'p@)nwtip@), i.e. functions satisfying boththe conditions (1.26) and (L'27):

ll/11",,,:

In this case, a necessary and sufficient condition for (1.25) to hold is that for parameters p'g satisfying I < p < g < oo we have /o

(l

l/(c)to(c) l" or)''

"

"'

(1.32)

s@):e(b):0.

The connection with the weighted Lebesgue spaces in Sec. 1.1 is evident:

E"(.):

tr"(tu")

L"(.)

introduced

.

Of course, in this case the weight conditions involve the finiteness of

u(t) dt)'

r",,llt,,r l(1.'

A

,'-o'{r)at,

weighted

L"(w)

defined as the set of all measurable functions

Remark 1.3 (two-sided conditions). The Hardy inequality (1'25)

{1""

(1.34)

Lebesgue space

Here ur(c, y) \s a two-dimensional measurable weight function defined and positive a.e. in (o,b) x (a,b).

* (*i,,

and sufficient conditions for the

fob

,t-l(t)dt})"/] .

*

(r.33)

to P. GURI(A). A modification of this condition as well as a condition for the case p > q will be given later.

::

A(a,b;uq,uP) <

e

etc. For instance, inequality (1.25) with u and u replaced by ue a,nd uP, respectively, for functions g satisfying (1.26), i.e., the inequality

(see [OK, Section 8]; this result is due

(1,'

"awtc)rqac)'/n

. c (l,o v'aw"ole)

,

26

Hordy's Inerynlitg and Relotd, Topics 27

Weighted Inequoli.t:ies of Hard,g TVpe

holds

if and only if

due to the fact that

,::1, (l',c(r)d,)'" (1"'

a-e'

odt)''o . *

forl
(iii)

(trf)(r)::

u(c)

[' 7p1r1t7ar,

d@) : f (r), s@):0 for s@): (IIfX'), g'(x) : - f (r), g(b) :0 for g(r) : (fi il@) ,

(1.36)

it

is more convenient to deal with the Hardy operators acting between weighted Lebesgue spaces.

(iu) Formulas for .4 a"nd ,l'in the case p can also be rewritten as

(1.37)

Ja

as an operator between unweighted Lebesgue spaces

o'

i":

,2?:I]+Lq.

(1.38)

An easy calculation shows that the corresponding necessary and sufficient conditions for ll,2f tllo < Cll/llp to hold have the (more symmetric) form

yr/t 1 p

forp) swith

i:i-i.

,: ^.

r.31y1ry:: \"".

u(r) fu Jx,

(1,'

> q (see (0.9) and (0.13))

'oo,)''o ur,ro*)"' \ r/p

(l"u (l,u ,'-o'tror)'''' (1"" u(t)d.t // ) u(r)dr

t/"

\ I

since integration by parts shows that

*

Nowye can replace the conditions A < oo and .4 < m by A* < m and .A* ( oo, respectively. The advantage is that these conditions are also meaningful for g : 1 and useful for 0 < g < 1 where d < 0.

1.4.

Similarly we could proceed with the opentor

,'n'toor)

F

o: (t)''' o. and n: (t)"'

forl
(l^' (1,' uolqor)"' (1"' *'uro,)'/o' ,o'1,10*)''' .

(1"' (!,"

a,nd

a^nd

/ fi \t/p' ( | uq(t)dt) ( / ae'U)dt) .* a(c(D \./c / \J o / sup

,:

II

,fr,

yp1,1r1or.

Clearly these notions are equivalent. But since we will be interested in the d,iffercntiol form of Hardy's inequality (1.25), which is connected to the inequalities (0.6) (or (1.a)) for I{ and (L.7) tor fr

Hardy's Inequality for Some Special Classes of Functions

lnner polnt and mixed conditions The differcnfial Hardy inequality

(!"' uavu@ds)''o . c (!"' w' or,{4a,)'/'

(r.se)

Hord,y's Inequolitg ond Reloted Topic.s 29 Weighteil Ineqnlities of Hadg TVpe

28

was considered for functions g e

Ae

satisfying one of the following

conditions:

9(a) =

0 or

g(b)

: 0 or

s@)

:9(b) : 0.

(1'40)

These additional conditions are natural since they avoid the meaningless case when g is a (nonzero) constant'

In what follows we will

consider classes point condition inner satisfying either an

of functions g e

:0 with c € (a,b),

sk)

Proof. (i) Necessity of $.a$ and (1.4b). Suppose that (1.39) holds for every g satisfying (1.41) and choose 9 such that 9(c) : 0 for n € lc,b). Then inequality (1.39) attains the form

Ae

/ fc

\r/q / fb / fi \l/p ( I ls@)lqu(t)d.r) s cl I ls'(x)lpv(c)drl, (r.42)

Irg(o)*129(b):g ca,n be

rewritten in the form

(1.43) e(o)+le(b):9, A#0. However, now I f -1 since for .l : -1, condition (1'43) is satisfied

/

meaningless.

g(")

Theorem L.5. Let c be fwd, a < c < b. Then inequolity (1'39) hotd,s for euery g e ACo satisfyi'ng (1.4L) il and onlv if (1.44)

oo

A(c,b;u,u) < m. Let

for

(1.45)

0, A * -L. Then inequolitv (l'39) holds

e P. be fweil, # ^ (1.43) i/ and only ACo safisfging

^ ge euery

(1.48)

(1.4e)

/

\Jc

which is the Hardy inequality on the interval (c, b) and consequently, the condition (1.a5) is satisfied. (ii) Sufficiency of (1.44) and (1.45). Suppose that conditions (l. ) and (1.45) are satisfied. Then inequality (1.48) holds for gr e ACo(a.,c) with 7rG) :0, and inequality (1.49) holds for 92 € ACo(c,b) with gz(c'):0. Taking g e Aco(a,b) such that

inequality (1'39) is = const. (# 0), for which

A(o,";u,u) <

\r/p

(1.41)

\Jc

with,\1,,\2 nonzero constantsFirst, let us note that condition (1.42)

/ tc

which is the Hardy inequality on the interrral (o, c), a,nd consequently, the condition (l.aa) is satisfied. Similarly, for g such that 9(c) = 0 for c € (a,cl, (1.39) attains the form

c fixed, or a rnised condition

by any function g(o)

\1/c

SC I I ls'(x)lpu(r)drl, I\JcI ls@)lqu(r)hl / \Ja /

:

( gr(*) for r {

I gz(") for

€.

(a,cl,

c e [c,b),

we find that 1b

| Jo

1s1x11t"1"1a"

:

f"'

br@)lqu(r)d,x +

f"o

lor@)lqu(r\d,r

if

A(a,b;u, u) < oo

(1.46)

A1o,b;z,u) < oo.

(1.47)

="'l(1""b't@)lpa(t)0,)o'o*(1"'bi2@)lpu(r)0,)'''l

t"'l(1"'b'@)rpa(r)o*)o''*(1"'f (c)rpu(c)*)'''l

30

Weighted Inequolities of HordV

/ fi

Hardy's Ineqnlitg ond Reloted Topics 3l

hpe

:2Cq (./" lo'(")lor,(r)tu

1c/r

1.5. The Role of the lnterval

)

The interual (a,b)

(ii,i) fi we define the operator ? as fif

(ril@):

*

r{r)or1""

- ;frH,

In the previous considerations, the interval (a, b) was arbitrary, -oo < a < b ( +m. In the classical case (0.4), G. H. HARDY

i.".

* l,u ,{r)or,

(1.50)

(1.5r)

llp,".

: (0,*).

(a,b)

then 9(c) : $f)(r) satisfies condition (1.43) and d : f . Conse quently, instead of the Hardy inequality (1.39), we can consider the inequality

llr/llq," < cllf

considered the particular interval

Since

For this interval, the necessary and sufficient condition for the validity of the general Hardy inequality (0.6) or (1.25), i.e., of the inequalities

(1,*

([

r(qd,) u6ya,)'/o

with/)0and il"/ilc,"

=

l#l

ilHflic,'*l*l

ilrr/il',",

conditions (1.a6) and (1.47) are obviously sufficient for (1.51). For .\ € (-1,0) it ir fi ) 0 and -1fr t 0, and consequently,

/ f*

=

"

\l/s

(lr*

/ f@

Sc 1 1 I\J0I ls@)lqu(r)d,el ,/ \J0 with g(0)

for/>0wehave

:

0 reads, for

A:

.4(0,

HI sir+r)Tf, frf <^f,rrr. l)l - '' Thus, for / > 0, the conditions (1.46) and (1.47) are also necessory since if (1.51) holds, then the inequalities

(1.52)

I(

fp(n)u(r)dr)"o {r.rr)

tl/p ls'(x)lpu(a)dal /

p < q
o; u,v)

::r"p f / c>0 \Jo

t/o

,1r;at) /

r l/p'

/ tx

' (/o ,'-r'@dr)

<

oo.

(1.s4)

Let us show that all other possible choices of the interval (a,D) be reduced to this special case.

ca^n

llr. f lln,, <

c(l

+ 1)ll/llp,, ana llFll;0,,, <

c(I + 1)lll-1ll/llp"

(i) Consider the case (a,b)

also hold, which implies (1.46) and (1.47).

For I ( -1 or I ) 0, ? is a combination of I/ and If with one positive and one negative coefficient. Therefore, the foregoing argument cannot be used and the necessity follows from the more general case considered in Chap. 2 (see Theorem 2.3). tr

: (--,

oo)

.

Then the inequality

(/* U:-/(')dr)

,@*)'/o . c (/-

rp(x)a(x)da) "o1,.uu;

32

Weighted. Inequolities

holds for all .f

ol

Hordy Tlpe

Hardg's IneEuality ond, Relotd, Topics

> 0 if and only if

A(-*,oo;u,u)

(ii)

= ::R(l-

u@dt)"'

r

(!:- ur-p'(t)dt)

3i:t

Consider the case

r/P'

o:0, 0
Indeed: Let t : lns and r : lny. Then dt : Lds, d,t under these substitutions, inequality (1.55) becomes

(f ([

:

([

f,dy, and

(1,' ra)',)'

holds for all .f > 0

t(tr")r")q "(rtiloo)''o

f-,,,(tnfiay)

T,

which is (1.53) for only if

:$ (/i.e. if and only

l,i,D,

and

t : * *d , : h.Then

r'-o'(t)dt\'/' /

dr

: -ffids

a,nd

. *,

,

=

"

(l:' (#) . (#) ;*,,,*)''' il(a)

"(#)

Ttyl

b \ b (y+L)2' '\y+7) '/

if

\/"

f

ur-p'(lns)

,^\'/o' <- oo' *

/ \ro ,-o')

Making the above substitution, we can write this condition as

yr/q 1 ilnr :,UP' / f* u(ildt) ( o., (t)dt) u'-p' (/_* ;$ (1"" and this is (1.56) with

r = lnr, since s € R for r > 0.

:

(*) ciy*)'"' (#) 6**',)"'

If a(s)ds)''' (1"' 7r-o'1"1a")"''

dc

and under these substitutions, inequality (1.58) becomes

U- U:'

it holds, according to (1.54), if a.nd

( :" ^..^( [*.r(1r,")ar\t'o :lE

. c (1,* |'tot (iloo)''o

(1 s8)

(1.5e)

-ffiau,

6(y): /-ru(lny).

Then (1.57) can be rewritten as

(1,* (1,' i(")r") ,tildy)'lo

sup ( [ur1r1or\t" (f \Jc ,/ \JO

0
(1.57)

Again, let

/(l"y) \ u(lny) fir-.t -, i(y): fu): sF,

. c (1,' raw@*)

if and only if

A(o,b;u,u):

t" (I* (ry)"

u@dr)'/n

i(y)

#TP'

=,(#) (#)'=,

then (1.60) can be rewritten as

(1,* (1,* i(")d") ,(ilda)''o

. c (1,* nuv@dy)

(1 60)

34

Weighted Ineguolitics of HadV Tbpe

Hard,y's Inequality and Relotd Topics 3b

But this is the conjugate Hardy inequality fot l,d,d, which is satisfied if and only if

*; i,A

:

i.e., if and only

if

A10,

[ d(s)as),/ ''o (\Jtf "'p r>o \./0 (

ar-r'1s)ds) ''o'

/

. *,

The substitutions t

: s*a

and then x

: U *o

tr/p / fb-o/ fu \c \l/q / ft-o
:$(/""(#) Gfy'")"' * (1.*,'-o (#) (#rp;(r-rxr-r') \rr

o")''''

.*.

,.::l_.

/ rb

\r/q / .-+

\

o.ilf_"

l/P'

/ fb-" 1r/o 1 p u"(s)d.s) (/ (/o

\l/p'

"l-ztslasJ

(I'-",(s + o)d")"' (I'

o.ill.u

(eii) Finally, consider the case

(l:.",t

10r)"0

.*-p'(s +

(1"'*" ut-p'(qdt)''''

Remark 1.6. The uniuersality of the interval

it forp )

(!"' (1"" r(r)dr) ,@)dr)''o =" (1"' no,@*)

1.6.

Gompactness of the Hardy Operator

(r.61)

,f > 0 if and only if

.

oo,

*)

was shown in p< q. By thesame approach one can also 4 ( oo, I < p < oo).

prove

"::?,

(6,

(0,

Sec. 1.5 only for the case

Again we can deduce from (1.53) and (1.5a) that the inequality

/rb

o)a")'/'

: r * a.

and this is (1.62) with s

-oo
:

oo

l.e.

*p ( /-. u(t)dt) ( 1'*' u'-o'U)atl ( oo, f>0 \.r-+ I \"ro / (0,b). which is (1.59) with ":;h, since for r ) 0, it is c e

A(a,biu,v)

<

or

Making the above substitution, this condition becomes

holds for all

show that (1.61) can

be rewritten as

S (0

<

Motivation

yUt1p

(l u@dt) (/"

t l/P'

ur-e,(t)dt)

For O a domain in RN with (smooth) boundary 0O, let Wol'e(O) be the Soboleu space,

w;'of:")

(1.62)

For this purpose, we write

hs(x) := h(n + c).

lt

:

{s ,lvgl e rp(n), glao : 0}.

is well-known that the imbedding

wot'o(o) c+ rq(o)

(1.63)

36

Weighted Inequolities

d

Hordg fupe

continuousif 1 < s N). Hence, the value

<

#+

is

Hard,g's Inequolitg and Relotd

and. compact

if

1


ffi,

(1 < p <

(1.64)

If o is the ball {c € S(lrl) :

(

( l" tn6y,N-tdr) \r0/-\"lo/

<

c(

l^ lg'(,)lrrN-tarl

the inequalitY

rnt"lr u{,1a,)'/o

is continuous for q for q> p'?

(

=

p*?

"

(lr"

b'v)lpa(idr)

p', compact for g ( p*

(1.66)

U(r)::

ft

(1.67)

tR

1

V(r)

r 1R, (1.68)

f(t)dt,

.E: we can reformulate the problem just mentioned for the operator (P,u,a) p' p* esponent critienl a = Does there exist, for p > 1 fixed,

::

(1.6e)

0, U(.R) < m, V(0) : m, V(R) = 0.

(1.70)

u(t)dt,

f,*

assumption (1.68) means that

U(0)

If

:

we further denote for l.

( p( g(

:

Bo@)

oo

gr/o1r1Yr/o'1r1

then we know that the inequality (1.67) holds - i.e., that the mapping H : U(a) -+ Lq(u) is continuous - if and only if

0
(

oo.

(1.71)

If we want the mapping H to be compoct, this condition has to be shzngthened.' the corresponding necessary a,nd sufficient condition then reads

lsaotr)

for the conjugate HardY oPerator Jt

for every

u'-o'{1ar,

Jo

sup Bo(r)

And since inequality (1.66) is equivalent to the Hardy inequality

1fry11r1:

simplicity we will assume that

rl(0,R), ur-P' e Lr@,n) bur vr-p' #. rr(0, R) .

and does not take place

llH/llo," < cll/llp,,

1. We shall consider functions defined

u€

(1.65)

with very special weights: u(r) : u(r) :1N-r' Hence, the following natural question occurs: For a fixed P, P ) l' a and for given (general) weight functions u a^nd u, does there exist parametlr p* J p* (p,u,u), P' ) p, such that the imbedding described : ior difierentiable functions I with { e I}(a) and with g(R) 0 by

(/"

trp(u) into Z{(u) compactly for q 1p*, continu-

Using the notation

\l/P

/ fR

37

and that the Hardy inequality (1.67) does not hold

on the interval (0, R), .R oo, and for the weight functions u, u satisfy

RN |rl < n} and if we consider rcdiol g(r), 0 < r < tr|, then we can rewrite the

\l/q

q>

1p*

Notation. Let p be fixed, p )

p+, continuous for g S p* aod does not

imbedding (1.63) in the form of the Ha'rdy inequality

/ rR

F *upr

(

is the secalled critiul etponent of the imbedding (1'63)' i'e' the imbedding is compact for g take plocefor q)P'.

that

ously for q for

*Np p=G

: functions, g(x)

such

Topics

=

J$a01"):0.

(1.72)

(For details, see [OK, Section 7].)

Remark 7.7. (i') Due to assumption (1.70), we immediately have Bq(n) = 0. Hence, we need only to take care of the first condition in (L.72), namely the behaviour of Bo(r) for r small, r > 0.

38

7W

Weighted Inequolities ol Hadg

Hardy's Ine$nlitg and Relotd, Topics 3g

(ii) We also immediately see that i/ the mapping fr z I/(u) -+ Lq@) is continuous for some i > p, then it is compoct for euery

q
:

It is still not clear what can happen if q p*. We will see later (see Examples 1.13 and 1.14 below) that for an appropriate choice of the weights u, u, all possibilities can occur: the mapping

fr :IP(u)+LP'(u)

Indeed, we have

:

B o@)

and since Ba(r) is bounded

(1.71)) and U(r) -+ 0 for (we have t/q - l/0 > 0).

r

Bq(r)tlr / o-r /n V1

in

)

(0,

ca"n be compact,

R) due to the continuity of .F (see -r 0 for r -+ 0

0, we obtain that Bo(r)

Assumption 1.8. We wiII assurne throughout this section that there ed.sts at least one | > p such thot

o!|!"Ba(")

<

or continuous but not compact, or not continuous at all. But first, let us show that for p' we have the formula

o.:p,limrSrf

wb\:: '. \' / q:

q.

Wo

Definition 1.9. Let us consider the set

::

{s

)p

sup B"(r) < oo}.

(1.73)

:

ie

Bf(r) S. ConsequentlS

:

U

Remark 1.L0. (i) Assumption 1.8 is reasonable since for general weights satisfying (1.70) it might happen that sup'
Lemma 1.11. Let p )

for every q > p. The couple

u(r):ta,Q,)

1; u(r)

- r2(P-rlexP+. r

for R < oo '- as an example. that the number p' from (1.74) could serve as the critical exponent for the mapping ff. Indeed: For g > p*, the Hardy inequality (1.67) does not hold due to the definition of the set .9 - we have sups." q according to the definition of ,5, and the mapping Il is compact due to Remark 1.7 (ii). ulay scrve

(it) It

-

seems

V(r)

(1.76)

'

limi5rfW(r).

(1.77)

:

(y(r))(c -p' w (r)) lp'

(1.78)

and (if Wq is finite)

limsup(q

sup,S.

log

::

(r)Viln' g)

(r.74)

P'

lloeu(r)l

Then obviously

0(r(tr|

Due to Assumption 1..8, the set ^9 is non-empty: we can define the number (possibly oo)

(1.75)

For this purpose, we introduce the notation

-'

i.e., that the (conjugate) Hardy inequality (1.67) holds for

5

fffi#.

-

p'W (r))

: q - dwo

(1.7e)

.

I and suppose that the weight functions u,v satisfy (1.68). Then Assumption L.8 hotds iJ and only if Wo > p l. -

Proof. Assume first that p-l < Ws I e, and let g be such that p < q < p'Wo. Then it follows from (1.28) and (1.29) that lim,-_ys B[(r) : 0 and thus, q € ,S. Hence, the whole interval b,p,Wo) belongs to ,S, and Assumption 1.8 follows. The

proof for the case Ws

:

oo is

similar. Conversely, a^ssume that Assumption 1.8 holds, i.e., that S f A. Then there exists a q > p such that supg
- /W(r) 3O for r ) 0

sufficiently small

40

Eardg's Inequolity ond Releted Topics

Weighted Inequalities ol Eardy fupe

we find

and hence

wo

: rimrf f.w(r) r-

#, fr

that

: o - t.

Bo@)

:

con$[r(d+1)p'

Theorem 1.t2. I'et p > L ond suppose that the weight functions u,u satisfy (1.68) ond Assumption L.8. Let p* be giuen by $.7a). Then fonnulo (1.75) holds and the mopping H : I/(a) a ['0(u) is

cnrnpactforllg
If p'

not

hold..

Moreoaer, for q - P', the mopping H : U(a) + U' (u) is unti,nuous i! ond onty if U(r)Vo'/c'1r) is bounileil forr neor zero, and it is compact if orut onlg if limr-oU(r)Vn'/n'(r) :0.

Proof. LetWo be given bV (1.77). Flom the proof of Lemma 1.11 we know that (p,p'Wd e ,5, which implies that p* >- dWo.Thus, the result follows if W6 : ee. Let now W6 < oo. If.q > prWs,then g f ^9, i.e.limsup"-e Bfl(r): oo, since choosing a sequence {rn].r rn -+ 0, such that l;lz(r'n) + Wo a.s n. -+ oo, we find that

. P:

f (w(r")) : q - dwo > o and thus, due to (1.78), limr.-p- BE(r") : oo. Consequently,

p*

:

p'Ws must hold. The remaining part of the proof follows from our previous contr siderations (see Remark 1.7 (ii)) a,nd from [OK, Section 7].

u(r)

:

ra,a >

l,

<

u(r) :

Rp+\l

/e'

(* +L)d

-p-L'

u'(").

Example 1.14. Let u(r)

: tN-I, 1 < p ( N, and let lc be any

number such that k > p

1. Assume

U(r) for

r

:

-

lo'

u(s)ds::

that

rk(N-o'11@-r)mb)

near 0, where m is a positive smooth function such that

lof -(t) : ti* r-+0 log r

o.

Conditions (1.70) are satisfied, and since we can show that

N-p. "io llogrl p-l' ri*r?gY(T) _

we find that oo,

a'nd u(r)

-

wo:lTlw(r):$ffi:ft2

rP-r,

we have Bot) : @a+r /(a+ l))r/e (tog *)"/o', and hende Bo(r) ari r -) 0. Thus .9 = (p,m) and p* = so.

(ii) If we choose, for R <

-

The next example shows that the operator .F need not necessarily be continuous in the criticol case as a mapping from .Lr(u) into

lim (q -

we choose, for trl

(rp+r

Consequently, the operator fr : IP(u) -+ Lq(u) is continuous for q 1 p* and compact for q K p'. (iii) For the particular choice u(r) : u(r) : rN-I, i.e. o : N - 1 and p - (N - l)i (1 - p) in the foregoing example, we obtain for 7t* : Np/(N - p) the critical exponent from (1.64).

7l-+oo

Example 1.13. (i) If

/c

a^nd

Now, we are able to prove the main result of this section.

d,oes

4l

oo,

ra,a ) -7, and a(r) : ,9$-d with

P<

-l

-r

0

p-r,

i.e. Assumption 1.8 is satisfied due to Lemma 1.11. For p*

-

prtk we find that

4l@:

vk(r)u(r)

:

consr

(t - (i)t"-ntrto-\)k

m{,).

42

Weighted Inequalities of Hadg TVpe

Hardy's Ineqtolity ond Relotd Topics 43

Hence

m(r): llogrl, the Hardy inequality (1.67) - with q : p* -

-

for

-

does not hold, for rn.(r) 1, the mapping

-

not compact, for nr.(r) : llogrl-1, it is compact.

:

fr

:

U@) -+ U' (u) is continuous but

A limiting case of the discrete inequality (0.2)

consequence of our motivating example on pp. 35-36, we have considered in (1.66) functions g vanishing at the right end - g(R) : 0, which leads to the conjugate Hardy operator ,F1. Clearly, it is possible to consider the classical Hardy operator H : (H f)(r) : JJ f$)dt, and also an arbitrary interval (a,D).

Remark 1.1.5. As a

1.7.

Some Limiting Inequalities

-

1,"

where e is the best constant (see

do

1e

t{*)0, fo*

(1.80)

limit, for p tending to infinity, of the classical Hardy inequality (0.3) used for the function .;tlln,

a,

Indeed, the geometri,c rneon of f

s(#)"

lo*

rtdo,.

rn

/(t)ar)

(l lo' r'\/o(t)dt)o : "-o (l Io' tn IG)d') -

(a' >

0;

,

(1.81)

n=1

again with strict inequality sign whenever the sum on the right hand side converges.

Several authors have given other proofs, generalizations or applications of (1.8f); let us mention G. H. HARDY [4,4], K. KNOpp [1], L. CARLESON [1], J. A. COCHRAN and C. S. LEE [1], H. P. HEINIG [U, E. R. LOVE [1,2,3], Y. BICHENG and L. DEBNATH [U, H.ALZER [1], P. YAN and G. SUN [U and S. KAIJSER, L. E. PERSSON and A. OBERG [1]. For more references and information see [MPF, Chap. 4] and the review article by J. PEOARIC and K. B. STOLARSKY [U.

In his ,

satisfies

(see [HLP, p.139]) while [pl(p

o=l

Carlesonb resutt

,

*' (;l'

;*

!..) t/aPz...an I . Lon

original proof of (1.81) was based on the CARIEMAN [1]). According to G. H. HARDY [3, p. 156], the above limiting argument was observed. for the first time probably by G. POLYA.

K. KNOPP [1] or [HLP, p.250])

can be considered as a

I,- (: lo' r'/,e)at)'

@c|o

S

Lagrange multiplier method (see T.

The well-known Knopp inequolity

h/(r)d,)

A similar limiting procedure in (0.2) leads to the corlemon inequority

Remark L.16. Carleman's

Preliminary ResUlts

A limiting case of Hardy's inequality (0.3)

/*"*o (:

Moreover, we even have strict inequality in (1.80) whenever the integral on the right hand side converges. we also note that inequality (1.80) was known before the Knopp paper (K. KNOpp [1]); see Remark 1.16 below.

1)]p

+

e for p

+

@.

1954 paper, L. CARLESON [1] was obviously interested in finding an elementary proof of (1.81) but he formulated his result as an integral inequality which also proves (1.80). Namely, he proved: Let m(x) be o conuex function on (0,a) sotisfying m(0) = e. Then,

forp

) -1,

fo*

xoe-^(a)/'da =

**,

Io

f

e-^'@)dx.

(1.82)

44

Weighted Ineryalities ol Hordy TWe

Hardy's Inequality orul Related Toyics 45

Let /* be the decreasing rearrangement of / (see formula (6.2)) and let us apply (1.82) with the convex function

m(x)

: - lo" ,n l.{r)or.

=

*"*, (: Ir" rn/.(t)at) dr ssP+t lo* {f'(x)d,a,

which for pr:0 leads to (1.80). Similarly, applying (1.82) with the convex function m(c) defined as the polygon connecting the points (0,0) and (t,DLr ln(llai))' Tr.: 1,2,. .. (where {"i}Et denotes the decreasing rearrangement of {or}Er), we obtain for Pt - 0 that @OO@@

dW. -f< "fa:l , {aier.. n^ SL n:l n=l because on (n

-

1,n), we have rn'(r) rn(s)

z

: ln(l/ai)

ai

: eIo" n:l

and

:fn-ir e/o). n =*y)

Proof of (1.82). Using Hiilder's inequality and the convexity of rn:

m(ka)

>-

m(t) +

we conclude that, for any

rrA ' LP"-m(r)/a6'

kp+r Jo

=I" =

A>

(e 0,

- L)rm'(t), & )

IA Jo

"-

r

(k

(l r^

4r)"

,r"-^(,)/x < p(t+Dk/(k-D

m'

sp

(r,)

"-

- I vh

6r)

I

sp"-n'(a)4r.

Jo

Now, inequality (1.82) follows by letting .A -r oo and /c 1rk/(k-r)

-

-)

1, i.e.,

n

".

Remark 1.17. (i) Another generalization

which covers both (1.80) and..(1.81) was recently proved in S. KAIJSER, L. E. PERSSON and

A. OBERG [1]. In particular, this result implies the inequality

f*

J,

exp

( |

.. ..\ f, tn (t)aM(t)

(M,

Jo

f

)dM(r)

s

foo

"

Jo t@)dM(r)

with M(c) a right-continuous and increasing function on [0,oo). (ii) lt is easy to see that Knopp's inequality (1.80) implies Carleman's inequality (1.81). In fact, applying (1.80) with f (*) : on, x €[n - 1,n), n: I,2,..., and making some straightforward calculations we obtain (1.81).

1.8. Limiting Inequalities -

General Results

1,

A generalization

of inequality (1.8O)

Besidm the Hordy aueraging opetutor Hoz

(H"f)(x).::

(1.83)

l"' tg)dt

which can be called also the arithmetic mean operator, let us introduce the geometric rne&n oryrator G by

(Gf)(r):= Io^

m(a) / x

and hence

Then we obtain lo*

(l r^

rn

€xp

(: Ir' rnf(t)at)

,

/>0.

(1.84)

46

Weightel Inequalities of

Haily fupe

Hordg's Inequditg ond Reloted Topi.cs 4T

In this section we shall consider the weighted integral inequality

/ f*

\l/q

/ fe

\t/p

(1.85) 0, u(r) 2 0 and 0 I p,q ( €, as a limit of some

Hardy inequalities. Recall that u, u are weight functions and that Knopp's inequality (1.80) is a special case of (1.85) with p : q: 1, u(r) : u(r) 1,

or even equivalent to

(lr* rcrr"lnpytal{ar)o'0"

t"(#)]o'o

:

(*))\r/"

rt* (1 ./s 1' -v'1r1ar\'/ \c

/

c(/") : (c(/))", s)0,

:

G(f)G(d,

(1.87)

(1.88) (1.8e)

:

the scale of power means (H"(f")("))11" (* f"(qailt/o is increasing in o by Hiilder's inequality and, in pa^rticula,r,

Ii

G(f)(x) 3 (H"f)(x).

(1.e0)

By replacine I@) by f (a)u-r/v(c) in (1.85) and (1.89) we see that (1.85) is equivalent to the inequality

(1,*

e

o

n'(')u,(s)ds)''

=

"

(1,-

r av,)'''

llFllu'.,4_+Le(u\t

and similarly (1.91) means that

G : U -+

(iii)

g)

is the least constant C in the estimate

H : LP(u) + Lq(u) and C :

Moreover,

G(f

are normed linear

llr/llx s cll/lly.

"ljg*Ur"(/" a-+o+

X,y

(r.e2)

we will use this operator norm notation in the sequel; notice that the Hardy inequality (1.18) can be rewritten as llH,fllq,, S Cll/llp,, so that

(i) As already mentioned in Sec. 1.7,

f)(") :

ll"ll;-y

(1.86)

Some preliminaries

(G

where

f"(t)at)

ll"llx-y- :: supll#.!|y' r+o lllllx and

u(*).

(1,

spaces, the operator norm is defined by

For later purposes we also introduce the notation

u(r)::

",,"

foranys>0. (if) For an operator T : X + y

=

C:e.

=

(1.e1)

Lq(u) and

C

:

llGllpa6l_,y.

The equivalence of (1.8b), (r.91) and (r.92) rhus implies thar

llGll7,g;1-+r,"(,,)

:

llGll7"--+1,"1q

: nCil"/.o-7t,/t(n).

(1.98)

Moreover, (1-90) implies that we obtain sufficient conditions for (1.85) to hold by just using (1.90) and known Hardy-type inequalities for the case

Ho: Ls + Lca/e(u) with a suitable s (cf. (r.92)). In this way we also obtain an up

per bound for the operator norm llGll;r(,)-+r,r{u) (see (r.g3)). consequently, it only remains to find a corresponding lower bound and to show that (1.85) can be obtained as a limiting case of the corre sponding Hardy inequalities.

48

Weighted Inquolities of

Hordy's Ineryality ond Relotd Topics 4g

Hody fupe

The limiting Procedure

In view of our considerations in sec. 1.7 a.nd due to (1.87)' we study and thus inequality (1'85) - as a limiting ininequality (1.91) equality (for a + 0+) of the following scale of inequalities:

Pr*t.Assume that (1.97) holds. Apply the alternative form of Hardy's inequality described in Theorem 1.1 with a : 0, D : oor u(c) = 1, u(o) : s(s)s-o/o and p and g replaced by p/a and q/a., respectively, where 0 < a < p. Then (1.9S) and thus (1.g4) holds a,nd, moreover,

cNsffia". (1.e4)

d

By letting

cr

-+

0*

we find

which of course is equivalent to the Hardy inequalities

C < eupB since timo-1 (1.e5)

a and thus (1'85) We conclude that (1.91) moreover, artd, if lim6-ag Co m,

-

(

llGlf ;"1,y*2"1'y

:

-

holds for some

llGl[," -+Lq(u): ..5b llts."llt/J'

C<

-"cr'(u,)'

ft(x) =

i.e.,

(/-t"ll,t")u(c)ac)''o hokls for f >O Nf onil onlY il B

=

"

(1,*

:: supt-',' (lr' u,(dds)t'o

wi.th w giaen by (1.86).

Morvoaer,

il C is the leost oonstont

ro,o*) .* dn (1.85)

* h"(c)x14_y (c)

with a fixed t > 0 and with h.(c) that

)

0, e arbitrary positive, chosen

f,* Then

it

ng{da,.,.

follows that

(lr' .av,)"' r, * e)-t/e( c < m ,

a,nd, since e is a,rbitrary,

(r.e7)

u::g (/' u(t)da )\ Hence, the proof is

(i.". C

=

l/c

complete.

Remark 1.19. By arguing

(1.e8)

t-uPaC
in the proof of rheorem l.rg but usl.l) with @ = 0, D = oor u(c) = l, z(c) : u(s)s-t/o a,nd p,g replacedby pf a, q/a, respectively, to gether with (1.96) and the known estimates of.llE,1llprc+rtta(111. we as

ing Theorem 1.2 (instead of Theorem

llGllppy-r"@)), then

BSC<eLlPB.

x10,4(c)

so

g
Let0
'r,o.

conversely suppose that (l.sb) holds and apply the equivalent inequality (1.91) to the test function

a

(1'96)

* (if.-)"o :

p Now, we are ready to formulate the main result for the case 0 < S

Theorem L.18.

that (1.91) and thus (l.gb) holds with

50

Hardg's Inequolitg ond Relotd Topics 5l

Weighted Ineqtalities ol Hordy fupe

g ( p < oo' For details' obtain the main result for the case 0 < L. E. PERSSON and V' D' STEPANOV [U'

see

Let0 < g < p < x.'Lfr: llq-llp'Then

the

Theorem L.Zo.

t.

llG/llc," < cll.fllp,'' for

f

ZO

u.,:

Nf

and onlY

(l- (: I,'

\ "/p \ u@)at u,(t)dt)

tl" <

)

oo

(1.99)

with w giaen bY (1.86)' Moreouer,

with factors of

(u;

=

p

(see e.g. "byJ.

(*" Ir" r'-1ln/(t)dt)

,ho*r, "o-* the inequalitY

holds for

/

(1'100)

A. COCHRAN and C'-S' LEE [1])' Then it can be that obvious substitutions and by Theorem 1'18 llG"/llq," <

::

supt-

rcrr/c / f@ . / f@ (expsc-' / - f, r"-ltn co sC I I fo(n)rBdr,\/o tqdtl I / I Jo / // \ro \Jo \ holds for / > 0 with 0 < p 1q < oo and s ) 0 if and only if

(

r

* +p) c < sr/c-t/n"(r+e)/Gil ( (, -l s g * (t s"))-t^ \\ /p \ // (ii) In particula,r, for p: q: !, a: g we obtain the well.

known inequality

f6lP\f6 coexp

J"

(r"-" / t"-'ln f (t)dt )dn I

=

e\a+t)ts

Jo

r" f (r)dx

A. COCHRAN and C.-S. LEE [1] (see also H. P. HEINIG E. R. LOVE [1]). [1] and

due to J.

''' (lr'r"{')a')t/o

'-

1.9.

Miscellanea

The discrete case

Inequality (0.2), i.e.

where

,r"(o)

to,

cll/llp"

> 0 in the case 0 < p 3g < oo if and only if D"

:

and that, moreover,

and' q'

section can be extended Just a's a'n example' to some more general geometric mean operators' let us investigate operators G", s ) 0:

exp

1a1lc-1r1s1/8). Moreover,

r1lg)!_(!:!)1:0, \ s /c \ s /p

Remark 1.21. The results obtained in this

-

-

"UoD". Similarly also Theorem 1.20 can be generalized. Fbr details, see E. PERSSON and v. STEPANOV [1].

Bs

equiualence depending only on

(c"/X")

LrUs-Lu7rt/s;, ur(r)

Example L.22. (i) Applying the result just mentioned to u(t) u(t) : t9, a,g € R, we obtain that the inequality

if

llGllr"t"l-re

-

2" < llc"llle(u)-+r,r(u) <

inequality (1.85), i.e', the inequality

hokls

with u"(c)

t" (#)]o'o,,,,',

E (*E*)'=

(#)"i'o", 1(p(oo,

(1.10r)

52

Hordry's Inequohty and Relotel Topics 53

Weighted Inqualilies of HatdV TVpe

is the discrete analogue of the classical Hardy inequality (0'3)' The general weighted extension of (1.101) (i.e., the discrete analogue of inequality (0.6) with seEuen@weights {.r"}, {,'.}) has the form

fE* €"tl'/n for the whole index range 0 < q

.-c[E"o]'" (

(1.102)

m, 1 ( p < @' This inequality

can be derived from thegeneral "continuous" (i.e. integral) case using

step functions or using the "measure" characterization (i.e., the case *hl.e u(x)d,x and u(c)dr are replaced by rather general measures).

E.g., for 7


negative sequence

::r

{""}

H. P. HEINIG and S. HOFMANN [1]. They are expressed as combinations of the conditions for the cnntinuous case (on the intervals ln,n * 11, n. : 0,t,2,. . .) rnd of the conditions in the discrete case. For the remaining indices p,F,g,Q, the problem seems to be still open. Corresponding results for the differential form of the Hardy inequality can be found in H. P. HEINIG and A. KUFNER [1].

A special two-dimensional Hardy operator Let us consider the operator

oo, inequality (1.102) holds for every non-

(Hzf)(x,y):

if and only if

(i*)'^ (p,. r')

(lr*

In the case of discrete Hardy inequalities, the spaces under consideration are weighted sequence spaces tp. Fot amalgams, we need a combination of. I?- and fp-spaces. More precisely, a function J on (0, m) is said to belong to the weighted omalgam ot U(tp;w) if 1Flr'l

(a)leu(x)dnl

I

r/i
with 1 < p,F < m anduaweight function. Characterizations of the weights u, u for which the Hardy operator (and some simila.r operators) are bounded ftom lt(tp;a) to Ltpaiu), 1 < p < 4 ( oo, | < F < q < oo' were proved by C. CARTON-LEBRUN'

I*

ru, f )q (r, y)u(x,

il*or)"

o

sc(

(l.los) [* J0 f fp(r,y)u(r,y)drdy\'to \"r0 /

for / > 0 with u,u weight functions on (0,m) x (0,oo). For the case 1 < p < Q ( oo, it was shorn'n by B. MUCKENHOUPT [3] that the condition

Amalgams

ll/llp,p,.:tEL/"

f (s,t)dtds

and the corresponding inequality

A given bv (0.8). But not all results can be derived via the continuous case. Fbr some characterizations as well as for direct proofs see G. BENNETT [1,2]. See also L. LEINDLER [1].

lf

Ioo

(@t

which is a discrete analogue of the condition A < m from (0.7) with

(o rrn+l

lo"

/ f@ f@

sup (/

c>0,y>0

\Ja

/ Jy

\l/c

u(s,t)dtdsl /

" (lr" lou ,r-o'1,,r)drds)t'o' .

*

(r.ro4)

which corrmponds to the condition .A < m from (0.7) with A given by (0.8), is only necessary for (1.103) to hold. Then E. SAWYER [2] has found two additional conditions which together with (1.104) are necessary and sufficient. The corresponding full characterization of the weights for the case q < p or for N-dimensional analogues of I12 with N ) 2 seems to be open.

Hotdy's Inegnlity ond Relotd Topics 55

54

Weighted Ineqtalities ol Hotdv

7W

ut(n)uz (y)' o(c'g) : ur(r)uz(y)' For the special case u(r'y) = a successive use of two onethe full answer can be'gr.r"r, by S* l' eppnl,I, and A' KUFNER [1]' dimensional Hardv it"q""fiti*'

An N4imensional HardY oPerator

(i l,' roo)'

for functions

t

o'=

(#)'/*

!P(r)da'

1( P(

oo'

equivalent to the inequality Z 0 defined on (0' m) is

\e ,

fotdo, o" t* (]_ f" ?)" [* J-*" \o-t/ = t-,,,I(Datl

J--

(-oo' oo)' for functions t Z 0 defined on i""L* uv g(t) the ball {v e RN-: lvl 5 Now, for r € f" ni' CHnfST and L' GRAFAKOS lc[] and bv lB(r)l t'*-""f'o*"' defined by that the N-dimensional Hord'y operotor '?f,v [1], it is shown

RN,il;

#

I"r,rftu)ao'

c € RN

satisfies

t

vf,N I@)ledn

Jnr'

3

(#)"f"

ut")looc' 1 ( P 1@'

In P' DR'A'BEK' (h)o being again tle-u1! possible'inequotitv ex-was irniNld'i'a l' rtrNpR [1]' this Hotitv range whole the u' u and to

the constant

;l;.

wlights tended to general N-dimensiond oo' The (necessary a'nd 0 of the parameters P,Qt,l 1P < 1 validity of the inequality sufficient) conditions ior the

?' : '

(l*,w*r

(c)lqr(r)d') ''o

='(/

a1,1u(u)tut-'

4''' (l",,,o'-''

(v)dv)


"

ut'ln't')a')'/o

The difierentiol form of the N-dimensional Hardy inequality (l'.105), corresponding to the one'dimensional inequality (1.25)' ca,n be written as

(l*,an"(")d')''o

=

"(/

r""t"t

yv(ddo)

(1.106)

Here O is a (usually bounded) domain in RN with reasonable (e.g.lo cally Lipschitz) boundary 0O or the whole IRN and is considered to be from Cf (O) or from an appropriate subspace of the weighted Sobolev space !/l'r(O;u) where the right hand side of (1..106) is

I

121"1

(,ffiril@):

ft* ([",

The ditterential form ot (1.105)

The classical HardY inequalitY

f

exactly the analogues of the corresponding conditions for dimension one; hence, for I < p < q <m this condition reads a,re

(1'105)

a norm, We will not deal here in detail with general inequalities of the form (1.106) which are often called N-dimensionol Hardy ine4ualities or weighteil hiedrichs (Poincar6, Sobolev) ineqtalities. These inequalities a,re dealt with extensively in [OK]; see also A. KUFNER [1] or books devoted to Sobolev spaces (R. A. ADAMS [1], V. MAZ'JA [1]). Here, let us only mention that during the last decade, inequalities of

the form (1.106), mostly with power type weights but under rather weak conditions on the domain, have been investigated among others by A. WANNEBO [1,2]. Not

only Lebesgue spaces

Characterizations of weights (or measures) for which the Hardy op erator is bounded between weighted Orlicz spaces are in genenl not available. Partial results in this direction were given by a number of authors. We mention here only G. PALMIERI [1] and also N. LEVINSON [1] who has shown in particular that if the non-negative function

56

Weighted Inequalities ol Hatng

Hordy's Ineqnlity ond Relatd, Topics 57

I\pe

O defined on (0'oo) satisfies

o(r)o"(t)

p

- (t - 1) to'tt))', P) r,

(1.107)

I,* (:1,' rc,,)-o

then

*(: I,' t@d')*=(h)' Io*

I,*

o(r(x))dr

(1'10s)

for / 2 0 unless O(/) = 0. The proolis easy: If we denote !I'(t): (O1l;;th, then by (1.107) V"(t) > 0 and hence V(t) is convex. Thus by Jensen's inequalitY

. (l I'

ra)d')

and by Hardy's inequality (0.3) applied to

l- (:

lo'v1r1,11a,)o

o,=

![(/(o))' we have

'(:

d* s ct 1," Ia)d')

CARRO =^Vo'and

Q{'r{*)\o'

c*ct

holds if a.nd only all > 0 for exist positive constants Cs,Ct such that

for

/ 2 0 with positive

constants

'

P(r) + t

Ir'

Vou

L; Jo

"o: \o- 1, Jo/-u*o !"@)b.

a,n

inequality related to inequality (1.108):

be o positiae, oonues and strictly rnonotone function on (-oo,x). Then for euery mnsuroble rcol-aolud function f we houe

* (:1," I$)d,)*

=

I,*

o(t('))f

(1.10e)

if

there

,

scgQP&).

Ha"rdy inequality. Let us consider three special cases'

tp,P)

r-o{,)*,

Proof. By Jensen's inequality and F\rbini's theorem,

=

Example 1.23. A pa'rticular choice of the function iD or P,Q in the foregoing subsection can lead to interesting modifications of the 1, satisfies (1'107)' (1.108) is the classical Hardy inequality (0.3).

(i) The function O(4 =

fo*

f* oo fr f, I\E)at) ,p,1ar]o or. (+)"

lo

Io*

lo-

0,.(#)"

Proposition 1.24. Let ib

H. P. HEINIG [1] as follows: lf. P,Q are non-negative increasing functions on [0, oo) satisfying P(0) : Q(0) : 0, then the inequality

lo*

0 satisfies (1.107) for every

i.e., we have the classical Hardy inequality also for negotiue porrers. (iii) The function O(t) = et" with o ) L or o < 0 again satisfies (1.107), a,nd consequentlS for such o we have

Let us state

Now, (1.108) follows from the last two inequalities since iD

J'

)

A Hardy-Knopp inequality

(#)"/-t*trt'r))ptu'

This result was modified recently by M'

q

Compare with the Knopp inequality (1.80).

v(/(t))dt

=: I,

(ii) The tunction O(t) = t-q with > l. Hence (1.108) has the form

In this

case,

lo*

tat (1,*

i*) "

= l,* rct!' Now, (1.109) follows immediately by replacin1

f (t) by O(/(t)). o

58

Hardy's Ineqnlitg anil Relotd Topics 59

Weighted Ine4volities of Hordv TVpe

Example 1'25' (i) If we replace

/(t)

bV

Io*

ltt/(")'

choose

o(c) : ec in (1'109) and

then

supr r)0

we obtain

"*, (: I,'

h/(')d, *

wherc

r@)+ = I,*

1(t): J-

l-

(1.110)

) l'

which is Hardy's inequality (0.4) for e : inequality (1.110) can be rewritten into the usuol form (0'3)' i'e'

-1' Fbr the case p

l- (; lo'sav,)o o,=(#)"1- {@)tu,

(1.11 1)

s@): 71t(o-r)/o1'-t/n'

by some straightforward calculations; here,

Note that Hardy's inequality (1.110) holds for p

p = L, while in (1.111) we need to have p>

> 1, i'e',

also for

1'

tndefinite weights been Up to now, we have considered weight functions u, u which have VERBITSKY E' pisitiae a.e. Only recently, V. G' MAZ'JA a'nd I' conditions for the validity if1 nu"u derived necessaxy and sufficient of tfr" Hardy inequality for the case p : q :2 with weights which L their result reads as chonge sign. For the particulax case u(r)

=

follows:

hotds

lor

every g € C3"(0' oo)

u$)dt.

f@

r I l1@)l'dr: O(1) for Jr

r -r

0

if and only if

and r -+ oo.

There is also an extension with a (non-negative) weight u on the right hand side of (1.112). Then the necessary a,nd sufficient condition reads

:$(1"#)(/-nt'rr'#)

.-

1.10. Comments and Remarks 1.10.1. The historicnl remorks in Introduction are of course subjective a.nd do not claim to cover the immense amount of literature concerned with Ha,rdy's inequality. 8.g., let us mention the paper by E. SAWYER [1]. Instead of considering weighted Lebesgue spaces .L'(tl), it is also possible to deal with spaces .L"(tl) with u a measure where ut(r)dr is replaced by d.. A characterization of measures p and v for which II is bounded from I?(v) into ,c(p) (i.e. with dz a dpc instead of. u(r)dr and u(r)dt, respectively) was gtven by B. MUCKENHOUPT [1] in the case p : q > 1 and by G. SINNAMON [1,3] in the remaining cases of the indices. See also v. G. MAZ'JA [1]. 1.10.2. The alternative conditions of the r"alidity of Hardy's inequal-

The Hordy inequalitY

!o* btdt'u(s)drl

(1.113)

Moreover, the corresponding imbedding is cnnrpoct

(1'80)' Finally, replacing /(s) bv xl@), we have Knopp's inequality yields (lzj tue choice o(r) : sp with p 2 L in (1'109)

(l t,' rcto')'* = I"* r'@)*'

f@

/ l1@)l"drce Jr

=" Io- lg'(r)12d, tf anil only if

(1.112)

ity in Theorems 1.1 and 1.2 are taken from L. E. PERSSON a.nd V. STEPANOV [U. The proofs given here a,re new in the sense, that in the paper mentioned, the proofs are given for the internal

o),

while here, we have a general interval (c, D). However, at least the criterion from Theorem 1.1 appeared earlier: G. TOMASELLI (0,

60

Weighted Inegralities of

Hoilg

Hordy's

IW

(1.14) fot p = q, [1] derived the condition B < m with B from ihe result for I < p < q ( oo is due to V. D- STEPANOV [5]' For the case g < p see atso G. SINNAMON a,nd V. STEPANOV

(frzi,(,,il

(H'f)(r,y) :

lu.

There are also other criteria for the validity of Hardy's inequality' e.g., the finiteness of the number

x:{,ir.::?o

h 1""'rrl [lra

+

l"','-o'1!a,fo/n*'

a,

where the infimum is taken over all positive measurable functions

/.

This result is again due to G. TOMASELLI [1] for p - q a"nd to P. GURKA [1] for | 1p < g < oo (see [OK, Lemma 4.3])1.1O.3. The compactness of the Hardy operators IJ and F as well as the compactness of the corresponding imbeddings (1.29) a'nd (1.30)

are dealt with in [OK, Sec. 1.7]. Here, the results mentioned in Sec. 1.6 have been taken mainly from M. GARCIA-HUIDOBRO, A. KUFNER, R. MANASEVICH and C. S. YARUR [1]. See also A. KUFNER [7]. 1.10.4. The differcntiol form a,nalogue of the discrete Hardy inequality (1.102) is, of course,

:

Io'

Inqtalitg and Rnlotd, Topics

Ir*

61

f(s,t)dtd,s,

T,* TU* f

(n,t)dtds

etc.

1.10.6. The N-dimensional Hardy operator ,2f,y fromp. b4 was modified by H. P. HEINIG and G. SINNAMON [1] replacing rhe ball B(c) by a general domain in RN centercd at c and generated by a convex symmetric neighbourhood of the point 0. G. SINNAMON [b] extended these results to more general stor-shapd domains. Moreover, also results for the cnnjugote of. ,7fy,

.: I f t4*t., ')(c) FoT /e'1a1'l ffu)dv'

ca,n be derived.

1.10.7. A complete ctraracterization of the Orlicz space for which lhe difrerzntial form of Hardy's inequality provides an imbedding

wi*@) c+ tro is grven by V. G. MAZ'JA [1]. The function O is characterized in terms of capacities.

1.L0.8. There are several results for the case where Orlicz norm inequalities are replaced by modular inequalities of the form

.f fo

1.10.5. The two-dimensional Hardy operator H2on p. 53 is obviously the composition of two onedimensional operators If - one with respect to the variable y and the second with respect to the variable ti (Hz

t)@,y\

:

H"(Hc f)@, v).

Clea.rly, analogous results can be derived also with the conjugate operator .F, i.e., for

Q-'

'l

I Qfur(x)@f)(c)lus(c)dclI LJo I

< p-t

[lJ

,r,r,rrt')1,0(,)d,]

.

LAI [1] dealt with the case that P,Q are Young's functions such that Q-t o P is convex and u1 = ul E 1, the general case is due to Q.

by combining -[I

H. P. HEINIG and L. MALIGRANDA [U. The weight characteriza. tion when P is a Young function, Q "weakly convex" and q-l o p

62

Hordy's Ineqtality ond Rnloted Topics 63

Weighted Ineqnlities of Hodg I\pe

Lebesgue space ca'se not necessa,rily convex (which corresponds to the with indices q . p) was recently proved by Q' LAI [3]'

In fact, this equivalence holds for eaeryp ) 0, e 1p-L, e I -1 and the best constants can be found in J. BERGH, V. BURENKOV and L. E. PERSSON [1,2].

cases of the 1.L0.9. For a multidimensional version of some special and I' WIK limiting inequality in Sec' 1'8 see also C' SBORDONE ii; t. 6nAssx,-H. P. HEINIG and A' KUFNER [1]'

1.10.13. According to (1.114), the norms of / a,nd of. H"f in U(n") are equivalent, with (H"f)(t) : I@ f)(r). A simila"r property is possessed also by the operator

(1'98)' is the best 1.10.L0. The factor erlp lnTheorem 1'18, formula : u(c) : 1' For possible when p : g and is attained when u(c)

I-H".

O:n:lanalternativecriterionasthatinTheoreml'lSwasgiven (1.97) is differenr!)

n. p HEINIG [1, Theorem 1.4]. (The condlion F\rrther details For the case p < q the factor e|lp can be improved. can be found in L. E. PERSSON and V' STEPANOV [1]'

i,

from L' E' PERS1.10.11. The ideas and results in Sec' 1'8 a're taken of this type and V. STEPANOV [1]' Some previous results

SON for the case have been derived by B' OPIC and P' GURKA [1] q ( p and 0 < p S g, by 1,. piCX and B' OPIC [1] for the-case the whole for Uv U. p. UBtNtC, R. KERMAN and M' KRBEC [1] were ,"ulu p, g > 0. However, in these papers different approaches usedandthepapersdonotcontainestimatesoftheoperatornorms given here.

The results derived in L' E' PERSSON and V' STEPANOV [1] holdinfactformuchmoregeneralintegraloperatorstharrtheHardy V' STEPANOV operatorl see M. NASYROVA, L'E'PERSSON and M' NASYROVA [2]' [i] ana also the recent Ph'D' thesis bv results ca'n be found in supplementary some and Further facts P. JAIN, L. E. PERSSON and A' WEDESTIG [1'21'

l.lo.l2.Thecaseofdecreosingfunctionsisdealtwithindetailin (1'1) implies' Chap. 6. Here, let us only mention that inequality that for P ) l, e

0' we have

l-

(* to' rov')o r'd's x

r{')"* fo*

'

Namely,

/-

(lt't

-: l,' r(r)dr) rdx'- lo* ro{,)*"0, (1.11b)

/ 2 0 with p > l, € < p - L, e * 1. This equivalence was proved N. KRUGLJAK, L. MALIGRANDA a.nd L. E. PERSSON [2] a'nd by for decreasing functions earlier by C. BENNETT, R. DE VOR^E a,nd R. SHARPLEY [1]. In fact, for p - 2 there is equality in (1.115) even for more general weights than o6, see N. KAIBLINGER, L. MALIGRANDA a^nd L. E. PERSSON [1]; see also Chap. 5. for

1.10.14. The proof of the inequality (1.109) is taken from S. KAIJSER, L. E. PERSSON and A. OBERG [1]. Note that the same proof gives that also the inequality

* l,' (i l,'

rc*,)*

=

I,'

o(/(c)) ('-

;)

*

that 0 < 6 S oo provided .f ttd O a,re as in Proposition L.24.ln pa,rticular, by using this inequality for b < oo and with the special cases from Example 1.25 we obtain the following improvements of the Knopp and Hardy inequalities: holds for all b such

(1.114) Ioo

"*o(: I,'

h /(r)dr) a, s.loo

rt'l (r -i)*,

(1.16)

64

Weighted Ineqnlities of

and for

Hadv I\pe

p) l,

l'(;

fo'

otiat)o

a'

( o \p fh ^. , lJ" x,t

=(#)

1,t@-t)/r\ (.'-(;) )*,

Some Weighted Norm Inequalities

(r117)

p' 58' where bs : bp/@-r) and g(r) : 16b-r)/n1x-r/p as on The inelualities (1.116) and (1.117) have been also proved by A. erZrr,rpsile and J. PECARIC [3] (cf. also [1]) bv using a mixed(1'117) are mean inequatities technique. In fact, both (1'116) *d and refsharp and strict inequalities. For more information, results article review the [2] ur"rr"* concerning the technique mentioned see bv the same authors.

2.1.

Preliminaries

A general weighted norm inequality As already mentioned in Chap.

1.,

now we investigate inequalities of

the form

llTfllq," < cllfllp,"

(2.1)

or, mole precisely,

(1,' x,nailqu(c)ac)'/o where

?

is an operator of the form

(ril@): with

b(

lc(o, oo.

. c (1"' van u@'1a,)'/o (z.z)

l,

k(x,t)l$)dt

t) a given kernel, u, u weight

functions and

(2.3)

-m (c(

66

Some Weighted Norm

Weightet Inqrclities of Hodg Tfue

on the Pa^rameters We are interested in conditions on u, u and is satisfied for certain classes of p, g under which inequalitY (2'2) functions /. are provided Example 2.L. (i)The first examples of the oPerator.T is that I[' by the llardy operator I/ and its conjugate

:

(n/Xc)

fb -..

rz

(2'3) with the kernel k(n't) : These operators are of the form a' respectively' Neces: xtr,'l(t), t 1b, or k(c,t) x1",01(t)'1.t of the corresponding validity sary and sufficient "o"aitio* for'tttu and Chapter 1' inequality (2.1) have been given in Introduction (ii) A simple extension f,rovides dne rnoilifedHardy opetator '2f,

formulas (0.8), (0.9)) so that

2.2.

with g;, rlti, i

t?il@),:Q@) The operatot

,ff

{, 8d

'fr (2.5)

is again of the form (2'3) with the kernel

and an easy calculation shows

lltr

a
tllo," < cll/llpp

t)

:

X 6,,,1

(t) er (") 4r

(t) *

x (,,r) (t) p z@)rl,z(t).

Theorem 2.3. LetT if ond only if

be defineil by

(2.7). Then inequolity (2.1) holds

A(a,b; ulVrlq,ul{r l-P) < m

(2.8)

A(a,b;ulezf,ubhzl-P)
(2.9)

with A and A giuen by (0.8) ond (0.12), respectiaely, for I < p < g < oo and,by (0.9) aad (0.13), rcspecthtely, for0 < e 1p 1@,

1
<

a

'utlt-P) (0'9) for p > q (of-clurse' with A given by (0.8) for p < q *d !I holds with u replaced by uC *i ' by url-r1' An.analogolSr.result op*"to" of the fo:m '?f, ao.d if, hzve alreadv they acted between te"o *untioned in Chapter 1 (cf' (1'37)) where spaces') classical (i.e. non-weighted) Lebesgue

ilZin;;;;tu.t

(x,

ond,

that the inequality

A(a,b;uge

:

(2'6)

> 0 if and onlY if

[-

(2.7)

The main result of this section is

l'.ilorav''

k(r,t)=X1o,a'1(t)9("){(t)'

t

similafly

as

1,2, given functions defined on (o, b) (not necessarily non-negative). Thus, the kernel ,t in (2.3) has the special form k

with given (non-negative) functions 9,

holds for all

A Special Operator

(2.4)

VQ)t{Do'

t!

appears in A in the an integral sign, see can even vanish somewhere in (o,b).

(ril@)::er(r) [' *r(r)f (r)dt+e2(r) fu .,t"$)f(r)a, Jo Jz

defined as

(tr il@) ,: v(x) l'

67

Remark 2.2. Recall that the expression at-p form (u(r)r/-e@))r-e' : r1t-n'1a)r1f (o) (behind

The case T:,2f, +.F Now, we will deal with a pa,rticular operator ? defined

-

and @t)@)= J, f (t)dt' J- t{rlot

Inqtnlities

Prool. Denote

(Sr/Xr)

:= s,@)

(szf)(r)

:=

I: rh(t)f (t)dt;

er@l: 4)2(t)f (t)dt.

(2.10)

68

Weighted Ineryalities of Hardg

Then ? =

.Sr

*

Some Weightd Norm

7W

5z and consequentlY,

u

(i) Sufficiency. Due to Example 2'1 (ii), conditions (2'8) and (2'9) imply the validity of the inequalities llsr/llc," < Cllf llp,"

and

3 Clltllp," '

llsz/llc,"

I

i:1,2,

ilrlill,,,

p

: I Jo

max{u(r), l{r(t)lPe}'

We have u(c) S u'(c) and consequently, follows from (2.1) that

:

ll"fllq," < Cll/llp,'. Let

a,pbe

such that o

Then

/e

since for

< a < B
7, t f 0, I\n): t h/r(")lo'-ro!-e'@)

sgn

{r(c),

for c€ (o,alu[B'b)' for s e (a,9).

l](or)since, due to the definition of u, and l,

ll/llt,,.

:

:

(2.13)

.

4tr1t7

lt1atlq u(n) d,r

lo

Vr<,1V"@

(le

W,r(t)lP',1,-e'rr)dt) dt

(lu' w1(o)tqu(r)*) (l: bbr(t)tp'u!-e'$)dt)

r

> B we have I| rtrttlflttldr

ll"'

,t',ortqorlr-

lu

(r)u.(c)dc

=

"

:

0 and

w,{r)r,'ovr' (t)dt.

Using these estimates in (2.13) (for

(l'v,totcu(ddn)"'

fo l1tdlou-,{*1* JA

= lu Wr4lf@'-t)a!$-P)

lu' b, {*r' I' l

>

ll/llp,' S ll.fllp,o.' Thus, it

lTf(r)lqu(a)dx

, Iu' lrU, l"',t,t0 r tqdt + e,@) I"o,h,1t1fltyatlq u6ya,

ues

:

(r)lp e)t-p' d,t

F\rrther,

and consequently, both the inequalities in (2.12) hold. This implies that conditions (2.8) (for i: 1) and (2'9) (for i:2) hold' (iii) Necessity for general functions qi,*i'suppose again that irrequiity (2.1) holds and define, for e ) 0, a new weight function

a,(r)

wr{r)l'' (tt

er-p'(fl_c:)
(2'12)

Thus, inequality (2'1) follows from (2'11)' (ii) Nicessify for non-negative functions qi,rri'Suppose that inequality (2.1) holds and that t > 0' Then obviously

llsifllc," < llT/llo,"'

69

tu ,d.r{dto'rt-P' (x)dx

(2.r1)

llf.fllc," 3 llsr/llc," + llSz/llq," '

Inqtalities

/: /;,

we obtain

(l: l{tr1t1y'or-n'4dt)

(l: w,r(,)t'u!-,'(dd,)

/

70

Weighted Ineqnlities ol Hardy

Some Weighted Norm Inequalities 7L

I\pe

Thus, we have shown that conditions (2.15) a,nd (2.1?) are for (2.1) to hold, and Theorem 2.5 is proved for the

i.e.

necessary

(l' v,otc,(ddn)''' U: vh,(,)|t' ot"-o' 14*)''n = "' If

case

we denote

B(o;a,biu,v),: u:u@)tu)"' (l"u ur-o'p1a,)''o'

, 1'''07

u,lthil-P) S C

where C is independent of a and e' Now let a subsequence), then ue -+ u and we obtain

A

:

-r

a and e + 0 (via a

sup B(0;o,b;ul9ilq,ullhrl-P) S C <

a'

(2.15)

o
given by formula But this is condition (2'8) where, of course, A is g' p Thus' condition (0.8) which in fact corresponds to the case I ( (2.S) with p g is necessary for (2'1) to hold' iZ.f31, i.e., condition

Next, we denote

E(g;a,u;u,v)::

/fi

lJ"

vr/e 1P uAV-) llrvt-p'(r)dr) '

\r/P'

with the Then it can be shon'n completely analogously that

ur(r) =

f(") =

(

g.

It remains to prove the necessity of conditions (2.S) and (2.9) for the case p > g (and for general functions p;, rh).This will be done in the following, but before we need to introduce some notation and derive some auxiliary estimates, which we will do in the following

(2'16)

Remark 2.4. (i) Let us again emphasize that conditiorrs (2.15) and (2.17) are in fact conditions (2.8) and (2.9) with A a^nd A corresponding to the case p < q. Nevertheless, conditions (2.15) and (2.17) are necessary Jor euery choice of p,q, 7 < p < F, 0 ( q < oo. (ii) We now suppose that

0
l'hz(')lq'-r'!-o'(")

sgn

{z(c)'

t o,

(I (1,'"1t11e1t1yat)'/o

A{p,tlt) :

" (1"" ut-e' F)ll)g)lr' dt)

for c€ (o,9\,

Azk,rlt) :

otherwise

(

the following estimate holds:

"op o10
E1B;o,b;ul92lq,ul'lt2l-P)

1C

This is condition (2.9) where, of course, mula (0.12).

Is'

'f it oot

(2.r8)

choice

ma.x{u(c), l,l,z@)lPe},

(

tr

remarks.

then the last estimate reads B (B; a,b; ulPilq,

p

(2.r7)

gven by for-

|"' (l"" ultyleltyy at)'

* (1,' *

with|:i-;.

/

-r' (t)lt1t

1t11o'

4r)'

o.-p' (*)lrl,(dle'

dr),

/o

o'

r'-o'

1ry141r11o'

a*)'

/'

72

Weighted Inequolities of Hardg

Some Weighteil Norrn Ineryolities 73

I\pe

Then we have

Ar(p,rh+ /z)5(Ar(v,$) + At(v,'l'z)), Az(p, tt + rlz)S@z(v, rbi + Az(P, thl.))'

(2'19)

Ar(W,tb)

ultll-r) from Indeed: Since .41(9, ry') is nothing else tha"n A(a,b;ulglq' formula (0.9), we can rewrite it due to Remark 1' (iv) as

(+)"'

and llfll < C with C the constant from (2.1). We want to show that conditions (2.8) and (2.9) are satisfied, which means, due to formula (2.18), that

Ai@"t')

1co,

A1w,,t)

:

(l"'

(l:

up'11s1t11061)'/'

,

(1""

at-e' Q)lt

f

G)l''

dt)

/' le

,l(\rc[^'

*

(1"" ,t-p'(t)l't'z('lt''

: (c1,d), A2: (d,,d1), ,/tij:44XAi, i,j:L,2. Since ry'; : rbt * ,h;2, it follows from (2.19) and (2.21) that at least

o')'

|

and the second estimate

in (2'19) can be derived

If, e.g., A{W,rlt): oo but Az(pz,tz) ( m, then the operator .91 is not continuous while 52 is continuous. Consequently, the operator ? as the sum of ,Sr a^nd Sz cnnnot be enntinuous, which contradicts our assumption of its continuity. Hence, assume (2.2L). We now construct a sequence of intervals Iq,\l, i:1,2,..., by the following procedure: Take [cr,dr] : [a,b] and let d be its interior point, e L d < d1. Denote 41

,t-o'1t11l,bt4)lp' + l$2$)lp')dt)

at-p'(t)1,!t(rrro' or)''o'

one of the numbers

Ai(gi,ry'rr) and Ai(pi,{iz) must be infinite.

If

)'

A{%,drr) : oo,

a"nalogously'

Az(gz,thzt) <

a

or

The necessity of

conditions (2.8), (2'9) for p > q

inequality Suppose that, for the operator f defined by (2'7), holds. Thus, f is a untinuous operator'

T t LP(a) -> Lq(a),

(2.21)

Indeed:

u@)ls(x)lqdx)'u

\JO

[" Jo

+ Lq(u), i,:1,2.

A(pr,dr) : m and Az(pz,tltz): *.

,r-r'1t)l{r(t) +,br4)lo'dt) (zo'-,' S -\

(2.20)

We will proceed by contrailiction.lf. conditions (2.20) are not satisfied, then we c:rn show that necessarily

inequalities Now the first estimate in (2.19) follows from the obvious

(

a.

Notice that the first condition in (2.20) is necessary and sufficient for the continuity of the operator ,91 from (2.10) while the second condition in (2.20) is necessary and sufficient for the continuity of the operator Sz from (2.10),

Si: I?(u) where

Az(pz,tbz) <

Ar(pr,{1) ( m, Az(pz,rhzr): a, (2'1) then we take for

l"r,drl the interval

Ar(Pt,{n): o,

{c1,d];

if

Az(gz,{zz) <

a

74

Weighted Inequolities of

Hadg TW

Some Weighteil Nonn Inequalities 75

or

Indeed: By (2.15) and Remark 2.4 (i),

x,

At(pt,rltn) <

Az(gz,rhzz\:

a,

then we take for ltz,dzl the interval ld,dil, and stop the procedure. If none of these four possibilities occurs, then at least for one i (equal to I or 2), we have

: m,

a; in this case we take for l"z,dzl the interval 6l Ar(pr,{ri)

Az(gz,rhz;):

[q,drl)[c2,d2l:

[ca,d3]

%(t) #

(

0 for a.e.

f, we

have

t'o'

fr

,'-o'rt)11,r(qle'dt)

\"/c*

/

since ry'1s is zero outside (cx,dx), (see (2.18)) that

it

. *.

follows from the definition of A1

Ar(w,rb*) /o

dt)'

:

[tr, d1] and thex : rh;Xa, i : 1,2. Suppose that the sequence {6p} is finiteand let d,, be its last term. The operator ?ln, Denote dr

and since

= (l:-. (l'o'av"(t)tc

) ''''

follows that

yr/e 1 fir / rb r l/p' ( at-p'(t)ltht(t)lo'at) s c 16,, I\Jdr l. "(r)lrr(t)lcdtl | / \Jcp /

(2.22)

and continue the procedure just described, choosing an interior point d, c2 1 d I d2, and considering the intervals Arr : (cz,d), Azt : (d,dz). Now, Ai plays the role of the interval (o, b) and conditions (2.22) play the role of conditions (2.21). So, we have constructed a (finite or infinite) sequence of intervals such that

it

(T^D@): e{x)

['

,br*ft)f(t)dt + rr@) fo g2*(t)f (t)dt Ja Jz is the restriction of T to functions with support in 6r. and, moreover, it is not continuous, since it is the sum of two operators ,S1,n, ,S2r, from which one is continuous and one not continuous to due the construction of the interval drr. Hence, ? is not continuous, either, and we have the desired contradiction. If the sequence {d1} is inf,nite, then there is a point c, "

€

We now show that, for every

n

" (l:

a.e.

in (a,cj),

= o

a.e.

in (d3,0).

(2.23)

(1". *

-'

: (l',^rte,(qf x

/c,

d - p' (t) l,h r * rr;lo' or)''

u(t)te,(qy at)' = (l:,

bx,d*|.

/c=l

,h@) : o tltz@)

" (l :

o'

p' ut - (r)1,p r *

@)lp' dn)

/q

/

n)t{ L (t)tp' dt)'

o',' - o'

(,)g,

t

(x)r' ds)

dt)

(+ I:*U:

s (l'r"@te |otc dt)"'

,r-p' (t)lrbr(t)lo'

(l:-

dt)'

obo' (t)g,r(t)tp'

Or)"' dt)

76

Weighted Inequolities of Hordy

I\pe

Some Weighteil Nonn Ineryalities 7T

2.3. General Hardy-type Operators. The Fundamental

The construction of the interval lcx,d*) implies that

Lemma

At(pr,drr): m

Deffnition 2.5. if it has the form

and consequently

/ fi tr/c :-. llu(t)le1!)lqdtl / \Jcr

(xf)(x),:

Now, due to (2.15),

.* ,urrrr(t)far)"' (1"' ,t-p'(t)r,hr(tyltat)"o' =" for every x € (a,c1), and this implies that r/1(r):0 a.e. in (4,c1). The proof that rp2(r):0 a.e. in (d6'D) follows a^nalogously. In view of (2.23) we have

and consequently, the operator

T

a.e.

in (a,c),

a.e.

in (c,b),

Qzf)@)

= :

k(r,t)

x1o,.,1@)ez(-,

l:

${t)f

I:$2ft)f

€ (a,b).

:

(0,m), i.e., with

Io'

k(c,t)f (t)d,t, o ) o.

(2.24)

0,

t ( c, in c and decreasing in t

0<

is increasing

(2.25)

+k(z,t),

The conjugate operatot .k to

(t)dt.

llfrll = A{s,r/r), ilrzll o Az(pz,rltz). the norm of ? we have llfll S C 1cr, but on the other

x, k(a,z)

0<

t < z < s.

(2.26)

Such kernels are sometimes called Oinarou kernels.

(t)dt,

&s)@)

Clearly,

For

r

k(x,t)f (t)dt,

and

+ (T2f)(r)

d

,:

/c(c,f) >

can be rewritten as

x1",oy(a)rr(

is a generol Hardy-type operotor

The results for a general interval (a, D) will be summarized at the end of this chapter (see Sec. 2.6). We suppose that the kernel ft satisfies

where

(rrl)(r)

(x il@)

/c

Qf)@): ("r/X")

1""

I(

For simplicity, we will deal with the case (c,b) the operator

(l,u

,ln(r):0 tbz(r):0

We will say that

,:

I,*

K

is given by

k(t,r)s(t)dt,

o

)

o.

(2.27)

Remark 2.6. If the kernel k(t,t) satisfies the monotonicity conditions from (2.25), then it follows at once that ha,nd,

C{Az(pz,rbz) * Ar(et,{r)) = oo ;(ll"rll + ll"rll) 2

I

k(x,t) 2 j(k@,2) + k(z,t))

for 0 1t


r

1

llTll >

holds, i.e. the lourer bound implied by (2.26) is satisfied.

due to (2.2L). But this is a contradiction, and thus we have completed

the proof of Theorem 2.3.

D

Example 2.7. The following kernels satisfu conditions (2.25) and (2.26):

78

Weighted Inequalities of

Hady \pe

Some Weighted Norm Inequdities

for.l ( p,!_ oo_ boundedly with norm C, then analogously its : conjugate K from (2.27) maps Lc'(ur-c') into I?'(ur-e'1, -

(z) The @naolution kerrel

k(t,t) :9@ where

t')

fr , 70, 1ur_e,) +

g satisfies

e@+b) x 9@)+p(b),

(ii)

0

1a,6< m.

(

k(x,t): - (;)

lr* xr a (t)lo'

ut - t'

:

{xd{df @d,l

where O satisfies

(iii)

SUp

ll/llp."=r

xO(a)+O(b),

0

(

o)0,

c>t>0,

k(x,t): l"y?+, - t' P>0, r)t)0. : [' eft)dr, x>t>0, Jt

weighted Lebesgue spaces for a wide range of indices. This problem has been extensively studied in the primary literature (see, for exarnple, S. BLOOM and R. KERMAN [t], F.J. MARTiN-REyES and

E. SAWYER [1], R. OINAROV [1,2J, V. D. STEPANOV [1,2,4] and the sources cited there). (ii) Let us mention that an important role will be played by duality.lf. the operator K from (2.24) maps tre(u) into Lc(u),

K:LP(u)+Lq(u)

s(r)(K

f)(4dtl

where we have applied F\rbini's theorem, Hiilder's inequarity and the assumption that

llK

f llc," s cllf llr,,.

(2.28)

p. 13.) (iia) The particular operator .I(,

(See also

with a non-negative function g.

Remark and Notation 2.8. (i) In the sequel, weight functions will be characterized for which the operator K is bounded between

(t) or)''

ll/11","=r

(iu) The integrol kernel

k(a,t)

fo*

l/ 'd::='

a,b < oo.

The particular cases of (i), (ii)

k(x,t) : (r-t)",

Q|_o,y

with the sarne norm C. Indeed: by duality we have

The kernel

O(a.b)

Lp,

(xf)(,).-

#

1""

O

- qafe)dt, o ) -1

is called the Riemann-Liouville operator and its conjugate

(x

il@),:

#

I,*

U _ a)"

f(t)dt

is called the weyl froctionor integur operotor. Flom the results be. low it follows in pa.rticular that weights will be characterized for which these operators are bounded on weighted Lebesgue spaces when a > 0. The case -l < o < 0 is not covered here

k(r:4 =

(s

- f)", -1 < a < 0, is not an Oinarov

because

kernel. (Recall that

in Example 2.7 (iii), this kernel appeared only with o 2 bfy

80

I\pe

Weighted Ineguolities ol Hardg

Some Weighted Norm Inequalities

(iu) In order to find necessary and sufficient conditions on the weight functions u and u under which inequality (2.28)' i.e., the inequality

(1,* x*n@)rqu(r)d,n)"'=

"

(f

gl

Q) A Ks e Lq(u), then f*.-

.

f@

-.

l (Ks)q(r)u(x)dx N JoI Jo

ff@)lpa(r)d,)"o (2.2s)

s@){oi''odq-t1x11xou11a1ar foo

+

|

JO

holds, we need a technical lemma. To simplify the notation, we denote

s@)(frs7t-t1r'11x"11r1a.

fors)0

:: g"n11r7 :: (r"rrX")

lo"

I,*

u'1r,r1n1r1or,

u"rr,flh(t)dt

(2.30)

with /c an Oinarov kernel. Nolle that if s : 0, then (2.30) reduces to the Hardy operators l/ and -f,[, respectively. We will write

Kyh: Kh, krn:

frn.

F\rrthermore, let us emphasize that without loss of generality we can assume that

/>0.

(2.31)

Now, we are ready to formulate our fundamental lemma.

Lemma 2.9.

Suppose that 1

( q( a

and thot Kq,ko oo itefined

b'y (2.30).

(") If Kf e Lq(u),

Prool. since the proof of (b) is quite similar to that of (a), we omit proving the first pa^rt only. Denote the left and right hand sides of (2.J2) by .I and J, respec_ tively. F\-rbini's theorem and obvious estimates vield

it

r-

Io*

: T,* "@ (1,'

{*

ilo{r)u(r)dt

([

k@,t)r(edt)

dn

ry

f* rcl l,* rr,,t)u(r) (lo' ur,,")f (4d")q *

Io*

rc l,

k@,t)u(r)

(1,'

' o,o,

ur,,z)f (4dz)o-'

*0,

=:/o*fr.

then

!

k@,2)r(z)d,z)'-'

: l* ra I,* ur,,t)u(x) [(/'. |")ro,4r@d4o-' o,o,

Since z lo*

u(,) (lo' ur,,t1y1t1at)q ar

fo*

*

4t

1 x in the integral

f {dt*07'10-r(x)(Kqu)(n)dr

Io*

I@)6.7t-t({(fru)@)dn.

.le one obtains

k(r, z) x k(x,t) + k(t, z), (2.32)

and consequently,

from (2.26) that

82

Ineqnlities of llordg fupe

Weighted

.16

Some Weighted Norm Ineqnlities g3

:

x !,* to (1,* ooo,t)u@)ar) (1,' ,@0,)o-'0, * f- tri I,

:

fo*

k@,t)u(r)

(lr'

un,z)f

Io*

@az) *0,

Thus, we have the result for e

It q * 2, let us wrire

f g)(xu)(t)1x yy-r

,r:

1t1at

Thus f x J * f1, and since.Il ) 0, the lower bound of (a) follows. To complete the proof we must show that I1SJ. Consider first the case q: 2. F\bini's theorem yields

foo

f@

r@

f@

:

f,

J"

l@

I

f@

f@

J,

foo

f@

foo

h N Jo I tAllJt f@lJz foo

foo

| f(t) JtI f@ JzI

foo

fz

Io*

2.

f ft)rt(t)dt

16

/f@

I I@ |\JzI JO

+ k(z,t)

and

k'(r,z)u(r)drdzdt

k(x,t)u(x)

I,*

f@

f, *

k(z,t)k(r,z)u(x)d,xdzdt

I,*

(1," ur*,")/(")d")q ' I,' *r,,2)f

k@,

1,

o,

2)k(t,t)u(n)

(z)dzd,x

(lr' *rr,s)/(")ds)q' 0,0,.

f@

l"*

k2(r,z)u(t)

I,* rk) I,*

(lr'

k(n,z)k(z,t)u(t)

rr,,s)/(s)ds)o-' oro,

(1," rrr,")/(")d")q '0,0"

:' 4(r) + 1i(0.

/n@ ^

= Jo I f@lf@dtl Jz Jo +

rawee)gf)(z)d.z

Here we have again used Fhbini,s theorem. But since t < z < r, condition (2.26) yields k(n,t) x k(r,z) + k(z,t) and consequently,

rt|)x

r@

Jo

:

k(o,t)k(r,z)u(t)hdzd't'

Since t < z < r, we conclude that k(r,t) x k(r,z) consequently, by a repeated use of F\rbini's theorem,

+

=

I{t)::fr* k(r,t)u(a) (1,' rrr,z)f (4dz)o-'

: I f@l *@,t)u(x)l t'(r,z)l@)dzdndt Jt JO Jt

:

f*

where

: J.

11

tu)t*of)()(R2e(z)dz*

: J.

'

r$lwou)(t)(Ksf)q-t1t'1at

*

lo*

k'(r,z)u(r)dtd.z

\ rz k@,2)u(x)dol I k(u,t)f(t)dtdz ,/ JO

Thus,

,r=

Io

t(t)te)dt

* Io* l(t)4(t)dt =: rf + {.

(2.33)

84

Weighted Inequolities of Hardy fupe

If q >

Some Weightd Norm Ine4uolildes 8b

2, we apply Hiilder's inequality (with exponents q

l*) t" obtain rflr;

:

lr* rcl t,*

I

s l,* ta(1,*

(lr'

oru,

rr*,s)/(s)ds) ' *)'-""0-" o"

\ (c-z)/(c-t)

k(r,z)u(a)(K7'Y-r@d")

:

dz

and thus, applying F\rbini's theorem (twice) and again Htilder's inequality,

'(l

:

lo*

\ (q-zlk-r)

k@,2)u(x)(Kf)q-t@)d'-)

)q

(d*)k-

to*

)

(q-r)

J@-z)/(o-r)

2, we find that

t{Drlt)a,

tr* rcl f,* f {,)ot,,q I,

lo*

z) /

f)q-' @d")r/(c-r)

, (1,' k(r,s)/(s)d")n-'

:

4 / f@

u@) (K f

Simila"rly, using the fact that e

4:

qu)t/(q-r) Q)

"U

" (I*

: (lr*, t"lturu)(z)(Ks

t@

tt f6

k@,2)u(a)(Kilc-'@)ds) *)(q-z)/(c-t)

q' ur*,")/(r)a")

uor,,z)u(c))t/(c-r)

* (1,* k(x,z)u(r) (1,'

" (1,* r@ (1,

: (lr* tra6,qk)(Ksr)q-'14a,)r/(q-r)

*ot
(q-t)+(o -2)/ (c-r) ur / @)

,

-

na

Io"

ur,,tlro)

k@,2)u(a)

dad,zdt

I,

k@,2)u(x)

" U,' ft (c, s)/(s)ds)o-t *oro"

dzdt

ra)6ou1r/(t-t1 r", tc-z)/te-t)

" (1,*

k(r, z)u(x)(K ilc-t @)dr;

lo'

I {Doro,

s

fo*

taw il@ (1,* k(x,z)u(x)dr)'/to-" (c-z)/(c-t)

/ f@ r . (l k(n,z)u(a)(Kilc-t@)dft)

dz

86

Weighted Inequolities ol Hordg

7W

Some Weighteil Nortn Inequalities g7

Hence the result follows for q

Finallyifl
. (l-

f@

(1,* k(x,z)u(r)(Kilc-t(n)dr) *)(c-z)/(c-r)

4(r) : l,* rel I,-

: (lr* tt"lt* 7'so-t(z)(Ku)(4d")t/(c-t) " (/*

: (1,*

f

u(r)(K f)o-, o, (1"' k(n,

r (a)d.z)

z)

: *)

s

" *

=

C

x

Is

orrr,z)u(x)(lr'urr,s)/(s)ds)q

'*0"

J,

x kz_q(x,

z)d.ad.z

arrdsincexlz)t. (z)

(

x

y

1t

-

|

(/ (fre e) o

-

")"'o

" 7k'z)/(o-r) (1,'ur*,";y1";a")o

(lr*

' s (1,' ur,,")1(")a")o '

f()(koQQ)(Kof)c-t(z)dz I t/(s-t) ' '

foo

Jo fQ)$7'1t-t(z)(xu)(z)ar)

1k-z)/(t-rl

where the second inequality follows from the fact that in the second variable. Hence

lc

is decreasing

Jr/(q-r) /c-2)/(c-r)

S ClJrlk-r) since .I

2.

f* -. f@ / P \c-2 kq(o,z)u(a)\J, k(x,s)!(gds) f@ J,

k-z)/(c-t)

Consequently,

11

>

1Is

+

Jrl(o-z)/(q-t)

*.I1. The indices q - I and

fi *u conjugate, and hence

rft4 s

l,* r{4(1,' ,av,)'-

r@ (1,' r6ya,)o ' (froQe)azd.t

r1o

.16

= J, and thus

h ScsJ +4h, q- L l.e.

hSCs(q

-

1)/.

kq(x,z)u(x)dnd,z,

and F\rbini's theorem vields

(/o+/r) J It{n:wzo-tiG-mq' But we have shown that

I*

:

fo*

n&od@ lo r@(l:

: #I-

f

(z) (K ou) (z) (K s |

r61a,)o 'ata"

| )t- (z)dz,

88

Weighted Ineqnlities of

Hady 1lpe

Some Weighted Norm Inequalities g9

since

f" rat (1,' ,av,)'-' o, :

-*

f*

I,' #r(1,' ,av,)o-'

-...+ l^ fQ)gu)(z)(Kf)q-t(z)dzl rol

o,

: #(1,"11"P")o: frttrflo-'(,).

: ]q- -t. L

This completes the proof of the lemma.

2.4.

Similarly,

rf lr;

since

k(z,t)f

(z)

General Hardy-type Operators. The Case p Sq

*0,

Now we are able to state and prove the first main theorem of this chapter describing necessary and sufficient conditions of the validity of inequality (2.29).

I,* urr,z)u(r) (lr' urr,"y11"yr")o-' oro"

Theorem 2.LO. Let 1
: Ir* ur",t)lk) I,* urr,z)u(r)(1,'nrr,"111"p")o-' = I,*

'l

t < z < s and lc is increasing in the first variable.

Hence

(2.30). Then the inequotity

rf

(t* wt@)u(ddx)'/o . c (1,* raru{*1a,)'/' (2.t4) " (1,' k(z,s)f(s)ds)o-' lr*

n,l

Io"

Ir*

f

holds

for

> 0 if

all

f

,4s

::

(z)

(x

o-'

I,-

#

()

. *.

(2.36)

yyc-t (z) (K u) (z)dz

::

sup(.ks u1ut1t1

The best ennstant

r

(2.35)

and

A1

ll,-

a

orr,z)u(r)d,rdtdz

x rf+rf s

if

sup(Fq ult/o 1t11Xsrr-p')L/p' (t) <

and therefore I1

and, only

ur",t)f(t)

" (lr" k(2, s) f (s)d,")

*

o,o,o,

(R oe (z) (K s r)o- | (z)dz

(Ko,ur-n')r'u'

C in (ZB$

1ry

sotisfies

C = ma:r(,40,,4r).

(2.37)

90

Weightel Inequolities ol Hadg fupc

Some Weightd Norm Ineguolitiet

Remark 2.11. Due to (2.30), we can write .46 and .A1 also in

But since lc is increasing in the first va.riable, we have

the form

As

:

(I;-n'1{ds)

(1"* wnl'@)'(dd*)ttq ,

H(/-kq(s,r)u(s)ds)"' yr/t 1 p / f@ \r/P' At = sup{/ u(s)dsl | | kp'(t,s)ar-p'(s)ds) t>0 \Jr / \J0 /

, (f

(2.3s)

/ f@

H , i.e. if we take k(x,t) 1, then .4s a,nd A1 coincide and are nothing else than the constant .4 froni (0.8).

Proof of Theorem 2.10.

(i)

and

(Ko,vr-o"11q

,: [' JO

l{'(t,syur-n'(s)do <

m

ft(s) Then

it

::

X1o,t1@)kP' -1 1t,

r;rr -r' 1r;.

follows from (2.34) that

(1,* wnrpvul4a,)'/o

t

"

(I* yy1,1,14a,)u'

: (lr' t{'(t't)at-e'P1a')l "

= C(Kp,ur-n 7r/o$).

it

*)''

o

1

o,

follows that

(2.g9)

for some , > 0. (If not, then u 0 a.e. But if (2.3a) is satisfied then = the left hand side in (2.34) is zero and hence u 0 a.e., and the = convention 0.oo : 0 implies that (2.35) a,nd (2.36) hold.) If (2.39) is satisfied for some t ) 0, define a function fi by

-r' 1s,o,)'

\r/q / rt

/o

Necessityof conditions (2.35), (2.36).

Suppose

r s)&e' - (t, s)ur

, (1,* u@dr)"" (lo" kp'(t,s\at-p'(")d") : (k ouyr 1t)(x ur -o' 1ty

:

If K is the Haxly operator

"O, (l:,t(r,

(Rsu1r lo 1t11Ko,ur-t' 1Un' S C. Hence (2.36) holds. To see that (2.35) is also satisfied note /f6

U,

that (2.8a) is equivalent to

1frs|o' 1dt)r-o' (dd')'/'

sc

(lr* sa'(t)uL-e'(rr*)''

(2.40)

with the same constant C as in (2.34) (see Remark 2.8 (ii)). As above we may assume that

(fr.gft) ,:

Ir-

&q(s, r)u(s)ds

<

oo

for some t > 0 and for such t define a function 91 by

gt(s)

:

I Xp,-y (c),tq- (c,

t)u(c).

92

Some Weighted, Norm Ineqnlit:iea gg

Weighkd Inequolities of Hordy Tlpe

Now by the upper estimate of (2.32) from Lemma 2.9 we obtain

Then (2.40) yields

(xo,'-o'1're' (r)

:

(&u)

C)

(1""-"'t') ('[

= (l'ur-p'(s)

I :: [* Jo

(lr*

up'11x1q@)d,r

s (\'ro[* x,lt*oy1o-t(fl(kqe@)dx N

kq(x,t)u(!dr) o,)''n

* Jo/ [* f@)(x y1o-r1r)(ru)(c)ac) or',s)ke-l(c,t)u

t*lo')n o")''o'

::

C(Jo +

J) .

By Hiilder's inequality

s

(lr* 6 n,)r' (s)ur-r' (")a")'/o

=

"

: so

"

(lr*

oI'

f@

Jo :: I JO

61u'-o'1da,)'/

(1,* kq(n,t)u(do,)"0' = c (ko,)"o'

(r)

f(r)(xoy'1t-r(r)(Kqu)(x)d.a

/ f@

\l/p

\J0

/

' (/If

that

ut-r'@)(Ko1;r'(c-r)(c)( frou)e'(x)d,x)"o'

(ko,)''o

1t;

(ro"-o')"'

{')

=

"

=

rr-n' @)(k

qu)P'

(r),

/ ,*

/ f.

1t'(c-l)

( | yltlatl d,\r/r' | | .,(") I "\Jo-" / \Jo J

In particular, (2.42)

i.e., the lower estimate in (2.37) holds. (ii) Sufficiency of. conditions (2.35) and (2.36). We may a.ssume that / has a compact support in (0, oo) and that 0 < ll/llo,' < oo. The general case then follows via Fatou's lemma.

(r)

then the second factor on the right hand side in (2.48) takes the form

and hence (2.35) holds.

man(As,A)SC,

(2.43)

we denote u

(2.41)

.

and can be estimated from above by

*_, (I*

u @)

p f (x) dalrt

-

rt r n

via the standard Hardy inequality (cf. (0.6)) if and only

c' ': i$

/ f@

if \l/p'

lr/@'(q-rD t fi u@)an) i-n'@an) < oo (l \Jo

94

Weighted Inqualitics

ol

Hordg

7W

Some Weighteil Nonn Inequalities 95

(cf. condition (0.7); the application of the Hardy inequality is correct since 1 p). Moreover, wehave C2 x Cs. But by (2.35),

f€foo-

I u@)dr : JtI

Jt

S

,q{'

(lo*

or-o'O)(frep'@)

I,

: (/-

ov/(r)(Koa!-P')-q(c)dr abP'(r)

(lo'

o'-o'{4a")-o a,

(1,',r-r1s)as)'-'

l,*

:

l:

Cs
- / t@

\r/P+(q-rl/P a@)fe(x)da) 3cqAlllflll'. llo

(2'M\

*)"o'

rt-r' {O(fr

u)p'

I"* at-p'(x) (1,*

(l- {-t' (x)(fu)p' (r)(K f)o'ro-')(')e) "o . 1r.ou1

uu,4u1lat)e

an

=

(/-

"u,

:

vp1o1Qa,)'to

(t)dx

.*-p'(r)kp'(t,da,)'/o or)' "u, (1"

:

(l-

*-' ,fl(fren'@)a*) a1rc ry're-rr1",)1/r'

(/-

=

Similarly,

talwvlt-t(fl(fre@)dr= fo*

(1"*

(I

: (1"* u(t) (K

Therefore,

rr ::

Or'tr-r)1s))

=

and hence

sctr-'

(lr' or*

By Minkowski's integral inequality and by (2.36) we have

s #(l:,r-r'1s)ds)

to

K/ is increasing, the second factor on the right

at-n'61(Kou)P'(r)dr

= $' l,

=#

Since the function

hand side in (2.45) can be written as

ol'

(1,

oi'

(-d

o,

aFp'(x)kp'(t,da,)'/o or)' ur -t' |

/ e'

(fl dt)

uo)(ksq-L/c(qdt)e

f,* alkoult/c'(qdt)

(q')p' A!' (kou1e'/o'

1sy .

Hence

Jt S q'Arll/llpp

(lr* &*lo'/t'

1s)d(K

I',0'o,o'

6))"o

,

.

96

Weighterl Inequalities of Hardg

Some Weightd Norm Ineqrolities 97

IW

and by Minkowski's integral inequality (we have d p 3 d it follows that

Jt <

Atq'1rf,,",'

(l-

/d > t

u(sx/r/)q(')a')'/o'

since

holds

for oll g > 0 if

le :: (2.46)

if

sup(Kqulrh1t11frsat-p')Up'(t) < ,>0

a

(2.4e)

ond

i1 ::

Thus

I :

ond only

sup(Keu7t/t1t11kr,uL-p')r/p'(t) < m.

(2.50)

t>0

llr/ll[,"5("ro + 4)S(ll/ll$,, + llfllp,'Itlq')

The unstont C is the same os in (2.34).

Coll/ll$,, +Collfllo,'Irlq' '

ProoI. (i) The proof of the necessity part is the same as in the proof of Theorem 2.10, and it is therefore omitted.

Using Young's inequality ab b: Il/'t' , we obtain

1

{ + #, q )

r scollf|L"*ffVW,,+

l, with o:

Collf llp,''

(ii)

i,

Sufficiency. By

llkgllc,"

which implies

:

sup | [* &nlrqf @)aal lJO

ll/llo,,.r -q' =1

:

/5ll/llg,",

duality we have

sup

ll/ll"',.'-" =l lo*

I

o@)txf)(4d,1

Hiilder's inequality yields

l.e.

llK/llq,"5ll/llp,'

,

from and this is the required estimate (2.34). Moreover, it follon's with together hence (2.A4) and, (2.46) that Cima><(As,'A1) and tr (2.42) we have (2.37).

Now,wecaneasilyprovethecorrespondingresultfortheconjugate operator

K,

defined bY

(xg)(r),=

L*

k(t,x)s(t)d,t, r )

o.

(2.47)

Theorem 2.L2. LetL < p 39 < @' Let fr be the a$tt'gote Hatd'y' giaen by type opemtor defned by fonnula (2'47)' I'et Ks and K" be formulo (2.30). Then the inequality llfrsllo," S cllgllp,,

(2.48)

< llr* o{,ltx/X,)dcl llello,,llr

fllp,,o,-,,

.

Finally, since As, .41 correspond to As, .41 with u, u and p, g replaced by ur-n', ur-c' and q',p', respectively, Theorem 2.10 implies that llK

I llu,,,, _,, S cll/110,,,,,_*

holds if and only if conditions (2.49) a,nd (2.50) are satisfied. Combining all these estimates, we obtain (2.4S).

ct

Example 2.13. We state the results for the Riema,nn-Liouville op erator and for the Weyl fractional integral operator, mentioned in Remark 2.8 (iii).

Letl
98

Weighted Inequalities ol Hardy fupe

Some Weightd Norm Inequatities

2.5.

(i) rf

General Hardy-type Operators. The Gase p

Notation and Remark 2.14. (i) Let

(Kf)(r): ['@-t)df(t)dt, a)0, r)0, JO

Define

r

fE (/

("

111 -::--rqp and introduce numbers

> 0 if and only if

-

t)oqu(s)d,)'"

(lr',r-r'1s)ds)"' . *

and

''' s)ao'or-o'(")d") "o' . * :$ (/-,(s)as) (lo' ,, (?i)

.

Bs

:: {I-

l{xou),/n{t)(Ksut-o')r1o'(r)] rr-r'rrror}"'

81

::

f(kodr/o{t)(Ko,ot-n'y/e'(qf'

{I-

Gd@): Jr[*$-x)ds(t)dt, o)0, r)0,

llkgllo,, < Cllgllp,, > 0 if and only if

(2.s1)

If .fif is the Hardy operator H, i.e. if we ta.ke k(r,t): L for t I x, k(x,t) : 0 for t ) x, then 81 is a multiple of 86 and Bs coincides

max(Bs,

r

r/p'

s)'qu(s)d'")"' (f "t-c'1s)ds) :lE (/'(t -

<

oo

(1,* u - t)ap'ot-o't,)r") "o

. *.

Br) <

@

(2.52)

uhere Bs,B1 are defined, in (2.51). Moreouer, the best constant C in (2.3A) satisfies

C

and

'"

.

1(q(p0 rf ond, only if

then the inequality

H (/',(")d")

ullat}'/'

,

with the number ,4 from (0.9). (ii) R€call that for the Ha,rdy operator and its conjugate, a characterization was given for the full ra,nge 0 < g < p < oo, p > 1. In what follows, we first restrict ourselves to the ranje

If

is satisfied for all g

us now consider the case

by

llK/llc," S cll/llp,"

/

q

01.

then the inequality

is satisfied for all

>

c

ma:c(Be, .B1)

.

(2.53)

Proof. (i) Necessity of (2.52). Choosing u and u appropriatelg we may assume that Bs ( oo, without increasing the constant C in

100

Some Weighted Norm

Weighted Ineqralihes of HordV lbpc

in (e'-Lle\' (2.34) (e.g., replace u and ul-p' by us and u]-p' supported coni > b). Once the result is proved in this case, the additional general the approximating ditions on u and u can be removed by weights by these oppropriote weights' Now define / bY

f (t) : ( K

1l (x

oat

- o'

and substituting

f

ou1'

/ @tl

Tnen a[/p: llfllp,' by assumption, we have c n5/r

:

1'

/ (ps' )

/ tt _ \c-l \ \t/q k@,t)f(s)ds) a.)at) (/, '

:c'|lq' (1,* tn ru,")$)(If(")d")o

into (2'34) which is satisfied

: c'/q' (1,* ral&,qpl x

: (l-

(1," uo,rrr(t)d,)q u{4a*)'to

^o

(1,' k(r't)f(t)dt)

: (l-t(') (/

k(r,t)u(c)

(l"

>

k(c's)f(s)d")n-'

4'''

' (lo" or',"1f1";a")o *) *)"'

( [^* f (r)( [.* 01,,''7u,",(l'k(c,s)f(s)d")o-' \Jt \JO

that g where we have used F\rbini's theorem and the fact

s ( t ( c, we have lc(c,s) = lc(c,t) + t1t,s) lc(c, s) 2 Czk(s,t) for some Cz > 0'Hence c n5/n

2 c;ro' (

/-

or)"'

o')o)''' > I'

Since

and in particular

lttl ( /- k(r,t)u(r)

(1,'&,u1*od6y

(Ksur-n' 1, ltoo) Tsyur,n'1s;a")

clh' (lr*

q1

or)t'o

6r"lr+t(q-r)/Q4) (r)/(r)

/ ft(Kout-o' \c-r \r/c . (/, o, 1'1too') 1s1ur-n' g)) ) since ^Rou is a decreasing function. F\rrthermore,

fo'

.

'

(t) at - P' (t)'

cilIllp," > llKfllo,"

: (l-

Inqualities

t*

oot-o' 1' / @o) 1s1ur*r' 1s)ds

:

fo'

,r-r'1",

:;rch :

(/"

,'-o'1r,1ar)r/(et') o,

(lo','-o' t,r")

firWoar-P')'/@'d(t)

"'*

l0l

102

Weighted Ineryalities ol Hordg TVpe

Some Weighted Norm

(+)','

"tn

(ii)

u)r+r / rno'' " (l-,4 /

("r+)' (K

x

sur-

rr)

f (txrour -

o'

( !o* {x r'

Y

/ (p' q'

109

Then Bllo' : llfllq,,rrr-o, and we proceed as before, using of course the dual inequality. The result B;SC proves the necessity part.

so that

Cn[/o

Inqualities

\

+'

n1,1t+r

/ @c'

/

)'

t

(nq)+t / rod

I 1t7ut - t'

(*r+)'/o' (lo* 6nu1le

p'

il (il dt)

tp'

Sufficiency

(2.52). Note that by Lemma 2.9,

r :: Jo[* u611rcry@)dr

1t)

s

",

/q

1t1

of.

(1"* f (,)(Kof)q-t(r)(Kou)(r)da

*

at)L

::

p1

Ir*

I @)@ 71t-' 1r11R uy1,ya,)

Cr(Jo +

Ji

.

estimate Js, we apply Hiilder's inequality for the product of three factors with exponents p, p/(q - l) a^nd r/q. We obtain Tlo

x

(K

sur-

n'

Y

h' 1tlar- n' 1t) Or)t'

Jo

= (*r+),n (I* = (",*)'"'

:

J,1t1,6at)'/o

f@

J, lf (r)ur/o 1ty11r[-n')/n 6)(r
s

B[/e.

llf llp,,(

lo*

,'-o'

1r)ld,r

1x)(Krur-n'1-p@)(Kof)o1r1ar)k-t"o

"8.

Now, we can use the Hardy inequality

Therefore

1 ot-rr/t'B'te-ttP: , un, - / ('""oll\r,o' "rl"r';) " i.e., Bsf,C ( oo if (2.34) holds. To prove an analogous assertion for .B1, recall that (2'34) is equivalent to the inequality llKgllr,,ur-p, ( Cll9llo,,.,'-"'where K is the conjugate to g

K, with the sa"rne constant C as in (2'34)' Now

(t)

:

(k su1' / @c' ) (t) (K o' ar - n' )' / (t' q' ) $) u(t)'

(lo* ,'-o'rx)(Ksut-e'1-@ S Cp

/ f6

I\J0 I

(I'

t&or)o

o,)"o

tr/P fP(x)u(o)d"

/

I

define

which holds since the necessary and sufficient condition for its validity is satisfied:

104

Weighted Ineryolities of Hordy

Some Weighteil Nonn Inequalities fOS

I\pe

g)(Koat-e' )-e (qdt)'' (l-,1r-n' xB

(l'

Let us denote the inner integral by J2@). Then by Minkowski's integral inequality we have

o'-o' (Ddt)'to

(r2(c))t/n'

: ",lg (ll r*'"'(/"r-r'1'Y')-' o)

:

(lo'

ur-n'1t1dt)

"-o'tlo')

cl-tlllll9,,B3.

to scl-tllfllli'-'.a3 =

rr :

ttdt* fo*

Io*

It is y' ( r,

1a

;l

[

(

K/

q

)

-I r 1

1

(

^k

u ) ( 11

u-

|/n

Js@)

pllat

lVllr,,Jlto' where

Jz ::

: :

V lo* lo*

I"-

o"

lllllo,

( [" \J0

,g' 1,10(x .11o'k-rt 1'))

/

and hence we can fix an s,

"o'

.

:

l,*

u(t)(Ko,ur-P')r/P' (t)dt

l,*

Uk

ou)t

/ n py

1Ko, ar -n'

x [(Rouy- r / n 1t1ur / {

| / o' 111ur/ s 1t1l

(fl]at

fl''k-t)@)(kQP'(r)ur-e'(n)dt

{x f)o'(q-D @)d (-

([

l,* ,u")p'

a.-e'

(1,* Xk *lr

$\d)

,uQn'g\at-o'1qo') o1*ff'(a-r)(r)

'

/n

61

(K o, uL - p' )u p' (t)1, u1qn

, (1,* & ou1-" / o

1t'1

(2.55)

y' < s < r, and use Hiilder's

inequality:

7Y-t@)(kQ@)d'r

u',r

{d

s

(2'54)

yidlds we first use Hiilder's inequality which

V

(lo" ur'rr,t1ur-r'e)dt)''o'

Jt

Therefore

[,

&(s,r)u(s)ds)o' or)"o'

Therefore

'*

To estimate

u(r)

(l

l:{x)-

t-l+i

(t-)

:(*)"o

(1,*,*o r, I,*

;-v'p1at) " (1" / t \r/P t " (/, ?:t

::

u1o

or)"

"'

)

106

ol

Weighted Inegrolities

7W

Hardy

Some Weighted Norm

:/@-r)

''"' ruourr,r'-r/s(x) : (:--,) s"/ \P

-

. (f

-

tA ul r /

f

p'11x

o,

f

*

t/' - o'

7r

/'p'

1t;at)

rrt1 " r,

y 1n' k

-

r) (r / n+ r /s

)

-

r

(

s

)

(ry)

d(r(/)(')

I

: :

-l/t.

-llp*1ls':I/d

{kouy-c lng)d(frsdg)

- l,*

s p'(q- t,

;\t*o"1-"'/p+r(t)

J,

:; p'( f* f/,

yr /o' (t)lE u(t)d,t.

/

ll/llo,'

/ re -,t,n qr-n' / " 1r1Jf,' / " 61a1x

(/,

71o'

to-t)

y'

tk od' -n' /'

1x1

= p'(q - r) x (K y

yr'

r/s

\1/P' @)

)

f@

J,

:

- p'), we get

f6_

/

" 1r)d(x 1y'

(o-t)

(")

f6

Jo-

J

y1",)

@))'

\r-P'l" u(t)(Kf)q(x)ar) t-

a\tt-t-,a\

Ja@)(x 1yk-r\/(s/p+t)-"/n' p)d.(K

\P'/s

f)@)''

Now, integrating by parts and using Hiilder,s inequality with expo.

Applying again the Hiilder inequality, now with exponents s/p', s/(s

f*

" (/, nents

"

1

^]

'-o'"1 o,

Ja@)(K f)(q-r)s(|+|)-s /o' 1r'1a1x

, (lr* &rq @)(K f)q -, 1x1a1x y

Then it follows from (2.55) that / n \l/s' r,s\e-)

(/

/)

"

Denote

t@ Jn(r):: ' l(xsqr tn (t)(Ko, ut-t'

Jo-

lr

Or,*

' Lt&r)'- o'/'(r)(K fl(q-r)(r-p'ls) f\dc Jz

and

I*

r',"

r

since

l6 ,I (ksu)-"'/oqt)u(t)d,t

Ine4talities l0Z

Goulr-t'/'1x1Jf,'/"

k- r)- I (r)d(r(tx')

1x7

and r /(r

J a(x) (K f

-

)(q

s), we obtain

-

t) /

f@/f@^

Io- (1,* tk "rl' x (K

- r) -

(s / p

/n

"

/ p'

1ty(K r,

(s)d,(K f

ut

-o'

1'

/ n'

L)(s / p+rl - / p' f )k 1t)d,(K f ) (t) s

:%#(1,*Go,Yro1,1 x (K r, ur-n' y"/e' (t)u(s)ds)

l-

)

(r)

6,1

u(t)d,t)

Weighted Inequalit;ies ol

Some Weighteil Nonn

Hady Tlpe

lo

exercise.

x (K o, ur - e' 1" / r' Tr!u(r)

1 qs(tls -

Ur)

/Jo

fOg

Remark 2.16. (i) The formulation (and proof) of the corresponding theorem for the conjugate operator .k is left to the reader as a,n

1 f- (K''Ya(t/s-r/')1s11frsu7"b@\ +;6-6 -

Inqtalities

(if) As

mentioned in Remark 2.f4 (ii), Theorem 2.1b (and its counterpa.rt for the operator K) was proved for g ( p with g > 1. The next theorem deals with the case

dr

(tror)" /e(t)(Ke'u'-p')"/p'(r)uv"("))

0
lc

of the Hardy-type

operator

6fl@): JO[' x (K o, ur -

o'

1,

/

o' 1ryu(r) d,r)

"'

ff

- u (r

Xn/

)q

(

Theorem 2.LT. Let

Therefore

K uo

C(p,q,s)llf llo,,BJuq' '

r S c{Jo * h) 3 cl-rct ll/ll$,,16 + c(p,q, s)crll/llp ,,Btlr/c'

s cl-tcrllfllfl,,B8 * W

Suppose0

( q < 1

be the operator Jrom Remork 2.16.

,: (I*

l(koulr/o

1t1(Kst)r-n'1r1c'

e)Yur-p'(Dat)'/' <

*,

then the inequolity

the However, using (2.54) and Young's inequality as at the end of proof of Theorem 2.10 (see p' 96)' we obtain

*

o, f > o,

we only suppose that ft(r, t) > 0 is decreosing in t, and no other conditions are imposed. on the other ha^nd, the sufficient conditions a.re different from the necessary ones.

d d")

: q'hBilll(Jill(|-sl')' h S C(p,q,s)ll/llp,,Brll K f ll1:,: :

*6,t)f (t)dt, r )

f llq," s cllf llr,, holds for all f > O uith C < Bo. Conaersely, il Q.56) holds for alt f > 0, then Bz S C llK

82

i,

:: ( [\J0

(2.56)

where

Go'ln' lo 7t1ut-n' 1r1or\"' /

Remark 2.18. (i) Observe that in Theorem 2.lT we require only one condition for the sufficiency part, namely the finiteness of the same 86 as in Theorem 2.15, a^nd also only oue condition for the

i.e.

rsll/llt,"(83 + Bl),

necessity is required.

and (2.34) follows with

(ii) For the proof of sufficiency, cS@$ + B!)1tto.

tr

called' leoel

we need the concept of the so, function introduced by I. HALpERIN [r]. The proof of

110

Weighted Ineqnlities ol HotdV

7W

Some Weighted Norm Inqtalihies

the following lemma can be found, e.g., in the Appendix to [OK]. Another proof was given independently by G. SINNAMON [1].

Lemma 2,L9. Let (a,b) be an intentol ond w o weight function on (a,b). Suppose thot 0 < tu(r) 1 m tor 8.e. E € (a,b) ond' [!w@)dn < a. Then for eoch measwvble function f > 0 therc exists o non' negatiae function fo (colleit "leuel function of | ") such thot f, f,

I !@dtS JoI [o(t)dt force Ja

(i)

(ii) ry w@)

(fi) To prove that Bs ( oo implies (2.b6) we assume without thut..I is compactiy supported in (0,m) and that

loss of generality

ar-p' e ^Ll. Property (i) of Lemm a 2'.lg

(xr)(,)

-@)dx

k(x,z)

(o,b),

k(x,e

(ffi)p = l"'

.@)d'*

(K

,to, p> L-

Proof of Theorern 2.17. (i) Suppose that (2.56) is satisfied for all functions / > 0. Then by the reverse Minkowski integral inequality we find

where

: Applying the duality

of.

U(u) it

:

follows that

:

d::=,

= (/-

(fr

lo*

r ax*ou)uc $)dt

=

r l/y' ouye'

/o

111or-t 1fidt

)

ll(Kou)t/qllp,,o,-t,

-

Bz.

lo" tt4o, * fo'

r"t{o"*

I,'

([

,(")d") datk(r,t))

I' ([

/"G)d") d2(k(x,t))

f')(")

/o is the level function of /. Therefore

l* Io*

wnq@)u(r)d,r

u(,)(Kf\c-t@)

s

I'

rrr,t)f"(t)dtdr

f* r"t4 I,* or,,t)u(n)L(1,' . I') uo,s)/"(s)ds]

= fo* f

C

that

fo- wilo{a)u(x)dx <

cll/llp,,

s-hows

: I' ur,,rr(l:/(")d")

is decreasing on (a,b),

(iii) I^' (#)p

lll

/(!

$)

I foo

Jo f @ J,

k@,t)u(n)ll' ur,,"lr"(")C kq(t,t)u(r)

/ ft

\ C_r

(/, r"f"U")'

since 0 < q < l and lc(c,s) >.tc(c,t) for 0 of Lemma 2.lg with u): ut-p' we have

(

o-'

o,o,

o-'

o*0,

drdt

s S t. By property (ii)

LLz

lIody

Weighted Ineqtolities ol

1t l^ f"{,)a, Jo

:

Some Weighted, Norrar. Inequalities

fupe

ft( f@ \rt-o,(r)d" ,,

llf"nl,"

/, \;r+6 /

:

|o*

u(r)(x f)q(r)ax

= |o*

, (lr',r-r'1s)as)q

fft11o,k'-r)(c-rt-q/p$)uq/p$)

foo

r-d' slpl|ld: l(f -L)/q'-t/pl': rlq yields -

(/-u"l . (l-

=l

i- i t.t k

(koQ'tt@)o')o''

(l-,r')e@),(dd,)q/e ( [* " \/o

ovn'@)(Kour-p')'/q'

wherc

k(t,r)

r)0, e)0,

int. If

i,s increosing

/ fe

I (xoor-r"1,/o' = (\"ro

\r/r

1t)(Kou1,/oltyur-n'

llatl' /

<

oo

then

. \c/r (r)(koQ'h(rlat )

/

: ll/'lll,,B3. The result now follows from the fact that due to property (iii) of Lemma 2.19

1

be d,efined, by

(frg)@): [*k4,r)s(t)dt Jz

(lo" ,'-o' {irr)t'

=llllll,". tr

B0

:

@)a'

The corresponding result for the adjoint operator .E could have the following form.

o1,),pya*)*e

ut-p'(x)

fP(n)u(r)dx

JO

Conjecture2.2O. Supposet
u(r)(K f)q (a)d'n

=

I,- {Pr)o't'-o'ro*'

s I,- {p,)o,,_,,1da,

1Kou11t1at.

since [(p'- rXq- 1) Hiilder's inequality with exponents pf q and' Qt/q)'

J,

(f")P(r)u(o)d,x

: [* (yrr-n,1x) Jo {'!:,),)o"-o'(")d' )

so that

r@

: T,*

113

llksllo,, < Cllsllp,,

(2.57)

Iorollg>0withCSBo. Conaersely,

if (2.57) hotds for

u, =

(I*

ott

g > 0, then Ez S C

(xoe//e plut-p'(!dt)

1a

where

LLA

Some Weightd Norm

Weighted Inewalities ol Hodg TVpc

2.6.

As

Some Modifications and Extensions

::

/ap11xoat-n'f/t'

1t)

Ineqtalities

<

x,

<

a.

lls

"1lr(freu)r The general interval (a,b)

A1 ::

Let us mention the results corresponding to the results of secs. 2.3 to 2.5 for the case of a general interval (o,b), i'e', for the operators

(xf)(r) : Jo['

r@,t)!(t)dt,

&d@) : Jtfo oe,t)s(t)dt,

(2.58)

( t 1s < b and satisfying &(c,t)>0 for rlt)a, k is increasing in r and decreasing in t , k(x,t) x k(r,z) + k(z,t) for a 1t 1 z 1 a 1b'

(2.62)

"1;10(&u)r/qe)(K,/ur-n'1r/n'7t) We omit the proof since we can reduce Theorem 2.21 to Theo rem 2.L0, i.e., to the case of the interval (0,6), by the same steps as it was done in sec. 1.5. For this reason, the proof of rheorem 2.2L us well as the formulation a^nd proof of the analogues of rheorem 2.L2 and Theorem 2.15 are left to the reader as exercises.

with tc(r,t) defined for c

F\rrthermore, analogously to (2'30) define operators

Strong and weak type inequalities (2'59)

K" and ^fr" fot

s20as (rr"h)(r)

:

!o'

(&aX') = n"{r,*\h(t)dt l,u

k'{r,t)h(t)dt,

(2.60)

Then we have the following assertion which corresponds to Theorem 2.10:

I < P S q < oo. Let K be the_operttor from (2.58) satisfying conditions (2.59) ond' let K" ond Kg be giaen bY forrnula (2.60). Then the inequality

Inequality (2.1), i.e., the inequality

(1"' urn*ofu(ddr)"0

.

"

(1"'

vt x,,1*ya,)'/o

with a rather general (integral) operator ? is sometimes called a strong type inequality or a strong (p,q) inequolity. Many authors deal also with the so-called, weak type inequalities or weak (p,q) inequalities of the form

Theorem 2.2L, Let

(1"'w ttr)u@)ax)'/o

. c (1"' rav{da,)'/o

i.e., the inequolity

llK/llc,' < Cll/llp," holds

tor oll f > 0 if anil

onlY

il

,

(2'01)

u({r e(o,b) : (r/X") > }}) <

"^-'(I'

V@)fu@)dz)

(2.63)

> 0 is arbitrary, C is independent of / and ,\ a,nd u(E) : [ra(x)d,r for any measurable set .E C (a, b). We will not deal here with weak type inequalitiesl for some results, see e.g. K. ANDERSEN and B. MUCKENHOUPT [1], v. KOKTLASHVTLT [2], r. GENEBASHVTLI, A. GOGATTSHVILI and v. KoKrrASHVrLr [1] and F. J. MARTiN_REYES [1]. where A

116

Weighted Inequalities of Hodg TVpe

Some Weighted Nonn Ineqnlit:ies Llz

A more general case some of the papers just mentioned deal in fact with strong and weak type inequalities for the more special operator ? defined as

(ril@):

:!B

ll*or",Y,t)r(u)dPl

(2'64)

with (X, d, p) a general measure space with a quasi-metric d, a mea' sure p and Ic a positive measurable kernel on X x X x [0,oo)' For X - (o,b), dp : dY and lc depending only on c and y' we have the operator investigated in this chapter.

2.7.

2.7.4- The result of rheorem 2.10 extends further to the index range 0 < g < p,p t 1. To see this we have to apply the modular inequality of Q. LAI [3, Theorem 1] with P(r) : sP, p ] 1, and e(r) = xq, 0 < q < p. (See also Comment 1.10.8.) Then this result, together with schur's lemma, shows that inequality (2.34) hords if a^nd only if

for all covering sequences

{rx}*rz (r* Q. LAI

P,ll,'-'.'

f,'j

EI/"*'

l,')-,"-o'rqdtl

uo'

_,

r,

o,

t)ur -n' 1t1drlt' "'

used also for 0 < g S 1, P

> 1)-

2.7.2. The general Hardy-type operator in the form from Definition 2.5 was introduced and investigated by S. BLOOM and R. KERMAN [1] and independently by R' OINAROV [1]' The result involving the Riemann-Liouville operator (see Example 2.7 (iii) or Remark i.s (iii)) was proved by F. J. MARTIN-REYES and E. SAWYER [1] and independently by V. D. STEPANOV [t]; in fact, in the first mentioned paper, a somewhat more Seneral kernel was considered, namely that given in Example 2.7 (i). The proofs of the results given here follow closely those of v. D. STEPANOV [2,51. More information will be contained in the book by V' D' STEPANOV: Voltena integral operotors on semiutis (in preparation).

2.7.3. The case 0 < g < 1 < p < @ in Theorem 2.17 requires the concept of the leuel lunction and its properties described in Lemma 2.19. This lemma was proved by I. HALPERIN [1] and used first in connection with the Hardy inequality by G. SINNAMON [1'3]'

r/c

tr4o4
and

Comments and Remarks

2.7.L The proof of Theorem 2.3 follows the ideas of P. A' ZHAROV but the idea can be [1] (where only the ca^se p' q > 1 was considered,

[3]), the conditions

are satisfied, where

*

:

*

kq(x,rk)u(r)Or)


- i.

2.7.5- As in the case of the Hardy inequalities, the question of sharp constants is largely open. However, certain estimates for the constants can be given in terms of the operator norm. In the case of the Riemann-Liouville operator, such estimates (which for specific weights are sharp) have been given by V. M. MANAKOV [U. 2.7.6. The weight characterization for the Riemann-Liouville operator, which we rewrite here in the form

(/,/X")

:

fo'

{,

- q'-r !g)dt, o ) o,

has been given in this chapter only in the case o ) 1, and similarly for the conjugate Weyl fractional integral operator. The case 0 < o < I does not follow here since then the kernel k(r,y): (c- g)o-r is not an Oinarov kernel. Properties of the modified operator

Q,il@)

=9 Io" r* -r)o-r/(r)dr, 0 >

o

118

Weighted,

Ineqnlities of Hailg

7W

as a mapping between nonweighted U and, Zq with 0 ( p,g ( and p ) ma:c(l/o,l) are considered by D. V. PROKHOROV [1].

m

3

2.7.7.Integral operators, in particular fractional integrals, in more general homogeneous spaces (see Sec. 2.6) are investigated by many authors. Let us mention at least the importa,nt Georgian school, rep resented, e.8., by the paper of V, KOKILASHVILI and A. MESHKHI

[l] and the references

The Hardy-steklov Operator

there.

3.1.

Introduction

In this chapter, we will deal mainly with functions defined on the (sta^ndard) interval

(0,

*).

Let us start with an example.

Example 3.1. The classical Hardy operator tions / : f(t)) 0, f € (0,m), as @

f)(r) ,:

g)dt, Io' f

o<

r

<

I/, oo

defined for func-

,

(3.1)

is obviously related to the triangular domain d : {(c, t) : 0 < t I t l oo) (see Fig. 3.1). This can be modified by considering the

operator

Ql)@)::

ft("| J,

l$)dt, 0
(3.2)

L2O Weighted Inqtolities

d Hodg

The llordy-Sfr"iloru Opentor

TVpe

t2l

i.e. the inequality

(/-

rttrl @)lqu(r')ds)"u = "

(lr* ff(n)lpa@)e)'' ,

(s.4)

can be easily reduced to the investigation of the Ha.rdy inequality for the pair of weight functions U, u where u (a)

:

(3.5)

u(b-l (c)Xb-t )'(r)

with D-l the inverse function to

b.

Indeed: We have Fig. 3.1

lo*

{rr)otdu(c)dx

:

I,*

: :

(1,^o

!@d)

I"* (1," ilqdr)

u@)dr

u(b-rtu)Xb-')'tu)av

[* @il'(o)u(ildv

JO

by the obvious substitution g:b(c), i.e. o: D-t(y)' consequently, the necessary and sufficient condition for the validity of the Hardy inequalitY

llH/lls,u S cll/llp,, in the

Fig. 3.2

where b(c) is a strictly increasing difierentiable function on

(3.3)

of pararneters L < p 3

;$ (l-

[0, oo]

satisfying b(0) = 0, b(oo) : oo' The operator ? is related to a ( 0( "perturbk" iriangular domain A(b) : {(t,t) : 0 < c o, t < b(c)) (see Fig. 3.2), but the investigation of the weighted norm inequality

llr/llo," 3 Cll|llp,o,

ra,nge

can be rewritten

:!B

-

{ ( m,

i.e. the condition

,oor)"' (lo',r-r'1t;at)''n . in the case of inequality (3.4)

-

(l-,{")a")'/o (l-",'-n'1"1a")''o' .

*

as

*.

(3.6)

t22

It

The Hodg-Stekloa Opentor 123

Weighted Inq:aalities ol Hady Type

is easy to see that the conjugate operator

(rn@),:

with

l";!g)dt

fr of.T has the form

a(s):b-r(')

operabr F. This motivates the introduction of a new operator

I":;'

(3.7)

ru)dt

entiable functions on

[0, oo]

b

:

b(r) be strictly increasing differ-

(3.11)

:

I

)r,

r",

b(x) = s, u(t) =

a(r)

:

,9

.

(3.12)

Thus we compaxe two operators:

f,

t/2

(3.8)

t(

?r is the Ha.rdy operator, and the inequality ll"r/llc," < dlllllo," satisfied if a,nd only if

0 a*l :-:-+pqf

oo, by formula

from (3.7) is bounded,

T z U(a\ -> Lq(u),

(3' ) holds for p,g such that

0
1

(fl;' r1,u,),(")*)''o' c (1,* ruw(o)d')"o(''')

is

(3.13)

a < -1 (i.e., 0 < p - 1; see Example 0.3). To verify the validity of the inequality llTzfllc," S Cll/llo", we must show that condition (3.10) is satisfied. For our choice of u a^nd u we use the estimate and additionally

A : ,:l&, (1,' n*)'"

(1,),,,ar,-ota,)'to

Theorem 3.7) that the inequalitY

(f

t such that

and a(c)
rn

Remark 3.3. (i) Similarly as in the foregoing chapters, it is our aim to characterize weight functions u,u fot which the mapping f

i.e., for which inequality

and

Q1l)@): J0 I f(t)dt a,nd Q2t)@) = JI !(t)dt.

b(-).

The operator ? defined for / - /(t) > 0, 0 ( (3.7), wiil be called tbe Hady-SteHou opemtor'

r

(3.r0)

(ii) To see that the class of admissible weights u, u defined by (3.10) is strictly larger tha^n that of (3.6) take

satisfying

o(o):b(0):g' o(r)
0
.*

U"i:,r-r'1s)ds)''o'

where the supremum is taken over all

o(r)

where the functions o and b are defined below'

Definition 3.2. Let a: o(r),

dl / > 0 if and only if

A := sup (1,",av,)'''

(the inverse of b). The weight characterization of this operator follons the same lines as the chararterization of the conju$ate Ha,rdy

(rrx,) ,:

holds for

:

$.!rE{!.J,II! '

O'9SUpC c>0

r

t'

;

L24

Weighted Ine4talitics of

Hody

The HodySteHoa OPemtor 125

IW

finite if (3'13) holds' the last expression (and consequently also 'r{) is we have no oililitionol enn' Hence, forlhe Hardy-Steklov operator Tz

ilition'ona(andB)andthus,abiggerclassofadmissibleweights. Let (iii) In Tireorem 3.? we will deal with the case 1 < p S q < m' with (3'9) is trivial : o, *"rrtioo that for the ca'se p : q f inequality u(c) =

fo-'(t)

l- (/:;' ,(,)*) u@)dr= Io*r,, (fi;',{do)or, :.1.' On the other ha'nd' the i.e., (3.9) becomes equality with C euery droice i*ai inequality with p : q: L and u(c) : * fails for case u(c) = 1' of u, as the following example illustrates for the there is Example 3.4. The foregoing considerations indicate that

the Hardy-Stcklov a substantial distinction Letween the Hardy and again a(r) and operators. To grve another example, let us consider : weighted norm bic) as in (S.ri), but u(c) : o-Q a$ :r(t) 1'. The inequality lla'rdy the inequality ior the op"rtio'? from (3'2) is then for the averaging oPerator:

which

is not sotisfidif

="(1"*

we take p: q:

r'(d*)'

/t@

t c>0 \Jc

(see, e.g., [OK, Lemma

"o*opl"ding

5' ])

and the equality sign, as ca,n be shown by

The Ha,rdy-Steklov operator gives rise

to the moving

auercging

#@ l"l.' ,n or, t

>

o

(3.14)

This operator in its various forms is of considerable importance to the technicnt andysts in the study of equity markets. These technien'l onolysts try to predict the future of the stock price or the future of an equity market solely on the base of the past performauce of the stock price or market valuation, respectively' For example, some analysts consider a,n equity (stock) / whose price at time t is /(t)' a recommended "buy" if (Sl-zoo/)(t) < /(t) while the reversal of this inequality is a "sell". Similarly, if t(4 represents, say, the Dow Jones Industrial Average in New York at time t and if (Sl-zoo/Xr) S t(t)' then it was observed that the return of the market in the year following t gained 12% while a reversal of this inequality showed that the market lost 7% in the year following t. The introduction of weights and the pointwise estimate replaced by mea,n estimates with good control of the constants may therefore yield additional quantitative information in the study of fina.ncial markets.

The Steklov

u(t\dt

\


does not hold' On the other ha'nd' the

inequality (3'9) for p

2

1, since the corresponding

necessary and sufficient condition'

supl

log

An application to financial markets

(s:/x"):

Ja_,r,)

(1 1," rco)'o')''o

:

opemtor

u(t)dt.

Indeed: As F\rbini's theorem shows,

fi-

is satisfied with C F\rbini's theorem.

: g:1,

i'e'' the inequality

l- (: I',,'n'o4d''
oprator

For functions / defined on (-oo, ertor 5" is defined as

m) and for 7

(s"/x") = I,':: rg)dt.

)

0, the SteHou op-

(3.15)

As a consequence of our results concerning the Ha,rdy-Steklov shift operator, we will deduce later a characterization of weighLs u, u

126

Weighted Inegolities of Hard,y

The

I\pe

9.2.

for which

(/_ u."::

todt)q u{da*)'/o

="(l:

\ te(r)u(r\dn

r/p

Some Auxiliary Results

Definition 3.5. The functions

o and b introduced in Definition 3.2 are strictly increasing and differentiable, and consequently, the inverse functions o-1 a,nd b-r exist and are strictly increasing and differentiable, too. We define a sequence {*r}rrz as follows: For fixed m > 0 define m6 : rn att6

)

is satisfied.

An application to fractional order Hardy inequalities Although these inequalities will be dealt with in detail in Chap'

: a-l(a(rnk)) if ,c > o, mk = b-1(o(rn1a1)) if &<0.

rnk+r

5'

let us here mention one particular result. The integral lo* 0

<,\ < L,

^

/

: /(oo):0

it is the Ha,rdy "(r-r)e

inequality (1.25) for (o'6) a.nd u(r) - t-&;, or bY

a(rnx+r)

and for (see

c, [* ff'(flf s(t-x'tn6' Jo (in this case, p: e, a(x) :

: b(mi

for

ke

V,

(3.1e)

(3.16)

The following technical lemma will be required in the sequel:

-

(o,o),

G. N. JAKOVLEV [1,2] or P. GRISVARD tll)'The question arises which one of these two estimates is 'better", i'e', which one of the expressions (3.16) and (3.17) is larger. The a,nswer will be given in sec. 3.5 using the Hardy-steklov operator. In particula,r, it will be shown that (see, e.g.,

Wtudv s" Io* v'@)los$-^)Pd$'

This result was derived ea,rlier (see, e.g., A. KUFNER [ ])' and in Chap. 5 we will grve a direct proof (see Theorem 5'3)' Here it is mentioned simply as an interesting application of the lla,rdySteklov operator.

all

Fig. 3.3).

(3.17)

Ir* Ir-

(3.18)

Clearly

v{"\lo'-xPd'x

e Cl(O,m) with /(0) <m, either bY L

can be estimated for

Hody-St*lot Opmltar l2I

m-2m-t m=?,fb Fig.3.3

128

Weightsl Ineryalities ol Hordy

The Hordy-SkHoo Operctor 129

ftp

\c \ r/c (1,- (I:;' f (t)dt //) u(c)ac I

(3.18). Then Lemma 3.6. rzs rn > 0 orul ilefine {my}*ez Dy tmy lrmy1;y for k e V' awl

lim rnl

Jc+oo

: es' - Iim

nnk

:

=' (1,*

'c-+-@

that nrt

l

follows Pznol. Since o?nk) < b(mi = a(rn*+r)' it implies the existence this a'nd increasing, for lre Z.Henie {*x}xrzis (0, oo] such that of. M- € [0, oo) a,nd M+ e

lim rnr M-, k-+-m --=

-lim m3

:

rmk+t

If

lD

and replace in (3.21)

M+'

Is-+oo

f

M- :0

M+:a.

and E

(3.20)

The Case p

<

(8.22)

will be used intercha,ngeably

in

q

Theorem 3.7. If | 1 p < g < oo then (3.2L) holds for all functions I > O if ond only if

Acm

u(b-r(Y))(b-t)'(s)

(3.23)

wherc

Then

u"(v)dg

: u(x)da and

a@\dY

:

u(x)tu

if

A::

sup

(1,'

ond the svpremum Y

(ii)

\r/c

\c

The first main result of this chapter reads as follows.

u6 bY

u"(a) : u(o-1(Y))(a-t)'(Y)'

=

(3.2r)

f(t)w(t)dt) "1'Pcf / /

Inequalities (3.21) and (3.22) the sequel.

3.3. Some notation

.ro(Y)

f >0.

.
o(M*)'

that But, since a(c) < b(r) for o e (0,m), we conclude

,

by fut, we obtain an eqtiuolent inequality

|\J0| (\Jo(c) l..

= jgga(rnk+l) :

)

= ul-P'

| 1a 1 p@)

b(M-\

(i) Define functions uo and

fe(a)u@)er

we denote

Since o and b are continuous' we have

b(M*) : J$u{-o;

\ l/p

O'

= o(x) and

Y: b(r)' resPectivelY'

As mentioned, we will deal with the inequality

,oo")"'

is token oaer dl n

0
(|":::ovn'1s)ds) ond

t

such that

and a(o)
(3.25)

130 Weightd

Ineryalities ol HodV

\Pe

The Hady-Stekloa

=

" (I:

ur,(s)wt - n 1,

/ rb(t)

\

\.ro(c)

/

Oprtor lgl

)r")

r/P


(3.27)

,

where we have used the equivalent inequality (9.22). Since ur, the last integral in (3.27) is finite a^nd therefore

Fig. 3.4

Morzover, if C is the leost constont

C

U*,,u,"G)ds)

for

which (3.21) holils then

x A.

/ f"

of (3.23). Suppose that (3.21) holds, fix t and o according to (3.25) and let {rra}Eir be a sequence of -Ll-weights such that torr(c) < u(x) a,nd u' t ur as n -) €. For a fixed n € N, let

(i)

Necessity

/(") :

X(o(o),6(t))

(s)ru"(s)/ur(s).

:,Uq u(s)ds

)


f u as n -> m, the Monotone

where C is independent of n. Since ur. Convergence Theorem implies

U Proo!.

\ l-tlP

/ fo lr/c / rb(t) (l u(s)ds)

/ rt(t) \ l/P' w(s)ds (/.,", )

<

CIij],,,(,p")

(,',rqo")''' o"{")a')'/o

= (1,"

U"i:t(t)u'(4d')

ASC


(ii) Sufficiency of, (3.23). Fix t

(3.28)

a,nd

write y

:

o(c), ut

(l- (lil;' r@u(qdt)o'{")a")'/o

ye6y*14a")'/P =" (!,-

=

ar-p' in

/ tu \'/o / ,u(r) \ r/P' ..Tq...1 l..uo(s)dsI II ru(s)dsl s.acm. c(t)
/ \/y / But this is nothing else than the necessary and sufficient condition for the validity of the Hardy inequality in the interval (a(t), b(t)) and with weights uo and u: A1t1

=

c

for all c and t satisfying (3.2s). Taking the supremum over all such s and f we have (3.2a) with w: ar-t' and,

(3.24). Then

Ift<s(c,then

€.tr,

:

A1o1t1,b(t); uo,

u) S,4 < oo,

(see (0.12) and (1.34)), and thus we have

(l^';'

(

I"" r

(Ddt)q

u"(")r") "0

=, (l^';' lp (s)a(s)ds)"o,

Ineryali?ies of Hadg

132 Weightd

The Hordg-StcHoa

7W

='(#' with rr :

ur-p' and with

(3.2e)

Similarly, if c is fixed a,nd v

:

U.L,

SA<m

A(a(t),b(r); u6'u) S A

(

f (t)w@at) ua!)ds

{r)

ar)'

\

/ rb(r)

\l/c

)

l/P

tr.so)

wirh C Sk@,q)A(x) 3k(p,q)A. Now fix m> 0 a.nd define {**I*.2 as in (3'18)' Denote

(mp,rnt+r).

o(r) < o(rnt+r) = b(rn1) < b(t) for s € E1, bk

(/- (2,",o, |"o)r,n -rqat)q u@)d* )u'

= b(*x),

a'nd

Iz.

For each c € (0,oo), only one term in the sum Dxrz under the integral sign ca.n be zero, a.nd since (0, rc) : Ut ezEr = U[znr,znr+r], the change of variable y : a(r) and (3.29) yield


a(rnr),

=

:: h*

oo,

\c

o*:

1r1

or)o u

. (/- (2r,.r,, I::,' 11,v,1,v0,)ou")*)'/o

which implies

Then

(Do,*",t', I"'u,r$)w(t)dt Io

and similarly as above this means that

Ep:

*)"'

b(t) in (3.24), then

"r,r"
(d;'(1""^,,

o')"o

+ | ,",@) [^'' 1 1r1r'cl+l

(here we

1/P'

:

1'v

k€2

\ / ra*,) \t/n I ro u?)as) uu!)as) (./,"-'

A(r)

: (l-

)

C < k(p,q)a(t) S k(p,ilA

used (0.14)).

I":;' v p,'

(1,- (

(l-:' (1"" /(t)u,(t)dt)or"{")ar)t'0, fe(s)w(s)ds

133

we obtain via Minkon'ski's inequality

which can be rewritten, similarly as on p. 129, into the form

\ r/P

Operdor

if we denote

/fbr \c \llc h : {t /fo.*r (/\Ja(c) | ..f(t)w(t)dt)."1'p') / \tez'l-' /

= (E[.' (1,'.' u''(')d')

o'"1u)o')

134

fVF

Weighted Ineqnlilies ol Hordg

The Hordy-Sk$oo Opemtor 135

where the last inequality follows from the fact that q/p> l. The estimat e of. 12 is similar. The change of variables g

:

where

b(r)

and (3.30) yield

12

: :

(*E t"'.'

(F,

I^'"':"

(ff'

A

r$\u(Ddt)ou{ta')'to

': :

(Io..' r (t)w(qdt)o'u'o'oo)''

*o (/"

'r-r'1s)ds)

"'

U:::)

\r/P' / ro-'(t)

/ fa

/ t6

and consequently, (3.21)

Remark 3.8. (i) We can easily see that

QI)@):

P@)

Jor,,

the operator

f

con-

tr

conjugate to

f(t)dt

then (3.21) holds for

fa-'(a) Ja_,@)

(

oo,

/

> 0 if

Indeed: Using Minkowski's integral inequality, we obtain

:

(/-

( I^'lo'

,n or)"or*)'''

f(t)dt.

(Use the duality formula (f ,Tg) = \f f ,g, and F\rbini's theorem') Since the operator f nas a form similar to the operator ? and maps,La'(ur-c') continuously into If'(ur-n'\ provided the mapping T : U(a) -+ Lq(u) is bounded, we immediately obtain from Theorem 3.7 a cha,racterization of admissible weights rewriting condition (3.23) into

r{


l/c \ ( 1 ["-'(') -.,^ A :: ess -..^ ub)as < o' ,#P,(D \/r-,u, )

nrfnc,,

has the form

(r/X"):

such that 0

t:pS{(F,

- follows with the stant 2C 32k@,q)A. The last inequality and (3.28) imply (3'26). -

t

,

we suppose that

: c( I /P(s)ur(s)ds)\l/c / \"fo Therefore (3.22)

\ llc

""o(l "r-t'1s)ds) (/,,, u@as)

the supremum being taken over all c and and b-l(c) < o-1(r).

(ii) If

,{'-o'x'-o'1,)r") "o

lo-

ili(l-;; u{da,)'/o a,

fo*

tt,,t

o

Io*

f* (;-;;,,(,),")''lo"

f g)a(t)dt: All/|h,,.

(

oo

136

3.4.

Weighted IneEtolities ol Hordg

The Case P

>

The Hordg-SteWor Opemtor f37

IW

D(*i

q

:

In order to provide a weight characterization for the Hardy-Steklov operator in the index range

*

0
onil C(mp) oitrnttton g.5. Let u6',u6 be giaen by (3.20). If C(m*) are the best onstants in

(I"',,'(l:,1(s)tu(s)as)n',',(r)or)"o -"(1,0,.'"{o)'tildv) (3.31)

\l/P 1{ \t/o =/ril e Io(v)'(s)dv (/"- /(s)'(s)ds) ""1ilda) = (/") '

(l^',"'

(1,",(id") \ r/'

(1"',u"14a,)'/o ""1ildv)

(3.34)

Remark and Notation 3.10. (i) Observe that D(nr.3) and D(nt1) in Propositiorr 3.9 a,re the numbers A : A(a*,b$ub,u) a,nd A = A(ap,bp;uo,u) from (0.8) and'(0.13)vith u): ur-P' (more precisely, the corresponding numbers.A* and A* from Remark 1.4 (iv)) whose finiteness is necessar5r and sufficient for the Hardy inequality and the conjugate Hardy inequality i.e., the inequalities (3.31) and (3.32), to hold. The equivalence relations (3.33) follow from the corresponding estimates e.g. (0.15). - see (ii) Before formulating and proving the second main result of this chapter, let us introduce a sequence {Mp}162, which is constructed as the sequence {**}x.z in Definition 3.5 but with Ms : b-r(l), and the so-called norrnalizing function

/fir/ril

(/.-

o

(t)

:: I

kEz

(3.32) respectiaelY, then

C(mi x' D(me) and e@ixfrQn1,) where D

(mt)

: (l:,' ( ^'.'G1a')'

lt'

" U:,.,01,p")'/o,u(r)or),

(3.33)

x1*,,* **,1 {t) *4 {a-t

o o)ft

(t)

(3.35)

where (b-1 o a)t for k > 0 denotes the lc times repeated composition, while for ft ( 0, (D-1 o o)ft is to be interpretea as

6'r$*-r.

Suppose 0 < e 1 p, | < p ( oo, i : i - i. Then inequality (3.21) (or equiualently inequality (3.22)) holils lor oll tunctions .f > 0 if and onlg if

Theorem 8.LL.

A:ma:r(Ar,,4z)(m

(3.36)

13tt

The Hody-SteWoo Operctor 139

Ineqnlities of Hordg TVW

Weighted,

Then

where

A1

:: (!,* {1,' ,(,(,)) ( I.'lu''u'o')''' u@)ax)og)or) " (I'u{qa,)'t'

(1,-

:: (l- (

[:' I""o,' (

,(

(3.37)

1,a-'(u('l))

(l::]

Io

{,,a')' t'

Io"'

uu1,1 a

: (21i,

ond,

Az

(

")'

* (1," utta,)'to ut*)o*)"@or)'''

ou)o 10 or)'''

(l^';' (l^"u,-t4a")'/o

.,67a')'t'

*

u o 1r1

/o'otilds) ( 1u""

uo1"'1a")'

*p-' "

o1o

1r1or)"'

(3.38)

m : (b-1 o dk(t) then t : (o-r oD)&(rn) - mk.Hence if. t : Mr+r then Mpal :'trtrks which implies that m: a-r(l). If t: M1, then (o-r ob)k(rn) : M* : (a-r ob)k M6. and since ,1140 : 6-t (t), it follows that rn: A-l(f). Therefore the last sum is equal to If.

withw=ul-P'. Proo!. (i) Necessitg of condition (3.36). Suppose that (3'22) is satisfied. Let {rr,}3r, {r"}i" r be sequences of Zl-weights such that un(a) < u(r), wn(t) < .(r) and u,. f u, un t ur as n -r oo' If -- un(b-r(ilXb-t)'(y) and u6(3r) : "(b-1(v))(6-r)'(v), then "u.,"(r) q,.(u) < uufu). ' Fix rn > 0 and define {*x}x.z as in Definition 3'5' For D(rn1) and D(rn1) from (3.34) define

D(m)

: (r*"o' r*r,) "',

o1*1

:

(Pru'

r^r)'"

Dt-),

ilgz(-)}

<

-.

(P /- -r'

(3.3e)

(l^',.

(f

"'.-

G).")''o'

, (lr" ur1,1a,)'/' ,xrlou) o*)"'

:

and suppose max{sup

.

E/--J'o'1^n1o^)"'

: (/:-;',*8,',^r,o*)"'

140

The Hordg-SteHou Opemtor 141

Weighted Ineqnlities of Hardy 1\pe

/ \r/' < (o-'(r) - 6-111;;r/r *t, (8"'@i) :

1a-1(r)

-

6-111;1r/r

ilooz(-)

(

* (1"",r0,,(s)d")'o uo,*rurou)'h

= b(r) shows that

oo.

oo' then In the same way we can show that if sup-tsD(-) <

A2 (en. . ,^ d^\ r Tocompletetheproofofnecessitywemustshowthat(3.39)holds. : Since rl$f) + t Write o6 - a(rnx), b1, : b(mp) md xr X(*,or)'

,lkp'),

k(E l"o,"' (1"u,(1"',."G)a')''*'' '"(t)dt)

'-

y due to (3.39). But now the change of variables A1

:

:

=

k

(E

I"',' (1"', (!"',.'

, (1,*,a,"(s)d")

/@\

^G)

(3.40)

a")''*''

.,g1ar)o ''o,,roroo)'^

.

we have \ f/p'

U:,

w"(s)ds

= =

((/: (l:.

)

U

Ds and

(a)

(

"1';a")'/ter'r)q

*

"/*''*' *) (1,',.-' 1";a")

= b/(w,). ,,,

n

(/: (!).-^oo")u*'',-(,)0,)

: (ft)' (l:.U:."1'1a")'/@')'"{t)a)' ( which together with the fact that t

EI: (f''-raa')"' '(lr'

(

ah <. t

(

y so that a; and consequently, the last expression is equal to llere for each k, ax 1

u''^1'7a')'to

y, yields

I l/c

",,@)dY)

hU,*

Dn,xx

I^', (l":..,1"1 a,)'

' (l

k(l- ( |

xx(a)

rr/c "r,fu)ao)

I (l*',r-^1qa')'

/ fb, \'/(lc) uo,^g)ds) (l "

bi,

/ @o' )

\s r'/(pc) / fb'uo,^@)ds) ye(t)w^(t)dt)

:

( t(

/

tu

)

"u.

\c y1,(t)w"(t)dt)

\l/c "4"@)ay)

since for each g only one term of the sum can be non-zero. Now, if u < br: b(mr), then b-r(y) ( m6 and "(b-r(y)) 1 a(mk) = skIncreasing the interval of integration (with respect to t) from (o3, y)

to (c(b-r(y)),y)

*d

replacing Xr by 1 we conclude that the last

L42

Wei.ghted Inequolilies

of Hady Typc

The Hatdy-Stekloa

by (3.22). But, since u)n l rlt the definition of

expression is not larger than

h

(1,- (F, I,'n,,,,,, U.'" ltut

, : ,

(1,^u0,,(")ds)

1";a')

(1,*,' aw @ *)"'

"/@ )

.r*(r)."(40,)

x

,u,*@)du)

k(l- ( I"iu'*r(E(1",'^1qa")'/o lo

(1,',,0,"(")d")

=

ro@)'/o,,(qdt)o ur,,(ilar)"'

1r-n,I

since only one term of the sum can be non-zero. Now take

r

Then the change of variables

W.

,. (l

b(t) shows that (3'41) is equal to

h (l'- U"i:r1'P1'P') ^{4a')" (1,- (fij;'

=rk(|,*

r1,v,1,v,,)

rot.tda")'/u

\ "/q

u6,n(s)d,sl

/

\ r/p y"(t)u"(t)dt ) o'

li.. (1,',r',G)a')''

(1,0.,0,^r,)0,) -^pvar)"o

u6,"(s)ds

\ "/P

\

)

)

uo,n?)d,t

t/'

where the last equality follows by integration by parts (see also Re mark 1.4 (iv)). Since u6,r, a"nd u)n alre in.Lr, the sum is finite, a.nd comparing the last estimate via (8.42) with the first term in (8.40), we obtain by dividing

that

o

o

k

(/ \"rt

/ fb*

o

=

as)

: (*)"' (EI.',' (1"',,-,1,va,)'/o

* (lr".ro,.(s)d") " ro(r)) "' v:

shows that

Eu":.,^1,va")'/o

/ fb*

: (E

(t) = (>*,(1",.,,1"p") "/o

F,U"::a,(g

1,t3

(lro' uo,,rr)a") ype)up.(t)wr-r1t1at)

(/><

: (1,*

/

Opemtor

u{d*),

(EI^',(l-',,-^G)d,)''o(l,o'ur,^("1a,)'/ouo,n(,)*)r/c-t/p (3.42)

=3,(t)""

lA

The Hardy-Ste$oa

Weighted Inequalities ol Hordy Tgpe

the-\Ionotone Convergtince TheNow, since wnf u and u6,r, f u6, holds with tl"' u6'" replaced by orem applies and the last estimate reads ur, uD, resPectivelY, which

ConsequentlY, sup

61*7 ut" finite. Starting with this rn construct a sequence bnt];*ez according to Definition 3.5. Hence if rn1 < r 1 rnsa.1, then a(c) < a(rnr+) : b(rn*) < D(r) and writing again a1 : a(rnr), bx: b(mx), k e Z, it follows that

(l"i;'t.,u(qdt)

show that also The same arguments with minor modifications necessitv of

=EI:,*'

(3.36). condition --(3'36)' Suppose that Ar and A2 from (3'37) and Sufficiencvof tinl (3.38) are finite. Then also

s

*'

;;;;Fi-t?

This proves (3'3e) and hence the

/.

(f (/:;'(fi,-t")a") " (f'

''' * .

::

(.9r

u@)dx

(fi, rft)tu(t)dt *

(2 ["'.' *p

,0,",r, 'o,r(r)rr) "u,r)

145

are finite. Hence D(rn) and,61^1 u. finite a.e. in (a-l(t),o-l(l)), and therefore there is a^n rn € (D-l(l),@-1(1)) where bothD(m) and

f

2(rn) < m.

Opentor

I-':,:,'

t(')u'(4d')

u@)dn

(l^uu,t oa](qdt) u@)dr

I)...' (fl:' r 6-670')"ov')

+,92).

(3.43)

In Sr, the change of variable U : a(r), Proposition 3.9 and Hiilder's inequality with indices rf q and p/q yield,

and

(f (#'(/'"t")0")

'o

,sr

/o

'

(fi,"'{")a") '"1'F') "u"')

But this means - as was just shown

-

"' * '

'

that

o't,'o')''' (l--"' o'@)a*)'t' and (l;,''

= D [" (\Jyf- ,6-(r)dr)/ hezJor

s

Fruor^*,

(1,','

u"(y)dy

*ul*@dt)

146

The Honly-Steklou Operctor 147

Weighted Ineqtalities of Hotng Type

(l-

and

s

(Du",--,;""

:

/ f@ 1e/n oo@)l I JpQ)u(t)dt) \J0

if u € trr, then

vo1t1'P1at)o/o

(1,- (l^'i,'.' rt')*)o 'o)o')'/n

,l

Indeed:

: Proposition 3'9 and Similarly, in ,S2, the change of variable y b(t)' p/q yield Htilcler's inequality with indices rf q and sz

:

*Drl:.':,'

(1.u..,11'1'6P')n

(1,- (1":;'

r t,p (,)

uu,,)dv

#

o,)o

uo

(l- (l"i:/(')'(') # : (l*

ro,tt)at)o/o. /

o')oo uoo

u{r-o)o' 1r1dr)'''

: I *td p-' : l/(l -

q),

t*,0.)

)"'

!"uo' rt0'{r1\u1,1aro,)

(1"*,t to,)o-d/q

: ( [* + ( y1,y,1,p,) 1u1;,/e' u(t) f.'),',1,p)

obtain inequality (3.22) Using the last two estimates in (3'43), we with a constant Cf,(Oo(m1 + Ds(n't))r/c

as

{,vu' - o (,) o,)"

and use Hiilder's inequality with indices p we can estimate it by

[* \Jo

Io f {r)u(t)tu.

If we rewrite the left hand side in the last inequality

, (/-

Dn(*)(

Auttutt,"o

o

(

:

'

.

Remark 3.12. Notice that if 0 < q ( 1:pand u'u satisfy

| f"-'(t) u(s)ds < oo A# := T,:J%O Ju-,r,,

tr

\/o

1.16-'11) /

/

where we use F\rbini's theorem. Now, the inequality follows due to

the definition of A#.

3.5.

Some Applications

First, we make a special choice of the functions a(r) and D(c) and formulate this example as a corollary.

148

Weightel Inegralities of Hordg

Corollary g.LL. Let a

b

onil

The Horlry-Steklor Opemtor 149

7W be

reol numberq 0

< a < b'

Then

inequalitY

\l/c \c /ra/&x I [* ( ["- yplar) "1'ya' ] / / \/' \"fo' / r@

" (/,

= hotds for all function't

/

,.i18,,,

U-

Mp:

rr/P' Stt (t 'r-o'(s)d's) ( 6r

max(A1, Az)

<

:

(alb)kt,

(b/a)kL6,

i:

k,

e Z. Therefore,

k e Z. Hence

if.

(;)-

t e (Me,M*+r), then we have

(;)- M1,
u**t:I

with o(t) the normalizing function from (3.35). Consequently, a(t) = lft, and substituting this in (3.37) and (3.38) with , - ur-P',the corollary follows.

o'

where

::

a)e(t;

*rru-'oo)&(t) = and

(ii)for0
A1

o

(3.44)

)

> 0 il and only if

(i)forI
o':

(b-r

\1/p

fe(n)u@)ax

Prnf. (i) If 1 < p < 9 ( oo, the result follows at once from Theorem 3.7 with a(c) : ac and b(r) :6a. (ii) If 0 1 e 1p, p ) l, we have to verify with the same choice of o(c) and b(c) the conditions of Theorem 3.11. We have b-l(r) : r/b so that Mo : b-r(l) : Ilb and (b-1 o a)(t) : b-r(a(t)) : at/b and

Now we apply the foregoing corollary to obtain a weighted characterization involving the Steklov operator .9., from (3.15).

/

(l-+ L,U::'r-r'1";as)

Corollary 3.14. Let (J and V

be wei.ght functions on

R.

Then the

inequality

" (l:,{,;a")"/o

,(r)drd)/

p+"t

/ fe / \c \l/c / f@ \l/P (l F(t)dtlup1axl scll Fp(r)v(r)dxl ll \J-m / / / \J-oo \/r-r

ond

A2

with 1 > 0 fweil hokls lor euery F > 0 with o c,onstant C independent ol F Nf ond only if

I r'" (l:,r-e'1s)d's)

:= (/-

(i)forl
|

, l:!-l with|:i-i.

u ,=jlfl*, (1,' u''0")' "

(1,",,{"1a")'/o,@)a,at)

C A fot ionofrn,pth" bert unstont C in @'aa) sotisfies = ronge' inilex temoining q < a onil C x max(A1, A2) in the

pS

: C(l) > 0

(ii) for 0
r 1/P' yr-n'@)ds) (1,::

p,L1p1cr,

L

ma:c(A1,Az)

(

m,

( oo,

(3.45)

150

Weighted Inequolities of Hardy

The Herdg-SteHoa Operctor 181

I\pc

with t arbitrary but fixed holds if and only if the weight conditions (i) or (ii) from this corollary are satisfied. For example, if we take p : q :2 a.nd U(r) : V(n) : 1, then condition (3.45) is satisfied since

wherc

sup (y+t- r+t)U2(a -U)r/z: t (

(1,'' A2::

uro")''' u

ol

1

gSr12t*y

o')''

it

follows that

"

(t"" (!,*" (l:v'l-n'1'va') *

and

ll,llz < ctllfllz, l

e.

(/..

(1," u{ia")'/P uov,)or)','

for every

with!: + - i. a: e-1, !: E, "y ) 0' u(x) : rp-rV(loscj, u(r): |U(logc) and /(t): |f(togc). Since F is the ,ro.r-rr"guti.," on IR if Jnd only if / is non-negative on (0, m)' o result follows after several changes of variables' with

rr(dd,)'/

The next theorem which may be of independent interest i nvolves a special Hardy-Steklov operator with a two-dimensional weight. More precisely, we give conditions on the weights under which the inequality

([ f

Consider the initial value problem

1iltt-lrxr : 0, t)0, o€R, ur(r,O) : 0' rr6(c' 0) : /(c)'

ce

IR

' by

nt"to,)'

*or)''' (3.46)

Theorem 3.L6. Letl < p < q < oo and letu(t) andu(x,t) with t € (0, l) be weight functi,ons on (0, m). I/

A:

w(x,t)::z Jx-t [" '11r;rr.

:

(/',..,1t ,,,(1," u(,,qd,) (3.47)

But this is *(fl/Xr) where 51 is the Steklov operator, and by Corollary 3.18 it follows that the estimate 1-^/r

(!,',

is satisfied.

d'Alembert's formula

.f(+\llfll

w(x,t)

sc (1,* rp7,1,'1a,)'/'

is well known that the solution of this problem is given

'r

. ct(/_

t > 0 with d independent of t.

A Cauchy problem

,,

,a2{,,0a*)'/'

A two-dimensional weight

Proof. We apply Corollary 3.13

It

69,

A/n/ry1

" (l: ,r-t 1fldr)o''' or)''o . * th.en in,equali.fu (3.46) h.olds

for aII fim.etion,.s q )

0.

152

Weightett Ine4volities

Conaerselv,

of

i/ (3'46)

Hardv

hokls

The Hardy-SteHou Opemtor L33

I\pe

for

oll g

20

(3.a6) with

then

U:'t U:'"'("'t)d' r eB \ 1/P' . (f ;-c' Plar) < oo '

o"':F'-

Proof' Fix Then

it

t € (0,1) and apply

C

o)'"

Conversely,

s@)

Corollary 3'13 with o

: f' $ = l'

cntl(1,* taw{da*)oto at, fo'

(/' (/- w(r,t) (1" or"ta")"") o')''o (/' "olryor)'n (/-n't"l,

@o)

=

,,',t)a")'/n

(!'

''l-''

tl")

\

/

k(p,s) (0."11?.,,

lr"t'' .t"'t)a"J

(/u

( rP,r-n'1s)ds) ' te/r'1r/q '(/: ) :

is change g 0' = alti the last expre*sion Here we have made the to 1S.aZ1. Consequently, we have obtained finite for a.e. €10,

t

rjl,r"

0, then for fixed

xp,B1@)ut-n' @)dr.

For this function 9, (3.46) yields

;-r'6ya,)''' : (1,* tav@) (l: " " - (/'u- w(r, t) (l*, t"la,)' r') or)''

:

where

s k(p,q),-.,prr, (f

:

9)

o

. (1,"'u (l;" w(r,t) (l**-'r")xr.,pr(s)ds)'*)

l.e.

c(t)

(3.46) is satisfied for all

(3'48)

l' (/- w(r't)(1" tav')"')"

=

kbt,q)A.

a,0,0 < o < p, let

follows that

s

if

:

r

(1,"'u

(t" w(r't)(l!*-"")'")

: (!:,r-o'1";a") (1,"'u (t" and (3.48) follows by dividins Or

o')'''

o')o')'"

w(x,t)ar) 0,)'''

(f ,t-r'G)a")'/' .

tr

Remark 3.16. Let us point out that (3.48) is not sufficient for (3.46) a,nd that (3.47) is not necessary for (3.46). For details, see H. P. HEINIG a.nd G. SINNAMON [U. The following result is an easy consequence of Theorem 3.15. It answers the question raised in Sec. 3.1 and complements some of the results which will be dealt with in Chap. 5.

The Hardy-Stekloo

Weighted Ineryatities ol Hordg TVpe

154

Corollary 3.L7.

ondf €C1(0,m)' Thenthe

Letlcp39
Example 3.18. a,nd

inequalitY

p:

lJ'(c)lPu(r)

(/' u=fo

o,)''o

(3.4e)

and

or)''o

. *.

(s.bo)

of variables yield Prool. F\rbini's theorem a"nd an obvious change

(l-l-4frJ# a*aa)'/ :g,* =

('

LIF#

dsdx*

I* [ "sa!;!fin

: 2,/c (1,'

f %$'

3.6.

€ (0,1)

([",-

^,

d,)

or)''o

Some Generalizations and Extensions

An N 4i mensional Hardy-Steklov operator similarly as in sec. 1.9, we can consider the following N-dimensional analogue of the Hardy-Steklov operator:

(trnfl@)

(')oo' c'v € RN '

(3'51)

"(v..{t.ur,,,rf where a,b are the functions from Definition 3.2. The investigation of

the inequality

/ f

\l/q

ll^..@nf)q(r)u(r)h) \JRx ,/

"a*at)'/o rr'axo")o

)

it

f*i#ooo,l''

dvd')

3z,tqfi'l". out-,,(|" tf

l,-

with

can be shown that this number is finite. Consequently, the corresponding inequality (3.49) is the inequality mentioned in sec. 8.1 and shows that (3.17) is dominated by (8.16).

:: (/',."'.%*,,,(lr''@fTl'") " (!:ovo'14a,)''''

o.o1f.o7,

" (l: *e-r)n'6,)o-'

holds prouid'ed

A

u(r): sl*lP, a(r): r(l-)h

q. Then the number .4 from (8.b0) has the form

ff-l-ffia,au)'t (l="

Take

Opemtor lbb

*o')"0

Applying Theorem 3.15 with w@,t) = a/u(a(L trj t, ihe-result follows'

'

- t)) for g(') :

/ |

\l/p sc(l\Jn'" !e@)u(x)dxl /

(8.b2)

for / ) 0 with u(n),a(x) weight function on RN can be reduced to the onedimensional case by using polar coordinates,

x:tT, te(0,€), r€Er,

156

Weighted Inequolities

d

HodV

The Hanly-Stekloa Opemtor 157

IW

: (l- (fi:

and the notation with Erv the unit sphere in RN'

U(t)

: Jtx t- u(tr)f -Ldr '

Y(t)

:

(

;-n'(tt)tN-''")/

t\JD"

Theorem 3.L9. Letg < q < F, it iold's for all f >-0 if anil onlY / ral

fte)

\q

1(p< oo' Theninequality (3'52)

satisfi,ed,

/ 16

\1/q

rt,la")'uplat) Itl/ / / \Jo\J"to is

(3.53)

\ ="(/.

lor allF > 0' The constonts

\1/P

P(t\v(t)dt) trsal

: (i) Sufficiencv of (3'54)' Fix / f (*)' ' .rr(r) :

fr,f

€ RN' and define

.

Necessity of (3.54). Suppose

F

by

(t )

:

and define

f

(1" ;-r'(to)tN '.)

t)0, r€Eiv. lr,f {rr)r*-tdr: ar-e'

(tr)tN-'o)''''

Now the change to polar coordinates (3.54) yield, together with (3'53)'

fi

: tT' U :

so and inequality

\ r/c

u(tr)tN-rdrdt)

(l"ir''r"u")

o'

F(t)

o

ov')''

: (/- ,,,) (l.i:' f,.rt ,),*-'a"a")o o,)"0 :

/(s)d,)'t"'*)

(fi1f"/t'o)'" ^*o)

and hence

(1,-

: (1r,, unu(tr)tN -t dr)'to v-'ro {t)'

(l-L

F(t)ur-o'(tr,

that (3.52) is satisfied for J

{rr)r*-'0".

F(t)

=

ro1t1v1t1at)'/o

"

and by (3'53)' Then by lliilder's inequality

([" [.',.,/n*',

u@o')''o

(tr){ -r dr) or)"' =, (I* (!,, r u"ta : (l*,!p(r)u(r)dt)"' (ii)

C in (3'5a) onil (3'52) orc

the sarne.

Prmf'

t" (I*

'r")'")o

U,* U,^

ulto1tv-Lao)

" (,fr: l,,tt"\'*-'a"a")o

o')"0

)

0,

158

The Hardy-SteHoo Operutor lS9

Weighted Inequalities of Hordg Typc

(i) tor L < p
: (["'(") (",,,',.,/n*,"', r oau)' u) =

=

r (.[" Ie@)u(ddr)

/

,.

(I"

fr,tN

" (I" / f*

:cl I \Jo

1fian(r

J b(rl)
(

,r-n'p)dz)

where the suprernurn is token oaer oll s,g e RN urrth lyl a(lcl) < u(lyl); (ii) for0 < g < p, 1 I p 6, +

oo,

-P')

0-r(a(cl)
x(/

\r/p' I \r/' aolarl

| \r/J/ c(lcl)

ond,

t

/

r

J "(y)( J "(")L lcl
or)'

t,,/p "@,)dy,)

lcl
/

"(

r

., ,. o(lvl)<

..\1/o FP(t)v(t)dt)

/

allows to characterize weights u

\"/P "fu,)dYr) (3.56)

r

/ f -..f (/o"

from (3'53)'

( f t J

(ts(,)l. f \.,Rx L J

(tr)

< lr;l ond

: i i,

I

1N-1,1-r'11id")

{-roL-n'11")0")-'

where we have used the notation

r

(3.55)

ur-p'(to)tN-,*)- lo,or)'''

"

\uq/

IRN,

tP

-',, {tr1lr

: (lr* (I" ""(,)

x,,y,z €

u(z)dz) (

'uo( J h,l
f' {rr)rN -t a(tr)dror)''' (l fr, r* "

: (l " r*

f

ond,

tr

Theorem 3.19 again that (3.52) is satisfied. Let us formulate the result emphasizing

the properties ifre functions a,D are defined on (0,m,) and satisfy : tr with t > 0 and described in Definition 3.2' For c € RN and c r € Ery, we have lxl = t' > 0' Corollary 3.2O. Inequal,ity (3'52) is sotisfied for all | = f (n) c € RN, if onil onlY if

\"/p' I \r/"

-,.

(3'57)

au)an) < oo J .u'-P'@z)daz) ly,lca(lrl)

are satisfied. Here,S is o mtLiol Junction defined by

,9(c)

and u for which

1

:,s(l'l)

:

lclr-trofl'l)/lrrl

witho the normolizing function defined by (a.Bs) ond lDryl the sur-

foce arca of the uni.t sphere in lRN.

Prmf. By Theorem g.1g it suffices to show that conditions (8.5b), (3.56) and (3.57) are equivalent to the conditions of Theorems 3.2 and 3.1'1 with u, u replaced by I),v defined by (8.b3). But this follows at once via changes to polar coordinates. For example, if we change

160

TW

Weighted Ineqtalities of HordV

The Hordy-SteHou Operotor 16l

: 8P' yL: srpr'' u2 = : to polar coordinates in (3'56)' taking r tT' 9 then we obtain (0'oo)' € s2p2 with T,P,Pt,P2€ Erv and t,s,s1,s2

I*

_t

L,tN

s

(t)d,r[11,,",r,

/ ft |

" (/" /"" "fl-'"(

" (fi:'

: (l-"r,r

_,

lr,sN

s1P)dP1ds1

[1-,,",,,,

Weighted inequalities for this operator are studied in A. GOGATISHVILI and J. LANG [1]. Let us mention one particular result which for lc(o, t) : t reduces to Theorem B.Z.

u(s dd,o

Theorem 3.21. Let 1 < p < g < oo. Then the inequality

\'/P

)

1,,'l-"'-o'(s2P2)dP'0") u(")

"]

*)""

(!,hokls

(1":;)

",',";

for oll functions f >0

xf orul only

(r/X') :

if

\Uc I P@) \ rlP' u(s)dsl 1ro'1r,s)ur-e'(s)dsl .*. I - "yp-. ..1\"/c cJs,c(y)
/ fc

.

\"/"1y1

as required' The arguments

to"

Monotone tunctions monotone functions. The methods used there are also applicable for Hardy-Steklov operators. For example, if ? is such a^n operator de fined on d,ecreasing functions /, then the characterization of weights u, u for which

n{',t)f (t)dt,

in a natural way extended to opera'

f

(3.58)

tors of the form

(K",6/Xc) -- .o,'k(t,t)f(t)dt J o(o)

T : U(u) -+ Lq(u), 1 (p,s < oo requires (in the case Jfl u(x)dt: oo) conditions on u, u

(boundedly) for which the inequality

(1,*

(I"

rrs1ft1at) (1,' ,roo,1-u utta,)'/o'

following conditions: where the non-negative kernel lc satisfies the and decreasing in

/

In Chap. 6, we will deal with Hardy and Hardy-type inequalities for

can be

r

"

l

Generalized Hardy-type operators

k(c, t) is increasing in

=

(lo* fe(x)u(r)dr )

and

But this is the expression 41 from (3'37) for (3.55) and (3.57) are similar'

(i)

\ l/p

"'

*@, t) 1 e)at)

/ fuu(s)tcq(s,b(t))d,s) \r/c / N@) \ r/p' slp....ll ,t-r'(s)dsl .* cSy,a(y)!D(c) \"lc / l/ \/o(v)

(f u1,,;a"')"/o

, ( f"' v.-p'(s)d,sr)""'0,)"" / \J"(o /

investigated in Chap. 2,

(ii) e(c,z) 1.

t'

(3.5e)

sc

(lr*

oc'(t)ar-p'

(t

*)"'

162

Weighted, Inequolities

ol Honlg TVw

is satisfied for 9 ) 0, where f tor (see Chap. 6 for details).

it th"

The Hadg_SteHoa

conjugate Hardy-Steklov opera-

In order to solve (3.59) we need the knowledge of the mapping properties for the operator I(o,6 given in (3.58) with some kernel k(r,t). However, there are /our weight conditions which characterize the weights for which Kop i U(a) -+ fe(u) (boundedly) if 1< q < p < oo and ttlo conditions for the index range \ < p < q < oo (see T. CHEN and G. SINNAMON [1]), so the result is unwieldy. It may therefore be appropriate to state here a simple (sufficient) condition for which the Hardy-steklov operator defined on decreasing functions is bounded. For details, see the paper jrtst mentioned.

Theorem 3.22. II rr,t) (ne weight functions on (0,a) such that Jf, u(r)h: @, then the inequolitY

(l- (f;'rt,p,) ,(')d")''o =" (1,* ratu{,)a*)'/o is

satisfied

for

oll decrzosing functions

:g (l"o''(")'") holils

in the

cose

'-'

|)

0 wheneuer

(1," ruts) - a(s))qu(")'")"0

1< p< q<

a,

'-

ond

G. SINNAMON [U. If ? is the Hardy_Sreklov operator (B.Z) or rhe moving averaging operator (8.14), then for the sfecial D(r; :,, a(r) : a/\, ), ) 1, the characterization of weights .r,u "*" f* which U(o) -+ La(u_),_l 1 p 1e ( 6, is bounded, wa.s given by T : E. SAWYER (see H. p. HETNTG and c. STNNAMOT.i

trli.'

3'7'2' The application of the moving

averaging operator to financial markets can be found in Journar of Finance lgbz *nicr focused largely on the Dow Jones industrial average from

lggz to 1gg6.

3'7'3' The weight characterization of the steklov operator (see corollary 8.14) was given arso by E. N. BATUEV and V. D.-STEPANOV [1J in the index range I 1p, g < oo. In addition,

they discussed there the Cauchy problem mentioned in Sec. B.b. 3'7'4' Theorem 3.1b can again be found in H. p. HEINIG a,nd G. sINNAM'N [1] rogerher with examples which show that (a.aifis not necessary for (3.46) and (B. g) is not sufficient for (g.46i.

3'7'5' In corollary 3.r2, _the.weight u(c any weight u satisfying u(r,y) = u(A,r).

-

3'7'6' Theorem 3'rg is a modification of the corresponding results for the Hardy operator in higher dimensions Uy C. SiNXAIAOI.U 1q.

3'7'7' As mentioned in sec. 9.6, A. GOGATISHVILI and J. LANG fact, they proved weight characterizations for which'Ko,6 isiounaed

(1,^' ,(")d")

between certain weighted Banach function spaces. As a consequence, a characterization of weights u, u follows for which .l Ko,6

,

(/'trr"l - a(s))qu(s)

or)"0 u(a(r))a'(r)dr < oo

hold.s

in the arse I < q 1 P < @ with !

3.7.

Comments and Remarks

: l_! qP'

3.7.1. Most of the results mentioned above, namelS the main Theorems 3.7 and 3.11, are taken from H. P. HEINIG and

fit

if I < p < g ( oo, while a characterization for the range 1 1 prq < m together with the discrete analogue was given bv T' CHEN and G. SINNAMON For related results see arso [u. ^Le(u) boundedly

,.

yl) may be replaced by

[lJ studied weighted inequalities for the operator.tr(o,6 from (3.bg). In

/o

l,*

0pemtor t63

E. LOMAKINA and V. D. STEPAI{OV

1r1.

Higher Order Hardy Inequalities

4.1.

Preliminaries

In this chapter, we consider the k-th order Ha,rdy inequality

(!"'vow,@d,)''o =" (l"o wn roleu@)ar)

(4.1)

for functions g from certain subclasses of ACk-r with lc > 1. More precisely, since inequality (4.1) is meaningless if g is a polynomial of order less than lc, we will suppose that g satisfies the following "boundarv conditionstt : n{i)@)

eo(b) where Mo,

Mt

:0 :0

a.re certain subsets

for i € Ms, for j e Mt, of the set

N1: {0,1,...,& - U.

(4.2)

f66

Higher Order Hordg Ineqtotilies 167

Weighted Ine4nlities ol HordU fuW

Thecorrespondingsetoffunctionsgforwhichinequality(4.1)will be investigated is denoted bY

ACk-r(Ms,M;1'.

Example 4.1. Not all

Ii

we take k

:2, Mo: Mr:

The P6lya

/

a locally integrable function is uniquery solaoble. Therefore, there is a Green function G(rrt) for this BVp suctr that the solution of (a. ) is given by

g@:

Mt

are admisthen ACI(Mo, M1) consists

choices of the index sets Mo'

{1}, :0' The function of functions g € ACr such that g'(a) : S'(b) (4.2) but the right conditions g(c) : constll0) obviously satisfies hand side in (a.1) aonishes'

sible.

with

(1"' xrnoyu(r)de)"0

t if ieMo, ear=\I o if i(Mo.

(i

r:0'1'"''k-1' f("or+eri))r*1, i=0

(4'3)

i:1,2,..-,k.)

Remark 4.2. (i) As was shown in P' DRABEK and A' KUFNER for u e ACk-r(Ms'M) if [t], the Hardy inequality is meaningful u"a o"ty if. (Ms,M1) satisfies the P6lya condition' (ii)Thereisanothermoreimporta,rrtreasonforintroducingthe (4'3)' the boundary P6lya condition. Namely, for (Ms, M1) satisfying value problem (BVP)

= 0forj€M1

4.2.

(

.2).

Some Special Cases I

The case Ms : N3, Mt

:0

of

Ms:

0,

Mr = N*

If all boundary conditions appear at one end-point, then the p6lya condition is obviously satisfied and the Green function G(t,t) of the corresponding BVP (4.4) can be easily determined: we have

G(r,t): (4.4)

(4.7)

i.e., exactly k boundary conditions appear in

g(k) :

/in(a,b); ,(i)(o) : 0fori€Mo,

(4.6)

Let us denote bv IMJ the number of elements of the index set M; 0,1). We will consider the following special case:

lMol+lMrl:k,

(This condition states that there are at least i l's in the first i columns

90)(b)

. c (1"' van ,@1a,)'/o .

:

The pair (Mo,M) is said to satisfy the P6lyo conditionlf t

(4.5)

consequently, instead of the Hardy inequality (4.1) we investigate the equivalent weighted nomt inequolity

condition

(Mo' M1) as the Let us introduce the incid'ence matri't E of the pair : 0, l, " ' 1 defined as 2 x k matrix with elements eai, a: 0, 1, i 'k-

of ,Efor

I: G(c,t)f (t)dt (: (r/)(c)).

ifMt:9*6

ffi

X(",,)(t)

168

Higher Order Hordy

Weighted Ineqnlities ol Hodv Tvpe

Ineqnlities

169

(ii)orl.
%5x1",0v(t)

(1"u-

(1"' if.

Ms:0.

in the first case the form Consequently, inequality (a'6) has

ch

U:l/(c)l'u(r)d ')''o

')o')'o

ut-"'(t)dt)/o' ,r-r'@)a,)

(6, (4.e)

t1k-r U:ll:, - YPlatlo't")")"0

tt

, (I'

c;(t-r)c"1

(['" -

(1"'

lP

, (I'

,

and analogouslY for the other case:

t1&-ttn''r-P'

u@dt)

(odt)

u(")r")""

.*

"ithl:i-i. Pa^rt (i) of Theorem 4.3 is precisely Theorem 2.21 with the particular kernel k(r,t) : (s - t)e-l, while part (ii) corre sponds to Theorem 2.15 with the same kernel and the interval (o,b). Conditions (4.8) are then conditions (2.62) while (4.9) corresponds to (2.52). Using the notation from Introduction and Chap. 1, we ca,n

Remark 4.4.

#il (I'W"-")*-"(t) =

o'lo

'{do')''o

" U:ff@)tpu(r)d,)'''

rewrite conditions (4.8)

of certain weighted norm inBut these inequalities are special cases 2'21)' and so we can investigated in Chap' 2 (see Theorem "oJ;i;" result: immediately formulate our first

g€ all Theorem 4.3. The inequolity @'l) hotds for otunctions if onlv g(e-r)(a): if ond ACk-r satisfying g(o) J'nt1a)-:" ".(i) eitherLz-PSqlcn ond

(l t - xy(k-rhur,u,)''' (l""ur-e'(')dt)''o' (

"::?, "::?,

(l

u(r)dr)'''

(l:o-

tv(k-r)n', '-'' (!dt)

.--',n.r,

o'

'*t

A(a,b;(t A(a,

b;

a.s

-

c;(t-t)au(t), u(t))

u(t), @

-

t)

(r

-r)r"11;;

(see (0.8)) and conditions (4.9) as

A(a,b;(t A* (a,

b1

-

r;(t-tlc u(t), a(t))

u(t), (a

-

t)

(r-t)r"11;;

(see (0.9) and Remark 1.4 (iv)). Of course, in a similar way we can describe necessiaf,y and sufficient conditions for the case of functions

g e ACk-r satisfying g(bL:_g'(b):...:g(&-rl(a) - 0: we only have to replace A, A* by A, A' .See also Theorem 4.44.

170

Higher Order Hatdy Ineryalities L7l

Weighted Ineqnlities ol Hafllg Tgpe

also Remark 4.5. The lc-th order Hardy inequality (4.1) can be (1'25) succes' investigated using the (first order)."*tdt inequality taking k:2 example' sively ior the functions 9,9t,"',9(&-t)'For

and investigating the inequalitY

(/out"lt', p1a,)'to

,"

(1"'ff'(t)lpu(r)o')''o

'

(4'10)

:0,

g'(b)

:

o

is sat(i.e., taking Mo: {0}, M1: {ll so that the P6lyacondition tuo inequalities: isfied) we can try to solve this problem considering

the inequalitY

(l' wax,'t")d')"o =",(!"u v't )l''t'!a')'/' for functions

I

(4'11)

satisfying g(o) = 0' and the inequality

(4'r2) (/o u't'lt',@1a*)'/' s c, (1"' ls" (r)leu(r)o*)''o weight for functions g' satisfying a'(b) : 0' with some "intermediate"

parameter r' But besides the presence of function tu and ar, "u*iliu'ty and from the point of view of the initiol these rather undetermineJ this approach has inequality (4'10) redundant auxiliary data tu'r'

anotherdisadvantage:itcannotbeusedforeverychoiceofboundary conditions.E.g.,theconditionsunderwhichinequalities(4'11)and intersection' See also (4.12) hold siiultoneouslg may have an empty the following examPle. (a'10) with Example 4.6 [(o,b) : (-oo,oo)]' Consider inequality p -- s: o : -L, b : *oo and-u(c) = edt, a(r) : "0r' where a,PeR. If I eACr satisfies

9(-oo)=0, 9'(+m)-0'

that 9(-m)

( .ll)

holds

with

- if if and only

- 0

r:

p,

u(r) : eI,

and,for g such

?:c<0,

(4.13)

(4.14)

while inequality (4.12) holds - ifwith the same r,to and for g such that g'(*m) :0 if and only

-

1:B>A

for functions g e ACr satisfying the conditions g(a)

then inequality

(4.15)

(see [oK, Example 6.12]). The contradiction between conditions (4.14), (4.15) indicates that this approach i.e., the successive ap plication of first order Hardy inequalities -does not work for the interval (--' oo) if we consider boundary conditions of the type (4.r3), i.e., with Ms and M1 both nonempty. (Notice that the approach just described works if. we consider g such that g(-m) : gr(_oo) : 0, i.e. Ms: {0,1}, Mt:0: we have to take a:^f: p <0.) Consequently, this suggests that for (a,b) : (--,*), the choice of, Ms,M1 such that either Mo:0 or M1: 0 is the only reasonable one provided we want to use succesively first order Hardy inequalities.

The case (a,b)

:

(0,

*)

similarly as the infinite interval (-*, *), arso the semibounded interval (0, oo) plays an exceptional role. Namely, similarly as in Example 4.6, we can show that for some choices of the index sets Mo, Mt, the successive application of first order Hardy inequalities does not give satisfactory results. This approach works for the interval (0, m) if we choose Ms : Np, M1 : 0 (see Theorem 4.3) or Mo : 0, Mr = Nr (see Remark 4.4). Also the choice

Ms: {0,1,...,m-l}, M1: {m,m*1,...,k-l}

(4.16)

with 0 < n't, < k is reasonable as will be shown in Theorem 4.g below. we will illustrate it in Example 4.10. Moreover, the exceptional role of the interval (0, oo) will be dealt with in Sec. 4.g. of course, the formulation of analogous results for intervals of the type (a,6), a I 0, and (-*,0), b € R, is only a matter of routine.

L72

Higher Order Hatdg Ineqtolities f73

l\pe

Weightetl Ineqtolities of Hardg

Hardy inequality Example 4.7. Letus consider the second order

fo*

t"

btdlor..-zods

Io*

b"(')lo*"d'

uith Ms, M1 defined Dy (4.16) if and only

(4.17)

fo*

=", Io

fu'@11o'"-o6'

(4.18)

(cf)(*)

v

tdlo'a-pds

=

",

Io*

b" @)lo'"a" '

(4.20)

*; tryu(t), 1$-m+tltr 1111

(4.2r)

:

(4.1e)

g satisfying the other hand, if we consider functions

g(m)

g'(o)

:

o

this approach {1}, Mr : {0}): we carrnot use : i! and onlv if 0 g(oo) inequality (a.fgihofd' lo' g-"ttisfying y'(o) : 0 ir and

(i.e., choosinE since

:0,

Mo:

(a'19) hoids for. e satisfvins

is no urnmon aolue of a' ""i-fio:1,o)'while only if o € (-m,p - 1)' so that ^th11 for the c'hoice g(m) = Nevertheless, irr"q',tiity ( 'f7) holds g'(0) = 0 if the Parameter a satisfies

even

a>2P-t' see ExamPle 4.48.

1rn 1k' a= 0' b : o' Then inequality (4'L) allfunctionsg e ACk-r(Mo'Mr\ hotds withO (p,8 ( oo, p> I lor

Theorem 4.8. LetT

t

(rn-l)l(m-k-lr ,, 1s['@ -

11

-'

' (/-," - t)']'"-k-tilsyas) at

InequalitY (4.17) holds

if g(0) : 0, s'(0) : 0, for a 2P-l Mr - {1} or Mo = 0' (i.e. if Mo : t0, U, Mt -- 0 or Ms: {0}' case corresponds to (4'16))' On M1 : {0,1}, respectively; the second

u717, 1(x

Proof. The function g : Gf where

a.nd

lo*

- nlo u 1111

m; 1(m-r)e

.l-10,

via two first order Hardy inequalities

b{*\lo*a-2pdr

(0,

.A

if

(4.22)

satisfies the conditions

9(0) : g'(0): -..:g(^-\(O):0, 9(-)(oo) :

n(m+l)(m)

: ... :9(&-l)(oo) :0,

i.e. conditions (4.2) with Ms, M1 from (4.16). Moreover, we have : /, and inequality (4.1) can be rewritten as the weighted norm

g(k)

inequality

(/-

rttllt x)rqu@)aa)"'

. (/* ut'lreu@)ar) . (4.2s) "

Obviously, it is possible to consider inequality (4.23) only for Now, by F\rbini's theorem we have

(G/X')

:

_( (rn-1)!(rn-k- l)! \ fo' *rrr,t)f(t)dt *

I,* K2@,t)f(t )")

/

>

0.

L74

Higher Order Hordy Inegalities l7S

Weighted Ineqtalities ol Hodg Type

Let us again emphasize that we suppose that (Ms,M1) satisfies the P6lya condition and that lMol + lMil : k, Mo * 0, Mr * 0.

where

K1(r,t)

:

K2@,t)

:

l"'

A

-")--t(t - r)--k-td',

ft

t" -")--1(t Jr-

.

--r.- s)--e-lds,

0


0<

( Fr

r
Definition 4.9. Flor a pair (Mo, Mi satisfying the p6lya condition and such that lMsl > 0, lMll ) 0, we define non-negative integers arb,crd as follows: Let a be the number of consecutive L's at the

oo'

beginning of the top row of the incidence matrix .E; 6 the number of consecutive l.'s at the beginning of the bottom row of .O; c the number of consecutive 0's at the end of the top row of .E; a,nd d the number of consecutive 0's at the end of the bottom row of E.

and it can be easilY shown that

K{t,t)

- ,tn-r1m-k , K2@,t) o rmlm-k-r .

ConsequentlS inequality

(

F\rrthermore, we define non-negative integers

'23) is equivalent to inequality

A : a-L, A=a, B : b-L, B:b,

( [* (r^-r [' y,-xy1t)dt+r^ [* ,--k-tl4)dt)'.r1";a")'/o / / Jt Jo \/o\-

=

"

(lr*

(4.24)

Y'1,'1,1*'1a,)lo

On the left hand side of (4.24) we have the operator

?

given in (2'7)

with

g{r) : r^-r,

,!r{s)

: t*-k , 9z(r) : '*,

Hence inequality (a.24) holds bv Theorem 2'3 and (4.21) are satisfied

4.3.

'ltz(t)

if

:

1m-k-r '

and only

if (

C:c-l C:c D:d-l D:d

if a+c:k, if o*cKlc, if b*d=k,

if b+d
Example 4.10. Take k :6, Ms: {0,1,2,4}, M1: {1,0} [i.e. 9 e AC,(My,M1) satisfies S(0) : y'(0) : g"(0) :9(4)(0) : d, g,(t) : g"'(l) :0]. Then the incidence matrix is

":(; l;l;;)

'20) cl

The General Case

The P6lya condition is obviously satisfied and we have o = 3,

(0, m), we have greater Unlike the unbounded intervals (-oo, m) and interval variety of choices of the index sets Mo' Mt for a bound'ed (o,b).Without loss of generality, xte assume from now on that the

c: l, d,:2.

(0,1).

Thus, the k-th order Hardy inequality

a*c1k, b+d
( '1) has now the form

D:

0,

Obviously, for any pair (Mo, M1), we have

interval is

(4.30) below.

A,B,C,D as

follows:

a*clk-1.

(4.25)

Higher Order Hordy Inequolifiies L77

o! Hordg TW 176 Weighted Ineqnlities

Theorem 4.12. Let 0< q < m, 1(p< m. Let(M6,M) satisfy the P6Iyo undition, lM6l + lMl = k, lMol > 0, lMrl ) 0 orul o* c ( fr, D+ d < k. Then the Hardy inequality

ls the o':"1':t-l:t::-ti*T we have mentioned that [1i] unctionwehave

,"tH""11i"3t5;;;;"espondingGreenf&-r k-l <
G(r,t)= ,o-lmtr 1r - t1t

-i +Ert*";; t)k-1x10'')(t) j=O

+i

\l/c / F ( I ls@)lqu(r)d,rl \./o ,/

(4.26)

i:0

ul3 depend on (Mg, M1). where the consta,rrts

If

we denote

hokls

( X{x't) for 0
for

oll g

$'27) A

"rtr,4

r rrl f' (/, lJ,

K1(c,t)J(t)

&+

I

L

K2@',t)f

::

=

"

(4.28)

(1,' lt(")lo'(")d4

ly# ::":!)y:f* Kt',Kz [email protected]) sotisfg

(Mo'{1) uv-,-v'r 4.rL' Let a'heorern *cLtc rheorem

kt lMrl > O' Then the fKs(c,t)l N sA(l -c)bt"(l

k' lMol >


ffiil;;ia" ;;

;n;

*itt' i'a 'C'D

:

*d o*,"r7,!^ ' ated +\r .qn he aoproximated and lG(c't)l can be approxrmi

is simpl"]f lttl Jtrara"teriuation characterizationssrnrrlt*uvur'E

u, l/,

0'

for 0

P'

(1

-

t)de'ut-e'

It

lc \l/q G@,t)tg)dtl u(o)h)

(0' 1) on the uhole squa're

x

(0' 1)'

/

(t)or)"o' < oo.

(4.31)

: /(c) and

/ | \r/P' r (\/, lf(t)leaoat) =

(0,

follows from (4.29) that G(c, t) does not change sign in (0, 1) x l), and so we may restrict our attention to non-negative functions

/.

Since

in our

case

lG(c, r)l

x f(L - x)bt'(t -

t)d,

the above inequality is equivalent to

(lr' **r' - x)hu(r)d'r)"'

special PolYnomial

s"(1- a)bt'(t-t)d

(4.s0)

if

Proo!. Writing 9(r) : fi C1r,t)f (t)dt, we have n(*)(c) the Hardy inequality (4.30) becomes

/ frl tr

result: is based on the following

(1"' *rr, - *)oou(ddr)

' (l'

\1/c

(t)dtl u(r)tu)

/u

\l/p

e ACk-r(Ms,Mi rf and only

(4'1)' i'e' of the equivthe Hardy inequality of investigation then the ale* *"igttted norm inequalitY

tq

/ i

sc(\"/o 1 1n(t)1c1pu1c)dc) /

"

(1"' yng,pyat)

(I'

t"(1

-

t1o

71t1at)

178

Higher Order llardg Inqualities l7g

1W

J(t) = a1-p'(t)l,€@-r)(r-t)d(p'-1), *or" precisely, A < C' - since then

(i) If

I

Weighted Ineqvali.ties of HordV

we put here

fl1r

l'

JO

,'(t -

t)d fft)dt

:

tw'(t

Jo

-

we obtain (4'31)

4@'ut-n'

ge

ACk-r(Mo,Mr) if and only if

(i) eitherL
,'ro'

(l'

,::?,

(I"

"

and also

1l

fr

--,

.

[' 'n6ug)dt: Jol_ P'(r - fan'rr-n' rrrot. Jo

,::?,

fl l',"(, Jo / il

(l' =

t)d

f g)dt

(ii)

-

tlbt,ltyat)

.*,

t)Dr'rt-o' ,r)ar)t/o'

(4.32\

(/"

-

rc(r

tcP'(r

or0 < q < p,

,,-. : l'rl ! {t)r'/n 61t"(L - qda-r/e $)dt Jo

/ rl 'l/P' fe(t)a$)at)''' (1,^ P'(t -;san'ur-n'Q)dt)

tq'(L

* (1,'

Thus condition (4.31) is necessary' (ii) BV the Hiilder inequality

-

tAq(r

ond

t1Bo,1t1at)

-

t)dn'or-o' rrrar)tlo

| 1p < x

"

'

(I'

xrAo

andfor/Z0wehave

tq'(t

,

ond,

- tlbtult'1at)

(1"' (1,' rc(r

:,1/P

.*

-

t)Dt'rr-o'

(! - x)bqu(r)dr)

t''

,r)or) . -, (4.33)

llG/llc," S

n(/) 3 /ll/llpp'

(I'

Thus condition (4'31) is sufficient' between In the foregoing case, it was not necessary to distinguish describes the the cases p S q and p > q' In the next theorem' which separately' possibilities general case, we have to consider these two

Theorem 4.13. Let (Mo,M) sotisfy the P6Iya condition' lMol + : k, > 0, lMrl > 0' Lct a,b,c,d' A'B'C'D be as lMrl 'in-befinitionlMol 4.9. Then the Honly inequolity (a'30) holds for oll

(1,'

" (l"t

- tlBoultlat)

rc(r

l"p'(1

-

t)an'rr-o' ,rror)"0'

x roq(r - dBqu(x)d,r)t'" . -

uithL-!-!. q pr Proof- As mentioned above, the Hardy inequality is equiralent to inequality (4.28) which again due to the esrimates (a.29) and to

-

Higher Order Hardy Ineqnkties f81

180

Hordg Tgpe Weighted Inequolities of

to not change sign - is equivalent the fact that the kernels Kr'Kzdo

-,f

(/' ("^,t

r

ro(1

Io

t"(1-

t)D f

- dB l,' ,"r'-

g)dt

tld11tpt)n u@)a*)'tn

Proof. Flom formub (4.26) it is clear that h has ft - 2 continuous derivatives and that tr(k-z) is not differentiable. In particular' none of h, ht,...,h(k-2) is identically zero. Thus if we denote by S; the (closed) support of h(i), then S; is not empty. Since h is a polynomial on [0,t] and on [t,11, ,Sd is one of the intervals [0,U,{0,t] or [t,1]. .. Let z; be the number of zeros o1;(i) in S;. Tbke .L; to be 1 16(i) has a zero at the left endpoint of ^S; and 0 otherwise. Take R; to be 1 i1;r(i) has a zero at the right endpoint of ,Si and 0 otherwise. Obviously,

t"'(lr'

fel)u(qd)l

zs) Ls+I{o.

follows from Theorem 2'3 for / > 0. The result now in the Proof of Theorem 4'8'

analogously as

o (4'32)-and/gr

conditions Necessary and sufficient expressions A' A' A* ' A' the of in terms (4.33) can again be '"*Jti"" 1' (iv))' More precisely' ( '32)

Remark 4.14.

(see (0.8), (0'12) means that

and'n"*-t

A(0,1; tAq(l -t)ku(t)' ,r10, t;

fq (t

-

rq(l - t)-Do'(t)) < 6'

t)Bqu(t)'t-cp(1

-

t)-deo(t))

If 0 < i < k - 2, then 6(i-t) i" constant oft ,9i so the z;-s zeros of 6(;-t) tr6u1 lie in Sr-r are, in fact, in .9;. By the mean value theorem,

- 1 zeros in the interior of ,Si. Thus (4.35) z;) zi-1-1*Li* Ri, 0 < i < k-2. lf. L;:0, then *'rn"t 6(t)(0) l0 or Si : [t,1.] and 6ti)(t) 10.

7r(l) 5as

at least zi-r

The latter is impossible since h(i) is continuous. Since h is a section of Green's function of the boundary value problem (4.4), h must satisfy the boundary conditions, so the former implies i ( Ms. Thus

while (4.33) is

A.(0,1; ./s(t -t)bqu(t)' [q(l A-10,

t;

tos(L -t)Bcu(t),

- t)-Dpu(t)) <

fcp(L -

oo'

Lemma 4.t5' Let

t € (0' l)

Thenfor0
be

fired and

= o if ond ontY d: .t onlv il i e iri4 r,tti1il =0 il anit

ii,;1 nttf 101

M:'

Mr

d'enote

h(r) : G(t't)'

Lt>-Xu,(i), 0S iSk-2.

(4.36)

R;>Xu,$), 0
(4.37)

Similarly

t)-dp?,(t))

be done proof of Theorem 4'11' This will What now remains is the Let us start with the Green function in several auxiliary assertions' coefficients ur6i' Ctt,ti a"* by formula (4'26) with fixed

(4.34)

Now the graph ,1 tr(o-z) is a line segment on [0, t] and another line segment on [t, 1] so it has at most two zeros in its support' that is zk-z I 2. lf. k- 1 € Ms, then h(e-l)(Q) : 0 so 1(t-z) it constant on [0,t]. Whether the constant is zero or non-zero, there is no contribution to zk-2 from [0,t] so in this c&s€ z1-2 S 1' SimilarlS if k - 1 e Mr then zp-2 < 1. It would violate the P6lya condition for k - 1 to be in both Mo and M1, so we may conclude that 2

) zp-2I xu.(k -

1)

+ xM,(k

-

l)

.

(4.38)

182

Higher Order Honlg Inequolilies lgg

Weighted Ine4uolities of Hady fupe

Adding the inequalities

(

'34)

(4'38) and using the fact that

-

&-1

: f(x,tr"(') + x,u,(i))

lMol* lMrl :

are in fact equalities' (4.34)-(4.38) , : "' = Sk-z: [0't] and' by Suppose that ,90 : [0,t]' Then 51 (a'37) implies .ontirruity, Ro : Rr : ' ' ' : Rk-2: 1' Equality in zero : it ut 10, i, . , k - 2l C Mv Also, since S*-z [0't]., *" only g(e-t)10; 10, so oi ntt:il in Sr-z is at r : t. Thus zk-2 = 1 and Mrand we have L e k k - 1 |Mo. Equality in ( '3S) implies that excluded' Mr: {0,1,. ..,k - 1}, i.e' Mo:0, which was explicitly : [0' 1]' tt,r" So I ll,tlt. similarly we can show that So * It'l]' so '9s interior in ( '34) implies that h has no zeros in the No* "q,ruiiiy of ,Ss, which is statement (i) of our lemma' If i € Ms, then standard properties of Green's function imply equality in tn* 6t0(0) : O. Ot the other hand, if I 4 Mg:,then This proves 0' (4.36) impiies that L; = 0 and as argued above' 1'$l S tr ,t.t"rn"rrl (ii), and (iii) follows analogously'

Corollary 4.16. a-"G(r,t) is o polynon$al for r (1 - c)-bG(r, t) is a polynomiol lor t Z t' 1 t' formula (4'26) Proof. Let t and h be as in Lemma 4'l'5' For r &-l h(c) =

-

c)-DG(c,t) tr

lc'

i:0 wegetthetrivialinequality2
becomes

Similarly, it follows from Lemma 4.15 (iii) that (f is a polynomial for r )_ t. The adjoint boundary value problem Let

M6 = {ieNp:k_L_i4Mo}, Mi: {ieNl:&-1_i4M}. The boundary value problem

9('t) : 9(i)(0) : 9u)(1) : is called odjoint to BVP ( . )

The incidence matrix E* of (Md,Mil can be constructed from the incidence matrix E of. (Ms,Ml) so that we reverse each row of E and interchange 0's and I's. For example, if

i=0

F}omLemma4.15(ii)andfromthedefinitionofthenumberoit : follows that h(0) = h'(0) : ' ' ' :h(c-t) (0) 0' i'e'

for i
t o),

":(l'1o \01010

0)'

then

u':(t o l o o o). \11010 rJ

*-r

ri $ tt j:0 wu?T.

f in (0,1); 0 for ieM6, o for jeMi (for (a,b) : (0,1)).

It

is easy to see

(Mo,

Mi

that the pair (M;,

Mil

satisfies the p6lya condition

does, and for the numbers d' ,b* ,c*, d. defined tor (M[, according to Definition 4.9 we have o* c, bt d, c* a and The Green function for the adjoint problem is

:

:

:

if Mi)

it :

b.

(t-c)t-t'-r*FS'., It G'(r,t):G(t,a' ;J : 6iI X1o,t1@). ,r*Huti iti. Corollary 4.17. If Krcnecker symbo[).

j 1 c, then lrr,i! : (-l)j+ldi+i 3_1 (with 6 the

Higher Order Hordg Inequalities 185

184

Weighted Ineqvolities

ol

Hardy

11ype

Prnf. For t € (0,1) fixed, set h*(r)

:

G*(r,t).Tf.

r
(we have shown in Corollary 4.17 that

then

tu(t): alW(0,t). Since lV(c,

h*(r):E#.EE'"';* : &' : cimply that h-(0) Lemma 4.15 (ii) and the relation ... : [+(c-1)10; :0, which leads to (-r)itt-r-i

H

ti

(k-1:T*E'";:o

for all This porynomiar in t vanishes ar e zer o - ConsequentlY'

h"(0)

1.t,r101

:

Corollary 4.1E. r-o(1

,,

ror i
is clear that

[(;*)'w(r,r)rt=o]

if i < k-r-a' (-1]t*tlo-t.l-' :['. : rim A- \"/-;'ib (k-l-i)! -t(-tJi-" if i:k_!_a.

t - t)-dG(t,t) is o polynomial for 3t'

*11,T:9J:111i -l1:lililTi Proof. weknow from corouarv ::'r9 i::": T:T.:T: and is also a pJvnomiar if c 3 t'

This proves assertion (iii) a,nd the i/part of (ii). At the other endpoint we have for i e Mi

{-ro1ry

:

tr

Now,weinvestigatethebehaviourofr-"G(r,t)asr_+0. Then for 0 3 i s k - L ue : Lemma 4.L9. Letur(t) Dl=lt"i!rr.. houe

(;) tu(t) *0 Ior t € (0'1)' ,;;i ,t;tol =0if onioitvif ieM;whenilk-L-a' (;tl1 ,tt-t-o)(0) l0' liri ,to1r) ='o if onil onIY if i € Mi'

: and so, the

LetT:

: n-o(",",, %#xto,"l(t)) = EE' "+';

m [(*)'w(",t)rer] :o },* l'-" (*)'G(c't)lt=']

i/part of (iv) holds as well. {i € M[, :i
is o) shows that & -l - a is the largest element of M[,. Thus ? has exactly one element less than Mf . Up to now we have shown that tu satisfies the boundary conditions rrl(i)(0) : 0 for d e ? and tll(0(1) : 0 for i e Mi.The remaining part of the proof is similar to that of Lemma 4.15.

zi

be the number of zeros of tu(i) in its (closed) support. Since ur(i) is a polynomial, its support is the whole interval {0,11 unless p(i) : 0. As we have seen, ur(&-l-d)(o) # 0 sour(i) I 0 for

Let

Proof. Defrne

w(r,t)

a), thus

1,,tor1s1

x"

the conclusion follows

j*b

i <

For each fixed r, G(r,t) : G*(t,c) is a section of the Green function of the adjoint BVP, and consequently if i € Mf,, then (&)tC(*,t)lpo = 0. For such an i we have

t € (0, r), thus all coefficients

t;"d" '*# ;:;!':l!.'T' ":""ll.T*f,? firl,="i;ilffi"il;:oc1''',i, easilY'

:

0 for

: m ['-" (*)'G(c't)11=s ffi]

,,*r,n- *T : o for i < c. il ""-- -

(-"t)'

t) is a polynomial, it

sii =

i
186 Weightd IneEralities For

i:0

of Hodg TW

Higher Order Hardg Inequalilies 1gz

Corollary 4.2O. IJa*c:

we have the obvious inequality zd

(4.3e)

2xr(O)+xMi(0).

If tr(i) I 0 and 0 < i S k - 1, then as in Lemma 4'15 we observe that the number of zeros of u,(i) in (0, f ) is at least zi-r - 1 and' adding the boundary zeros' we have (4.40)

- 1+ 1a(i) + xui$).

zi 2 zi-r

= 0, then we have i> k- 1-4, so i 4f ' Abo' both zi andzi-larezero'Itfollowsthat( '40)holdsinthiscaseaswell' reducing to 0 2 -1 + 1Y; (i). 11trl(l)

Since the degree of

tr.r

is at most

lc

-

L we must have

0: zi.

(4'41)

Adding inequality (4.39) and inequalities (4'40) obtain zfi

for 1 S i < lc - I

+ zi+ "' + zi-r2 zi + "' * "i-z-

(k

we

- 1)

&-1

+ f(xr(t)

+ xv;(;))

and, by virtue of (4.41), this simplifies to 0

u -(k

-

1)

-(k - 1) + (k 1) - 0' in ( '39) and in ( 'a0) for I S i S lc - l'

+ (lrl + lMil)

:

Thus we have equality is Equality in (4.39) proves that u has no interior zeros which assertion (i). -- ii ,(0 'f O, equdity in (4.40) means that tr(i) has no boundary zeros except those accounted for by the terms 1a(i) and xu;$)' Hence the onlg i/pa.rts of (ii) and (iii). hold when u'(t) # 0' 0' As If u(i) o, ti"" of couise both u(t(O) : 0 and

'(t)11;.= wehaveseen,i>k-L_ainthiscasesotheonlyitpartof(ii)holds : 1' vacuously. Equality in (4.40) when tr(i) = 0 means that 1v;(1) tr so i € Mi and the only if pafi of (iv)"holds as well'

=

then uo"

f

k, thenwrc_r

*

0, and i!

aIc (

lc,

0.

Proof. If. a * e : k, then Lemma 4.1g (iii) states 11ru1 rrl(c-t)1O) 0. # This means, due to the definition of u, that woc_r + 0. As was mentioned above, a * c : lc 1 cannot hold (see (a.2b)), so if a*c < ft we have c < k-!-a. Now by the definition of a* (which is c) we have c e M6, so Lemma 4.19 (ii) states .G)10; ; O, which means that wo" * 0. tr Now we are able to derive the estimates of the Green function. First, let us introduce another boundary value problem which will reduce the number of estimates required. The symmetric boundary value probtem For our pair (Mo,

Mr), let Ms

=

Mt, Mt:

Mo.

The boundary value problem

g(h) :

fin (0,1); e(i)(0) : 0 for t efro, eu)(l) : 0 for j efrr is called syrnmetri,c ro BV? (4.q(fq(a,b) : (0,1)). The incidence matrix E of (Ms,Mq) simply means that we interthe rows of the incidence matrix E of (Ms,M1). Thus the P6lya condition still holds and it is easy to see that cha.nge

A:b,6:a,e:d, i=c and indeed that

A:8, E:

A, 0

:

D,

D:

C.

The Green function G@,t) of the symmetric BVp is given by

G1x,t1: G(l - z,l

- t).

Higher Otder Hordy Ineqnlities 189

188

TVpe Weighted Ineryalities ol Hardv

the enough to prove this theorem under Proof of Theorem 4'11'i|.is restricted are (4'29) iflhe estimates restriction that r + t ! f since become of function?(r,t) the symmetric BVP they

,o ,ft" Green

lc(l-r,1-t)l

* sb(l- flAtD1t-t)c for 0<sctll-r'

for 0
- t)-du(t) is a polynomial in the variable 1 - t with a non-zero constant term. Thus, for c S t, t-"(L-t)-dc(r,t) is a polynomial in the variables c and 1 - t with a non-zero constant term. and this

so (L

proves (S4).

We prove (S5) in two cases. First suppose that aI c
lc@,t)

it is enough to prove the second A further restriction shows that ttt- first is just the second applied to the equivalence in (a.29) t1""" adjoint BVP. we must establish the second relation So to complete the proof r*t S 1' To do this' it is enough to in (4.29) under the

| ,"t"

_r*l : alcll

&-l c-l

.t t

boundarY Point')

(sD

; <;i( - r\-:nr-c(r -

with perhaps one exceptional

t)-dlc(o,t)l <

oo for 0

<

rsts

i=d i=c*l

0
t)-dlc(s't)l < m'

?.)::':,(.ris -bounded above a"nd below (S5) r-"(1 - r7-'l)!i11-'t)-dlc(s't)l of (0'0) when O I x 1t'

(s4)

0

< Iim1,,4+10,r) '-a(1.-

(away from ,u,o)

i"

a neighbourhood

that

0 5 r -< t < 1- c'-Corollary 4'18(l shows porv"onlial' and ln" fu""'ioo - t1-n'-c x-o(L - t)-dG(r,tfi' u at (0,0). The first statement follows' is continuous on this set "**p, bv Lemma a'15 (i) and The statem"ot, 1ii; *i tsgl foliow On the set

Lemma 4.19 (i), resPectivelY' of Lemma a.fg (i"i ana tle dennition

b'

(which is d) imply that

u(1):tu'1t;: "': u'(d-r)(l):0'

u'(d)(l)

I0

.

.ai-o ''v'

w,,t#T

pression becomes

?

o

H

By Corollary 4.17, u;i: (-1)j+t5.+i,&-l for j < c, so the last ex-

L-t.

(s3)

1i-c

ml

i=c*l

t-t k-l

when 0 3 r 3 - q-dlc(r, t)l is continuouri - Q-a;c 1lpo"sifty'tt ("lt) : (0'0)' (Strictly speaking | -o except defined on the opefl set" extends

(S1) r-"(1

= mean ttrat tfris function' we continuously to-it'" closed set

-i-.a

i=o j=O

'o'''i"tio" prove the following five statements: t

(EE,,#)

il

(fr

_ I _ ,).

i=o+r

,:Li

i=a

j:c*r

Now,0 < r < t and i -a2 0, so in the first term, ci-otit-l-i-c 4 1k-r-o-c ( t. The last inequality is justified because 0 < t < 1 and k- 1 - a - c ) 1. (Recall that a*c : lc - 1 is impossible!) In the second term ci-o 1r 4t since i ) a* 1. In the third term we have i-a) 0 a,nd j 2 c* 1, ri-o|i-c S t as well. Thus the whole sum "o multiple of t a^nd is dominated by a constant so it tends to zero if t does. This completes the case a * c < k. In the second case, o * c: k, we do not have continuity at (0,0) in general so the argument is more delicate. We note that a ) 0 since otherwise c: lc would hold a,nd lMol= 0, which is prohibited.

r90

Higher Order Hardg Ineryalities 191

Type Weighted Ineqnlities of Hodv

Recall that in this

ca'se

C

Indeed: For

: c - l" We write Ll Fl ,i-o 1i-c*r

r-"t-cG(r,t\:

:

Suppose

i=d j-O

fii-o

ft-l *-r

ni-o

tL+r("): so, for 0

gi-c*l

<s(

r,,(1)

:lt-]-?-i':fi^i each 4 t for :"; zero with t since si-oti-c*l to ;ilil#.t"",ffil,,a; ,I c --r r ^-* io !'^ttndpd is bounded ;;.sho$' tha:\tn: l;'*,,'"'* (0'0)' + 'ioi.l"i'uJt"* i" u'u'otot" value as (r't) using the hypothesis ,^- some c [0 1] a.nd and using -^-o so e [0' 11 s as st for

;Ii"ffi:i.l- ffi;* a

* c:

lc, the

oi-o

4.4.

:

o!(lc-1-o)!

I'

by showing so we may complete the argument

that (4'42) is

(- 1;t-""t-1-""t-r (s)

where

,

/_ryi_r,

rr,(s):(-1)"-otffi is

a

positiuefunction

on (0' 11 f"' d;;> t'

r' are decreasing'and (-1)l-" (

:

rll

it-tl"-t L- , -,

" .)

\"_i)

boundary conditions

holds

if we consider functions g e ACk-r(Mo,Mi, i.e. satisfying

boundary conditions of the form (4.2). The idea is based on the reduction of inequality (4.1) to a weighted norm inequality

llrfllc,, < cllfllrl

I

o.

-np. . g,

Up to now, we have given a complete characterization of weights u, u and parameters p,q for which the /c-th order Hardy inequality (4.1)

value s and hence bounded above in absolute which i" it "orrtirr,rotl:i1' bounded below in absolute value on [0,1]. To show ttttt iiit-tf"o is value the 0 s ,,o zero in [0,1]. At = is enough to show rh;; i; ia^,

/ 1\rc-a \-r'

a. Then

Some Special Cases ll

Some other

(4.42)

!t-r)o-';1*-q'

)

where the last equality is an identity for binomial coefficients.

first term simPlifies to

fr-l

n

: *(:-j)'0,

4'L7 to where we have again used Corollarv

Wrirt"t

1 for some

1, the functions

r,"(s) >

il -jl

i=o j=c

r,'(s) > 0 when 0 < s <

1k-i-c

I(-r)&-'+-FT=n T- Sa +llwij

o we have I ro(s):it0.

ttu'ij?"-i-jl k-l

n:

I

t

where

?/

is expressed in terms of the Green function of the BVP

(4.4),

(rflrc):

xr{*,t)f(t)dt+

xr{r,t\f(t)dt, f,u with either K{x,t): 6\p(c - t)e-t, K2(r,t): 0 (if Mr : 0) or K2(r,t): *fo(t - o)fr-r, Kt(s,t) :0 (it Ms - 0 see Sec. 4.2)

f"'

-

or where the either non-negatiae or non-positiue kernels K1,K2 cart

Ig2

Weighted Inqualities ol Hodg TVpe

Higher Order Hordy Ineryatities lg3

be approximatively factorized (see Theorem 4.11), which then allows to use (twice) the "classical" first order Hardy inequality' Of course, instead of (4.2), we could consider more general bound-

can be expressed as

g(d

:

Qfllo)

:

ary conditions of the form

k-l grigU)P) :0, i DorigU)(") +

:

0,1,.'.,k

- 1,

="

f,'

xr{r,t)f(t)dt,

: ftrr- g)h+6-tr), 0(r
(4.43)

and proceed in the same way' i.e. to solve the BVP {u(k) - f in (a, b) & (4.43)) and investigate the corresponding Green function (if it exists); but even simple examples show that at least it is not guaranteed that the corresponding kernels do not ctrange sign so that the argument with positive operators cannot be used' : 2 Therefore, we will give only some exarnples for the case k where we consider the inequalitY

(1"'

xrtr,t)f(t)dt +

K1(r,t)

w'axpa(ddr) g.M)

A::0.t-a(t+4#0.

+ oe1y'(a) + 0oog(b) + go$' (b)

orog(o)

* o119'(a)

+ grcs(b) + 0\g'

:

which is equivarent

0,

For simplicity, we will deal again with the special case a

b

. c (1,' van,o61a,)'/'

(a.4a) (for (o, b)

:

(0,

(a.a5):

(i) For the case 1 < p <4

(b) = 0'

:

0,

b: l'

(4.48)

The kernels Kt Kz are not necessarily both non_negative or both non-positive but they are foctorized. Thus we can use the results of chap. 2 (see Theorem 2.3) and obtain immediately the fonowing necessory ond' sufficient conditions for the validity of the inequality

(1,' xrnoru(ddr)"0

o00e(@)

r)) under the conditions

( 6,

r lt+t-*lqu(t)dtl (/' .::?, /

r/c

Robin-type boundary conditlons

:

: 'ye(l)+dy'(l) :

ae(0)+ Ps'(o)

(0, 1)) under the conditions

" (lr'

tut

- py'or-e'@dt)"o'

. *,

0, (4.45)

0.

,::1,

(/

lrrt -

Blqu(qor)"

The solution of the BVP

g"

: f in (0,1)

& conditions

(4.45)

(4.47)

provided that

for functions g e ACr satisfYing

consider inequaliry (a.aa) (with (o, b)

(4.46)

where

j=o

(1"' van,,{')a")''o

fo"

" (1,'h

+a

-

-ytlP'

rr-n' @^)"o'

. *.

194

Weighted Ineguolities ol HordV

(ii) For the case 0 < g < P I

(l'u'h+a " (l'

(/' ff'

6, 1 ( P (

can again be expressed in the form (4.46), but now with

oo'

K1(x,t) : Cr+(B-1.)t-BC, 0
-1ttqu$)at)

ldt-Blp'at-o'lqor)'

vt

Higher Order Hard,y Inequalibies l9S

1W

withB:/%,C:#provided

lr+o- ,*y'1'l*)''' t*'

A::(a+0)0+d)10.

- Ptqu(qo') r r/r

'u' with

lr + d - ,,tll' ur-P' (,lO')

lcrx-Blqu(ddrl /


! : l-1 qp

A counterexamPle

( un(4.a8) is violated, then the Hardy inequality '44) for Indeed: der boundary conditions ( '45) becomes meaningless' i'e', for the boundary conditions : L, g :0 and d

If condition

a:

- -1,

1

e(0) = 0, we have

a: gl - ah+d):0

e(1) - e'(1) :

0

and the function

s@): s satisfies

theseconditionsbut.Tt(a).=Oandsotherighthandsidein(4.44) is zero while the left hand side is not'

Consider again inequality

the conditions

CIe(o)+Be(l)

"Ie'(0)+ds'(l)

: :

(0, 1)), but now under

o,

0'

The solution of the BVP g"

: I in

(0,1) & conditions (4'49)

(4.4e)

(4.51)

(i) Again, the Hardy inequality becomes meaningless if condition (4.51) is violated. As a counterexample we can ,rru thu function S@): (a+p)r- p with a*f 10, which satisfies (4.49) (the second -g,t(r):0. condition with 7: l, d -1, i.e., 7 * d:0) and again, (ii) The kernels Kr, Kz from (4.s0) cannot be in generJfaciorized or approximated by a product of a function of c and a function of t, moreover, they can chonge sign in the corresponding tria.ngle 0 < t < s < 1 or 0 < s <, < l. Consequently, the approaches described above cannot be used in the general case. However, in some special cases we can give necessary and sufficient conditions. 8.g., let us suppose that B : 0 (i.e., g :0 in ( . g)). Then we have

Kr(r,t) : Cr - t, Kz(r,t) :

(C

-

1)".

If 0 < C t,we have K{r,t)> 0 for 0 <, < r ( 1 and,

moreover' this kernel satisfies conditions (2.25) and (2.26) ofchap. 2. Thus, for the operator ? defined bv

Q!)@)

Another examPle

( . ) (with (4, b) :

(4'50)

:

f'^

l^ (c" - t)I(t)dt + (c Jo

1r

- \x Jr-, / I f U)dt

(4.52) \

with C ) 1, we can use the positivity argument, consider ? for > / 0 and obtain the following tripte of necessary and sufficient conditions for the validity of inequality (a. a) (for (a,b) : (0,1)) under conditions (4.49) with our choice of B(: 0) and C(> 1); i.e. under the conditions

e(o):0, .ys'(o)+(t-r)g'(t)=0, l)

1

:

196

Weighted Inequalities

ol

IIigher Order HardA Ineryotities 19?

Hordy Typ'

(i) Let us suppose Mo - 0, Mr = Nr. As we have seen (see Theorem 4.3, Remark 4.4), the Hardy inequality

forl( pSqlcx.' .,'l/q / r'

/ i

\1/P'

't-n'

( [' O" - t)c"(qdt) U, -sup. \Jc O
1t)dt)

<,oo

rl/P'

(4dt) ( oo ' )''o t*' ,uo ( /'teu1t;at)''' ( for-e'(t)dt)''o' / / rr

,:t?, (/, o
\Jo

u$)at

\l/q / f'

(1o' *r,

"

)

-

'-' rr)P'or-v

(4'53)

\J'

Recall that the first two conditions

in

(

suffi'53) are necessary and

inequality cient for the validity of the

(/' (f

n

r - t)r(t)dt)n '1")'")'

'o

=

"'

(1"'

rot'p)a')'t'

third integral in (4'52)' while the

/ rr

\r/P

s"rllr' lP(r)u(n)dn)' />o and covers the rest of the operator

holds for all functions g e

g(m)

if

a"nd

only if (for

I

bounded on (0,m):

Aek-t (0,N*), i.e., functions

: y'(*) :... :

r(t-t)1oo)

T'

G.54)

satisfying

:0

< p < q < m) the following two functions are

: (1,'O - ty$-r)oulOor) (1,* ,,-,,rrlor) / tc \r/q / r* B2(r) : qqor) g (r - c;(r-r)r ',,-/ @dr) .(4'55)

B1(r)

/ Z 0, i.e., they cover the first for in ( '53) is necessary and sufficient "t"iiri"^ 'l/q fl 1cu@)a'.) /rr/ rav'1 (l'(" l,'

for

/f*...._ .-\l/c /r@ rl/p (/. ls(r)lqu(r)dx) = (/, lg(u)(,)loo(d*) "

'

,

ll,

(fi) . Analogously, inequality (4.54) holds for all functions 9 e ACx-r1Nr,@) (i.e., sarisfying ntnllOj = 0 for i : 0,1,...,k _ l) if and only if the following two functions are bounded on (0,m): E,1,y

:

E,1,1

/ f@ \r/q '' / r, : lJ,' "AVt) (1,' O - t1o'G-r)ot-e'e)dt)

(1"* U- c)c(e-r),, Odr)'"

(I'

a'-e,G)dt)

,

(4'56)

(ifi)

4.5.

Now, we will show that for some classes of weight functions u, u, the boundedness of only one of the function s 81 82 , @r 81, E2) is necessary a,nd sufficient.

Reducing the Conditions

Remark4.2l.whileinthecaseofthefirstorderHa.rdyinequal. a1d sufrcient) condition' in the ities we t u.'u o"lv ;;;'i;;;;"arv sometimes poir.of conditions (and

of a higher order we have a some spesome specral ca'ses - i'e'' for even more' see (4'53))' In condition' orue reduce this pair lo onlg cial weights - ttt'" "Ule to unfortunately' works only for Let us describe * upp'o*t' which' $ ot Mo = 0' provided that' (o,b) : (0,oo) J #*nu case M1 -

case

moreover,

L
Theorem 4.22. (i) Denote

tr(a):

Ir'O

and suppose that therc esists

U(zr) <

"oU(r)

- tyo&-r)u1ryo, a positiae constont cs

f or

every z € (0, m)

e.sr)

such that

.

(4.88)

198

Higher Otder Hordy Ineqtalities 199

Tgpe Weighted Ineqnlities of Hordg

$.5a) hold's for .ea"ry s e.ACft-l(0'N fum $'55\ is bounded'

p\ if arul onlv

v(r): Jo[" @ -t)e'(&-r)ur-n'$)dt

(4'59)

Then inequotifu

ii-i" i""non-nr1'1 (ii)

ond

since

in the third

integral we have

2s <

t(

oo, i.e.

*-ttt.

| > Z ana

Consequently, we have

Denote

suppose

thot there erists a positiae

c'onstant

g.!a)

hords

i"n"iencv'lt

[" u(t)at : Jo -'-'

(lr' "Oor)"' (1,* U - x)n'$-t),t-p'(qdt)

(4'60)

il orur onrg for,erery g e AC*-'(N*,0) ii'i" n"*,on-Er1'\ from Q'56) is bounded' holds' then both the funcPrnf. (a) Necessitv'-\f inequality ( '54) *d thus' so is It2 '(or B1)' tions "' .B1, B2 (or nr,g;"t"Uot"'a"a' follon's from (4'58) that (Al Then inequolifu

:

ca such thot

cov(r\ lm euerg c € (0'oo) '

V(2x) 3

Bz@)

['

dQ'-r)u(t;c-c(ft-r)d1

Jo

x s-(,t-r)

:

coq

(l] *-r riat)'/o' ,u-,

B{2t)

m) implies also the boundof .B1 and so, the validity of (4.54). So, we proved case Q), and the proof for case (ii) follows analogously, showing that

and thus, the boundedness of 82 on (0, edness

B2Qx)

4.6.

!

const

B{r).

Overdetermined Classes (lc

:

1)

Up to now, we have investigated the Hardy inequality (4.1) for functions g e ACk-r(Ms, M1) where the index sets satisfied F\rrther, V -c)P'(&-r)rr-

I* :

P'

lMol

(t)dt

+

lMtl: *.

(4.61)

In this

case necessary and sufficient conditions for (4.1) to hold are (almost) fully described. Now, we will investigate (4.1) for functions

t)"'t--D o'-'' (')'o'&-'\)dt I: (* -

. l,:(i - ')"'t--\

, I:

o'-o'(')d't x 6a'$-'1)

ovo 1t)dt' sP'(&-r)

s e ACk-t(Mo,Mr) where

lMol+lMl> k,

(4.62)

which means that in the incidence matrix are more I's that 0's. Again we suppose that the P6lya condition is satisfied (see Sec. 4.1). The class AC*-r(Mo,M) with Ms,M1 satisfying (4.62) will be called oaerdetermined.

200

Weighted Inequdities of

Hody fupe

Higher Order Hatdy Ineryalities 2Ol

solution of the problem to find necessaxy for the validity of (4.1) on overdetermined than in the well-determined case (a.61): the in some special cases and the conditions are very cumbersome. But first, let us start with the first order Hardy inequality (1.25). We will see that the and sufEcient conditions classes is less satisfactory full answer is known only

Thecasek:l Here, the only overdetermined class is the class of functions

I

e ACv

satisfying

g(a):

e(b)

:0

(4.63)

For simplicity we consider again the case (a,0)

holds for such functions g

if

a,nd

only if, for 1

{(

x min

{(1"'

*-r(')d')

fo'

@,

,,-o'1r1a,

= l,' ,r-r,1rydc < oo or

,'-o't4a,

:

a'\-p'(qdt)"''

f"t

,r-r'1*ydt = 6

t

(4.66)

ond, set

, : {o, f" n6ya,:

(4.65)

,(lo'

beweight

functions on (0,1). Suppose that there is a nurnber z (0,1) € such that

(4.64)

It r,t \l/c u(t)dt) ( . :ip ,. || \.rc/ (c,d)c(o,b) ''o

Letr

Lemma 4.25.

f'

. c (1"' w'ow,tda")'/?

(0,1).

our approach in the foregoing parts was mostry based on the reduction of the Hardy inequality (4.1) to a weighted norm inequJiry, *ring with some operaror ? : u(u) -+ Lcd)- n *tr"i-rai"'*i J" *ru use operators acting on certain special subspaces of lp(u).Therefore, we will start with a useful lemma. Let us recall that an operator ? is called positiueif it maps non_ negative functions to non_negative functions.

and as was mentioned in Chap. 1, the inequality

(1"' wt-1t,,@)dt)'/n

:

. }] -

(see Remark 1..3).

Example 4.23. lf. we choose (c,b) : (-oo,-), then inequality (4.64) holds with u(r) : eo" , o ) 0, u(c) : eo'"P/q if I satisfies g(-m) :0 ondg(+oo):0, but it does nothold if g satisfies only

oor*}

.

Let T be a positive linear operator. Then T maps I?(u) t\ Lq(u) iJ and only if T rnaps lp(u) into Lq(u).

(4.67)

H

into

Proof' The iJ part is trivial. To prove the onry if part suppose that T : HnU(u; * Lc(u). Si.nce Lrlll?(u) is dense in l?(v)it is enough to show that T : Lr n I](u) _+ 7o1u),. Fix 9 € Lr n Lp@) a,nd suppose, without loss of generality, that

['. lo@)lax

Jo

one of these two conditions (see [OK, Example 6.13]).

Remark 4.24. The result just mentioned cannot be used if p > q and also if. both integrals appearing in the minimum in (4.65) are infinite. In the sequel, both disadvantages will be removed.

I,'

(i)

Suppose

s

1r

J,

ls@)\a,.

that the first condition in (4.66) holds:

[' Jo

,r-r'1r1a,

= J, ft

,r-o,1x)d,z \-/

I x,

2O2 Weighted Inequolities of Hardg

her Order Hody Ineryolities 2Os

TYpe

Now

and set

h(c) := aur-p'(c)

u|-r'

(1,' VAXu, - lo' lntllot) xro,,r('),

is zero where a

llh,llp,,

an, so

: ",(I'

: [z ur-o'(r)dn Obviously h > 0, and it can be easily Hiilder's inequality and shown trrat lgii t e .ri.'rurthermore, using

where

f

rf-r)n1r1u(ddr)

lf a

" U,'

(4.66), we have

rlhllo,,

s

or,r,

I,lg(t)lar

JO

Definition 4.26.

(1, ave'$)dt)''o' (lo' ls(t)leu(t)dt )

Since g S lSl + h, the positivity of

llTslln,,

s

T implies

llr(bl + h)lln,, < cll lgl+ hllo,'

that 1z ot-n'(t)d" : [) positive integer n set Suppose

hn(x)

z: anuf,-n'1r1( "\J'

['

vayo,

t:r-p'(x)d'o: oo' For each

- Jo [^' btt)lot)/ x1o,,r(")

€ (0'1) : ut-r' where ,1,-o'(r) - ur-p'(x)'.xs,.(") with '9n - {c u (r) . ,,i, ind, Llan :'ft ul-e' (r)dr' Again' h' ) 0' lgl + n" e and 9 ! lgl + h' so that for every n we have llrgilo," <

ilr(bl+

h,,)lls,' S Cllgllp,' + lllullp'"'

For fixed z € (0,1), let ,g = ,gr + S, where

(&/Xr)

::

(1,' ,aVr)x1o,,y(c);

(szf)(,)

::

(1,' ,avr) x1,,ry(r).

Note that Sr and

,S2,

(4.68)

and hence .S, are positive operators.

g satisfies g' : f in (0,1), g(0) :9(1) :0,

Lemma 4.27.

h(x)

:

Suppose that

- xp,g@))f (x). Then s: Proof. Since 9(0) :9(1) :0 we have and, set

and case (i) is Proved

(;;)

VAXor)

For n -+ oo we have that a,. -+ 0 so finally ll?: silq," < Cllgllp,r, which completes case (ii) and the proof. cl

\ l/p

,,,o'

- lr'

.11

:, (lr' uo-p)e(r)a(')d") "' (1,' lse)ldt - lr' tnalv,) s

ls(t)ld.t

(x1o,,y(")

s@)

:

lo'

(4.6e)

Sh.

r@or: - f (t)dt I,'

and hence, for z € (0,1) arbitrary but fixed,

s@)

:

x(0,,)(c)

:

(&/)(c) - (szlX')

fo' Xil,a, - x1,,r1(r) l,' ,tilu,

:

(sh)(c),

204

since h(c)

:

(x10,"1(c)

h(t)

Higher Order Eardg Ineryalities 2Os

7W

Weighted Inequolities of Hordg

Suppose

x1's'1@))f(c) means that

-

c e (o' z)' r € (z,L) '

:- {'(:)' .to' l -ftrl for

Ic: Theorem 4-28- Ireto < q (

Now, with

llgllc,"

@, L < p < a'

Suppose

v'u

or\e

with some z e (0' 1)' functions ond. conctitiin (4'66) is satisfieil Then the first order Hody inequolity

(/ hokts

rr/q u@)ar) ut"lt'

for ail g e ACo

ond only

if

=

" (/,

, \t/o

ls' (r)leu(x)tu

(4.70)

)

(4.71)

either

,::?"

(l'

(4.72)

u$1or)"0 (1,' *-'ro') ,::?, (l' or

(ii) 0 <

q

1

( P 1e

(€t

(

@t

ond'

(1"" (1,' u(ildt)'le (lo'

(1,' U"' u{oat)'/o (1,'

{')o') '(")*) "-''

(r):

r*'too') '(")")

(4.73)


with!:i-i. Proof. We begin S,

(x1o,,y(r)

: ft

by showing that (A'70) holds if and only if ,9 the

operator from Definition 4'26'

x1",9@))h(r), g(4

yp1ar.

JO

I;

:

: ['

llgllo,"

- | n6yat:

0,

it

is clear that

usiig Lemma 4.27,

3 cllf

llp,"

:

cllhllp,,.

To complete the proof, we show that the boundedness of ,s is equivalent to the conditions in $.72) or ( .73). since ,9 is the sum of two positive operators, it is bounded if a,nd only if both ,91 and ,92 are bounded. The boundedness of s1: u(v) -+ Lq(u) means that there exists a constant c such that

l([

r (q dr)x

o

ro,,y

(,) |

on the values

u@ dr)" 2

" (I' ff@)

te u

@)

ar)

/ on (0,1). since the left hand side does not depend ofl on (2, l), the above inequality is clearly equivalent

to the inequality

(1,'ll,' r@drl u@dr)'/o . c (l' van ,6)a,)'/o for all functions on

ip(u) -+ ti(") with

-

for all functions

(Ft

rIf

g'.

Since e(1) f ltyat: h(t)dt llgllq," S Clls'llp,": Cllf llp,". Thus,

(1,'

llr

Cllfllp,o,

/

ll^9hllq,"

o'-o'tio)'

:

< Cllhllp,o:

gbv

(i)1
llShllq,"

for functions h e I?(a) sarisfying ff n@)a.r : 1) n6yan in order to conclude that ,9 : IP@) -+ Lq(u).Fix such an tr. and define a,nd

f

:e(1):0

:

Conversely, suppose that (4.20) holds for 9 satisfying ( .?l). Ac_ cording to Lemma 4.25, it is enough ro prove that llShllo,, S Cllhllp,o

sotisfYing

g(0)

if

/ tr

+ Lafu) and that g satisfies (4.69).

- X(z,r))f , we use Lemma 4.2T and the bound_

which is (4.70) since .f

wei,ght

/ rr^

(X@,r)

edness of .9 to get

1 reads as follows:

The main result for

first that S : U(a)

h:

(0, z). But this is the Hardy inequality on the interval (0, z), which holds if and only if the first condition in (a.Tz) (for p < s) or in (4.73) (for p > q) holds.

206

Higher Order Hordy Inequalities 207

Weight'ed Inequolities ol Hordy 1\pe

of 52 reduces to A similar analysis shows that the boundedness the

the.interval (z' 1)' which yields the conjugate Hardy inequality on (+.li) otin (a'73)' This completes the proof'

"""r"a ".ittaition

in

o

we have derived conRemark 4.2g. (i) As follows from the proof' (a.71) by glueing tog-ether the ditions of the validity of (a.70) under g(0) : 0) and on (z' L) Hardy inequalities on (0, ); lfot I satisfying also from conditions (4'72) li", n *trtting 9(1) :'0,' ttris is evident

and (4.73) which can be rewritten as

4.7.

lMol+lMil:k+t with the following particular

A(z,liu,u)

M!-t

-

(Mt-', Mf-')

:ft

in

(0,1),

A'(0, ziu,a) (4.73')

(4.66) is crucial for our considerations' Of course' be replaced by equality' claiming the first part of this condition could that there \s a z € (0,1) such that

has a unique

fo'

I,'

,}-p'(t)dt

(<

*).

(4.74)

not possible to find a z Let us note that for some weights u it is in (a'? ) is finite. and the satisfying (4.66) (e.g' if one of tle integrals condition (4'65)' As an orn", innnire). In this case, one has to use Iet us mention the weight u(c) defined as

"*t*pl",

',t,=

t

and the Parameter P =2'

1 for c e (0, |1, 2r-L for re[],1)

(4.76)

lMf-ll+ lMf-ll

satisfies the

i6lya

:

0 for

i e Mt-t,

goo) :

0 for

j

g(i)(g)

e

Mf-r

:

"orriitior,.

(4'77)

solution,

(ii) Condition

or-o'{ilar:

choice:

cN&-r :.{0,,1,:..,f 2}, i =0,1, -

1 and the couple Consequently, BVP

9(*-r)

A'(z,l;u,v)

(4.75)

Mi={M!-t, k-1} where

(4.72',)

t)

We will deal with the special overdetermined case

lc

A(0,2;u'u)

Overdetermined Classes (tc >

s@)

:

Io' "o-r rr,t)h(t)dt

(4.7s)

.

Now, the solution of BVp

g(k):f, g(i)(g):0

for

i€Ms, eU)(l)=0 for [email protected])

can be combined from the BVP (4.77) and from the (overdetermined) BVP h'

: f in (0,1),

t!(0)

: h(l) :

0

(4.80)

whose solution is, according to Lemma 4.22, given by

7r: S$

-

Szf

with,91,52 from Definition 4.2ti, provided there is a z (0,1) € such that (4.66) is satisfied.

208

If

Weighted Inequalities ol Hardy Typ

Eigher Ord,er Hatdy Ine4uatibies

we denote

F(r) :

(x1o,,y(t)

we have h

:

-

xf,,rt(s)).f(t)

:

( I@) for n € (0,2), { t -/(o) for r €. (2,1),

SF where 5 = Sr * 52, and the solution of BVP

(

2Og

Pruf.Suppose, .,!!y) _+ Lq(u) 1nq,/ and g sarisfy (a.29). Set F = (xp,"1 - x1,g)f . since. / 9G., = h satisfy (4.g0) we may apply Lemma 4.2T to get g(,t-l)3"4 if where .g is the op"ruto" fro_ = Definition 4.26, and due to (4.2g) wcfinally obtain

'79)

:

g(t)

is given by

Io' "*-rrr,t)(sF)(t)dt= (r.F)(c).

Thus

s@)

: t;

G&-r(r,

1,,

t)(.9

F)(t)dt

which is (4.82) since

Gft-1(c,t)

lrtrrl " fxro,,l(t) Jo

:

llsllo,"

fz/

r{')a' * x(,,r)(t)

J,

r{')a'lat

(4'81)

fr / f" t--'(" (l

J,

,t)dt

\ _.

"rra

.

)F(s)ds

f

kt 0 < g 1a ondl < p ( 6, letu,a be weight functions on (0, l) and let z €. (0,1). satisly (4.66)' Suppose tnoi i[-r, Mf-r ore subsets o/ N*-r, lMf-tl + lMf-tl : k - L and (M[-r , U!-) satisfies the P6lga condition' Then the Hardy

Lemma 4.gO.

inequality

i=

(4.82)

cllg(u) llo,,

functions g e ACk-r(Mo,Mr) with M. 0,1, if onil only if T : IP(a\ + Lq(u).

(xp,4@)

-

X1,,r.1@))h(z), s@)

:

cx-r6,q6h)(t)dt.

lt

:

Then the following assertion will be useful:

for

(r):

Similarly as above we obtain that 9(c) (Th)(r). The definition of g shows rhat 'sh and g satisfy nviiia.zzl so that g satisfies the boundary conditions

(r/X').

s

c;-i";;*';;

< Cllhlln,, forJunction y, e i(qsatisfying (4.62) in order " that ? : r?(a) _+ Lq(u). Fix such *; ;; a"doJl ;o#nclude

where we have used F\bini's theorem. Let us denote the last expression by

holds

g(hl. that (4.g2) holds. Since ? is a positive op erator (recat that due to Theorem 4.1r, lilill" on (0' 1) x (0,1)), Lemma 4.2b shows that it is enough to prove "iso that llThllq,"

\

fz

llgllq,"

f:

Conversely, suppose

to-'(",t)dt)F(s)ds J" (/" *

= lffFllq,,, < CllFllp,o: Cllfllp,,,

= M:-'

U {e

- l},

o(i)19;:0 for ieMt-r, go)(r):o for ieu!_r. We have also

g(e-t; :,SA so that using Definition 4.26 weobtain

e(&-r)(o)

:

(s1ft)(o)

: g,

Finallg differentiation yields / satisfy BVp (4.29). Thus

ItThllq,,:

llsllq,u

which completes the

gG)

s(e-t)(r)

-,f

(s2nx1)

:

g.

and we have shown that g and

< cllg@ilo,o:

proof.

:

Cllfllp,o

=

Cllhllp,o,

tr

2]iO

Weighted Inquo'lities ol

The caseMo

:

Hody

Nl, M1= {k

Higher Otder Hardy Ineqnlities

Type

Thus (k

- l}

We start with this simpler case since conditions are more transparent' Here' satisfy

the necesriaxy a,nd sufficient I € ACk-r(Ms,M1) has to

: 0 for i = 0,1,...,ft - 1, r(*-t)111 = 0, (4'83) : ft-r' Ul-t :9' According i.e., we have Mf-l : {0, 1, ' ' ' ,k .2} g(i)(0)

has the simple form to s"". 4.2, the-Green function G&-l(c,t;

Gn-' (r, r)

:

G5!

@

-

: L' ll"' ffiro,')(')d']

h(s)ds

x1o,,'1@)

Q2h)(x)

:

X1",r1(x)

(Tsh)(x)

:

x1",r,1@)

(Tah)(x)

:

X1,,rjr;)@

Ifr
("h)(c): If

r)

/" E#h(s)ds'

(rhXo)

= l,'

-

no

'{.-.' /' *' (",: (k-1)! J,

(o

-

h(s)ds

s)ft-r

o(r)0".

-

s)ft-rh(s)ds

- s)r-t -

K,

J,

Q2h)(a)

z)*-r

z)k-t

f,'

(c

+

("sr,Xs) +

;

-

z1k-rlng)as;

- (c - s1k-rla1s)as; otio".

is the Riemann-Liouville operator on (0, z) (see Remark 2.g (iii)). Using the Binomial Formula

(c-6;t-t -

@

-

21*-r

:

[@

-

r) + (" _s)]r-r _ @_ 71x-t

= $1,t-tt ,-? \ i )('-

z\x-i-r1z

-

s)'

'

T2 can be simplified to

Q2h)(r):

-

z)(r

X1",r1(c)'

-

S/*-' ) f'- z)k-i-t

fr\i

z)k-z < (c

for 0 ( z < s < o

$

, f" (, - ,)ft-t *J,T'e\o/'*

lz K, Jo

fi

(s

-i=l:'-')k-'

-

@

+

Io"

r,

- {irrg)as.

Using the estimate

z then some calculations yield @

J,

fz

'r

h(s)ds' rt[ 1"@_t)k-2 -, J, ll, (k-2)! x(o'')(t)dtj

Q1h)(r) fE

Q1h)(r) :

t)k-2x(0,')(t)'

However,sinceweassumethroughoutthissectionthattheweight there exists a z € (0,1) such that function u satisfies condition (4.66), the form tfr" op"rutor ? defined in (a'81) has now

("h)(r)

- r)tQh)(x) :

(Tah)(x), where

2ll

-

z)k-r

( l, we can write

esh)(a) x xp,r.y(x)(t

h(s)ds

-(c - s;t-r

-

zyh-z

< ft("

-

z)(x

_

21*-z

_ l,'A )h(s)ds,

where the last exRreslion is the Hardy operator on (2,1) applied to the.function (s - z)h(s). Finally, is ttre conjugate Hardy operaior

on (2,1).

"a

212

Weighted Ineqtolities of Hadg

Now, we axe able

7W

Higher Qrder Hardy Ineryalities 2lS

describe the necessary and sufficient

to

conditions: (0-'1) Theorem 4.3L. Let u,a be weights on (0, L\ onit let z €

tsly (4.66). Then the Hority S

i lCt'-r (i) 1< P
satisfying (4-83)

inequolity @'82) holds

it

ond, onlv

if

lor

"?p,

(

, "t?,

t

-

'

" (I'O - t)p'(k-trat-e'rodt) u{,)ar)'/' ( oo,

o- r)p'(e-r)ur -o'(qdt)'/'

<

oo

'

(4'84)

< oo, (4.85)

z)e$'2)u(t)dt)

(1,'

(t ,, - z,o&-z)urr)or)"0 (1,' u - z)p'or-, ,qdt)"/o x (t -

(1," u - z)P'or-o' ,ilor)t'o

. *,

rrr*)"' ( 6,

z)o'ar-o'

(1,' (1," u -

z)a@

- r)

(1,'n

-

o

l/'

(4.87')

(4'87)

Proof' By Lemma 4-90

it

T: U(u) + Lq(u). But ? (more precisely,

'o

is enough to prove the equivalence of (4.g2'), (a.gg) and the boundedness of

(e - l)!") is the sum of positive operators and,.thus their separate boundedness i" ne"e""ury '4,the boundedness of their sum. and sufficient for

Tt,Tz,?3 and

' or

(t0 1<

(lr' u q
t1t'

i't-o''oo')t'o <@;

(4'88) holils ogoinond'

(4.88)

Now, ?1 is bounded if and only if (a.g4) and ( .8b) (for p < q) or (4.84'), (4.8S') (f?r p > q) hold (see Theorem .3). it H*ay operator ?r is bounded if and only if (4.86) (or " 1a.aO;;1 hotds, and

withl:i-i'

( t^' ( [-'" -c)e(*-r)"1" o')''o \Jo \rc

'

(4.86')

/ fr \'/c' \ . (l ,r-n'1t1at) ,r-n' 61ar) < oo. the conditions (4.8a)

z7o&-i-r)"t'io')

ur')o')''

(4.86)

(l'( t - z)c&-ttuG)d,)''' (1,' ur-r'6at)'''' ' *,

,=ill--,

(4.8b')

ond

u@dt)tto (1""

(l'

u{,1a,)'/o

sal-

all functions

either

tilo') c;c(ft-r),,trpr) (lo" .:p, (l'(t "-o' ,::?" (l'

([ ([

(1"

ove'p1at)'o' o'-o'(r)or)"".

*,

(4.84')

similarly for the Hardy operator Ta and "orrdition ia.aZj'1o" 1a.iZ,yy. Finally using Hiilder's inequality it can be shown tir"t rz'i. bounded if and only if (4.88) holds (see Theorem 4.12). tr Rerr_rark 4.92. (i) Note that we have supposed q e (l,oo) a,nd not q € (0' oo) as e'g. in Lemma 4.80. The reason is that the reults for the (Rieman-Liouville) operator T1 arc available only for q , i.

2L4

Weighted, Inequolities

Higher Order Hardy Inequakties

ol HadA TW

a similar theorem for the Obviously, it is easy to formulate g e ACk-r satisfying Mo : {k -1}, Mr : Nf i'e' for

(ii)

""";

n(t-r)1g;:0,

n(i)1f;

:O for i:0,1,"',k-1'

OnehasonlytousethefactthattheGreenfunctionGk-r(''t)has

then due to the estimates

fz fz IJs Gk-r(r,t)dt x c"(t-*)B J"I tclt-t1dat if r(s;

f"'ck-t{''!at

N *A(r

the simPle form

-

c)k-2x1",r1(t)

the case after Lemma 4'30' and then proceed similarly as in only two or at (iii) In contrast to the urelt- determinedcase where here we have conditions appeared' most three rr""*".ry t ta sufficient lc' even in the very simple case lMsl

:

a collection of. fiue"ooii"oo" cumbersee' the situation becomes more lMrl:1' As we will just speaking' lMrl > 1' if we allow, roughly

"o*"

The case Mi:

-,r,

*ro(1

#tn

M!-r u {k - LI' M!-t *

0' i

:

(Mf-t,M!-t).

^L_'r

fz/ fz

L'

-dB I"'rt(r-t)ddt if s1z1z1

- d' I: r(t - t)Ddt ir

ti

I,' Gh-r(r,t)dt

ro(L-r)u

I,/,e-t)ddt if rSz;

rA(r

I:

E

l"' Gk-r(x,t)dt A,

- d,

r(L

-

*o.(t - 48 tc{t f," t,"

ck-r{r,t)at

we obtain

N

,A(r-do

I"

z<

t)Ddt

t)d if

z

1

:0

1 s;

.',0- t)Ddt if I 1 tt

that

Qh)(r) = ("1h)(r) + .. . + (rshx') where

Gk-L(r,t)dt) r,tr)a,

(frnXr)

:

Xp,,'1(x)x"(L

- d" I,

(1," Gk-t(r,t)dt) n1r;a",

Q2h)(x)

:

x@,i@)xA(L

- ,f Io (1,' *rt - q'

(rh)(r) = J" (/" *

rG - t)Ddt

oA(r

.\

FortheGreenfunctionGk-r(x,t)ofBVP(a'77)wethenhave' (A'29) and if we use thee esaccording to Tt'eorem 4'11' estimates in (4'81)' timates in the expression for ?

l:

l"' Gk-r(z,t)dt x

o'r

the P6lya condition' Let us suppose that (Md .l 'Mf-\ satisfies contrarv to the cases men;f ;-*t, itf-tf *lMf-t.! : l-t'blt 4'32 (ii)' let us assume that Remark tioned after Lemmt;'b0 t"a i" by a,b,c,d,,A,B,C,D Mj_t is non.empty for i :0, 1' Denote again time for the couple this 4'9' but the number" i,tt'oi"""a in Definition

zls

(1""

* r, - ,)'or) h@)ds or)ft(s)ds,

216

Ineqwliti* of Hadg 7W

Weighted

Higher Otder lfiardy Inequatities 2lT

(rehx")

:

x1o,,y(c)ro(1

eah)(r)

:

x(0,")(o)so(

t

-

,)B

(rsnXc)

:

x1",r1(r)rA(1

-

(1," f ')D 1,"

=

" " f x1"s1(r)r"(t - d" l" tl ,"

(rohX')

Qth)(x) =

(rahx")

:

- ')B Ir" (1," F (r -

I,'

(1," F

rl/ x1",e@)rA(l

-

x1,,ty@)rA(l '

-

")b

fo

J, lJ,

tldat)h(c)ds

(r -tldat) (L

-

(7

t"(1

,

f,

= Js|

t"(1

-

"zio."

-

tldat)h(s)ds

-

- \ t)Ddt)h(s)ds

?3 is a Hardy operator on (0, our main result:

z)

(see Example

occ<,

/ fx

\"fo

tw(t

( l'l' - t)Bqu1)ar)' / \J"

tAq(r

at)

. *,

- t)hu@at)

/

uith H2(a,t):

ts"(t -

(4.e0)

s)Dd,s;

Hl (t, z)t"c (r ,ZiZ, (1,'

t 18

o u1s1or)''

o

2'7 (iv)),

(i)1
a.*p'$)dt)"'

t1bt r1t1

" (lr' Hl'(t,4at-o'(t1dt)"o'. *

'

Theorem 4.33. Lct u,u fu weighk on (0, 1) and' suppose thot there is o number z € (0,1) such that (4'66) ii; sotisfied'-I'et Ms,M1 be ol the form M; ='M!-r U {,t - 1} wherc Mt-t, Mf-.r ore nonimpty subsets o/ Nt-r satisfying the P6lyo condition ond such thot lMt-rl + lMf-tl - k - L. .Then the Hords inequolitv (4'82) att lunitions g e ACk-r(Mo,M) il anil only if either hotds fi sup

(1,' HI @, tltAt 1r -

,2ir.,(1,'

'

" (lr'

etc- Consequently, we can formulate

\l/q / r,

s)dds;

, (I"

tlDat)h(s)ds,

[' (\"f" f t"(1 - tlDat)/ h(s)ds ' ")D "lo

t)Ddt

H1(z,t): Ii "c(1 -

h(s)ds,

Here ?r is the conjugate Ha,rdy operator on (0, z), T2 is a Hardy-type operator as investigated in Chap. 2, with kernel

k(r,s)

with

- (',qar-p'

r

l/P'

(t)dt)' * /

(4.8e)

(l'

at-p'

. *,

(4.e1)

"1or-r' lqor)"o' < oo;

(4.e2)

(qdt)"o'

t"'rt - t)Bqu4at)

" (1,'

,:::,

(/

Hor'

-c(r

(t,

- t)wu*at)/

" (1,' H!'(t,z1at-n'p)dt)"o' .

*,

(4.e3)

218

Higher Oder Hardg Inequalities 2L9

Weighted Ineqrnlities of Hordg Tvpc

Hl(n,t)t"o(l

,"::,.r(1""

-

t;ae"11; o')''o

* (1,' a.-p'(qdt)'''' . tq(L

"Z\2,(1,"

*,

I l/P'

Hor'(t,

flur-n'6at /)

(1,' <

*;

"(l\Je

r

ar-P'(t)dt) .*'

(4.es)

-t'pultvat)

H!' (t, t)ut-r' 1t'sat)

,Aq(L

(1,' (1,' Hl (t, lrs (t - t)no uplor)''

- r1*t{ryar)

< oo;

o

(4.e5)

/

(ii) or L < q < p < oo, (4'92)

, (I' (4.e0)

and (4'96) hold agoin and' uith

p'

(l,"tq(l

tAo(r

oo,

l/P'

, (lr' Hpl' (z,t)ur-e'(r)dt)''o' .. *:

([

- tlbo'1t1at)

a-p'G)dt)'o'

ur-o' (r)0")

"' . *,

(4.91')

*,rt - t)w'@at)/

(1"'

!r -!-! -g

t[

" (lr'

/

r tt

t)tAo 1r

(4.e4)

r l/c t)wu(t)dtl Hf(t,z ltAe 1t (1"

,::?,

HI*,

* (lo" ,r-o'1r1ar) ( ",-'1r1or)

- oaaurrrorlrrt

I

" (1.

(1,' {1,'

-

t.uo'p,at)'" (1," H!'(",t1o'-o'p1at)"to

xH!' (z,r)ur-n' (r)d*)t''

.

oo

,

(4.89')

(I"' (1,' tAo(r ,

,Ao(r

tluo,ltyat)

''

(1"' Hl'(t,,y,'-'' lryar)lo

- x)hu(r)d,r)t'" . -,

(1,' {1," Hf (', t)# (r -

(4.93,)

t)Bo ug)at)

" (1,' u'-o'rrror)''o' ,'-o'1r1ar)

(F,

220

Wei.ghted Ineryalities

Higher Order Hordg Ine.galities 221

ol Hodv 7W

(1,' (1,' rc(1-r)Bo,(t)d)

''' (1,' n{

1t,'1oL-o'(Ddt)'t'

x rq(r -r)Bou(r\d")t'' . -,

(l"' (1," r

'

(

H3$' z)rc (t

-

\Jc

(4.e4)

rem 4.33, we proceed as follows: (i) Each condition involves integrals over subinter'als of (0, z) or (2,1), so extend the range of integration to either (0, z) or (2, l). (ii) Use the fact that (positive or negative) powers of 1. _r, I _ t, 1 - s axe bounded above on (0rz) and powers of xrtrs are bounded above on (2,1).

(4.e5')

integrals.

Iq t1w u1116)'

\r/g' /^ ,t-e'1t1dtl ur-e'@\dn)

In fact, if we set , : *, then (4.66) holds so that it remains to verify the conditions of rheorem 4.33. we know that either A o = or .A : a - 1 and similarly either B : bor B : 6 l. Suppose A : a and B : D. Verifying the conditions of Theo

(iii) use the restrictions on a,g,-y d to e'aluate the remaining < oo.

/

we illustrate this procedure by showing that the second condition in (4.90) is satisfied:

the Lemma 4'30' according to which Sketch of Proof.Again we apply boundedness of T : U(u) -+

Hardy inequality is equivalenf b the operators T1' ' ' ' 'Te tc(u).But ? is equivalent to a sum of' positiue ,o it u't we may examine each 4 individually' operator on (0' z)' it is bounded Since ?r is a conjugate Hardy (for p 1- q) holds' ?2 if and only if (4.89) l?;r p S 6 o' 1a,ae'1 and Example 2'? (iv)' if is bounded, *"oraiog to h;";;t" z'rz (a.g0) or (4.90'), respectively,,hold' and only if the two clnditions ?3 is equivalent to. (4.91) The boundedness of the Hardy operator of & is equivalent to or (4.91'), respectively, and the boundedness (4.92) due to Htilder's inequality' conditionq ,Ttgive rise to the first frve In the same way as ?t, last five' Ts,...,?e give rise to the "''

ffi' Mt are as in Theorem 4'33 and set u(t) : to(t - t)P, ul-P'(t) : tr(l - t)6'

Example 4.34. Suppose

,c(L

o2ir.,(1,"

" (1,' (1,' s Kolp, (fo"

ur,

- qpdt)

- 4oa") '"(r - ttoat)

ro*,at)"' (1,' (1,' uo,)'' {dt)

which is finite because c ) 0, ?+1 > 0 and Aq*a *L - o4*a+l > 0. (/( is the consta,nt arising from step (ii); it depends onbq+ g and d.) similarly we can verify the remaining conditions even in the case p > q.lf. A = a - I or B = b l, we have to modify the estimates on a,0,'f ,6- It ca,n be shown that these restrictions are also necessaxy for (4.82) to hold.

(4.82) holds provided Then the k-th order Hardy inequality

4.8.

d+1>0, c*1* 4)0' P+L+ bq>0 u in Definition 4'9' where a,b depend on Mf-1, U!-'

&

'y*1)0,

- t)ht,(t

Overdetermined Glasses (Another Approach)

Deffnition 4.35. A couple (Mo, M) of subsets of N1 - U will be calld standond if it satisfies the p6lya

{0, l,

. . .,

"*ditioo

222

Weighted Ineqnkties ol Hordg

(see Sec. 4.1) and

Higher Order Hordy

lVF

if

lMol+lMl:k' we have established necessary Recall that for standard couples validity of the lc-th order Hardy inand sufficient conditions of the and that the Green (a.30) for functions I e ACk-r(Mo'Mr\ "oJit,, function G(x,t) of the BVP

true for (Mo,M).) We will show that necessary and sufficient conditions of the validity of (4.80) for g € ACk-r(M6,Mr) a,re the some as the corresponding conditions for g e ACk-r(fro,fr) prculded the weights u a,nd u sati.sty some additional ossumptions. A particular case We will explain our idea on the particular standard couple

Necessary and sufficient conditions of the validity of (4.30) for e are described in Theorem 4.3 (see (a.i) and/or

g.

!Ck-t(6o,fr)

(4.e)). (4.e7)

of the Hardy inequality is uniquely determined' The investigation of the weighted norm in.ii tt tlien equivalent to the investigation

To the boundary conditions described by

:9'(0) : " ' -

g(0)

:0,

(4.100)

where

M is a nonempty

jeM,

(4.101)

subset of N1. Then we have the situation

described above, with

1l

(r/x") = Jo

(t)dt.

(4.ee)

"@,t)f

problems for the Up to now' we have investigated overdetermined

Mo: fr0, Mt = M. For the operator 16.\t

special case

M;: M!-r {k - 1}' i : 0' 1' uuple' but with respect to a where (Mf-t, MI-\ was a stondord we will deal with i"Jo,rAity of order k - 1' In this section il;; -i"*"rAoverdetermined i'e' couples satisfying U

couples (Mo,

M),

lMol+ lMrl >

r'

add some neu' boundor! con' which are constructed as follows: We couple 1frs'fri' Uy ditionsto the conditions described "t*a*d " the same is since (fi0, fi1) satisfies the P6lva condition'

id;i;o

nu)11;:o for

(4.e8)

for/Z0,where

9(tc-r)(0)

fry,fr1, i.".,

we add conditions

inequalitY

ll"/llq," < cllfllp"

229

fr : ryt, fr, :0.

9(&) : I in (0' 1) ' 9(0(0) : 0 for i€Ms,

90)(t) - 0 for ieMr,

Inepalities

\

T from (4.9g) we have

QI)@)=

|

I

J, @ -t)k-tf(t)dt,

ffi

r € (0,1).

: ?/ obviously satisfies conditions (4.100) and the : / in (0,1). Moreover, since sulr)- (k,, (-l)' tn/,fr (r - 11*-t-' f(t)dt,, : 0,1,"',ft - l, - i-

The functiol 9 equation 9(&)

.

conditions (4.101) lead to the assumptions

I' r, -

tyk-i-rlqqat

:o

for i

eM

.

(4.102)

224 Weightd

Ineqvalities

of

Hardg

IW

Eigher Otder Hardy Ineqntitir-c 226

which satLet us denote by Fu the set of all functions t € I/(u) isfy(4.102).Rrrthermore'supposethattheweightfunctionusatisfies

rl

for i e M,

(fldt c a ['l Jo' -11(t-j-r)r'rt-f and denote by

(4'103)

Vu the linear hull of the functions

Pi(t): (1- 11t-r-tu-r(t)' i

e M'

V-u is a finiteObviously, due to condition (4'103), 9i e U'(a) and dimensional subspace of. It'(u) (of dimension 'n: lMl)' If we define a duality (',')., between Zp(o) and I?'(u) by (g,

h),

: l-1r JO

s61n1t)u(t)dt, s e

lt(a)'

Thus, if we denote bV Vi i.e. the set of all g e U(u) such that

we have shown

Jo

- qh-i-rdt- o

for

i € M'

inequarity under the overdetermined conditions (4.100), (4.101) Uu ,irr"ua to the investigatio" ..f_,i:3"dity ( .l0 on the """, subset Vft of Ln@) ) provided u satisfies (4.108).

(ff) Obviously conditions (4.108) may be replaced by a singte condition

f ft -

t)G-io-t!p'or-e'(t)dt < @

with lb:maxfi:jeM\.

(4.105)

-

for the particular choice .Ds

:

ry1,

te'-t)at-p,(t)at)q u@)tu < @. I' (I" O- t;t-r1r - r)(&-io-r) (4.106)

m' ard Fu is a closed subspace ot' U(a) of finite codimension So we have proved the following assertion'

f}' Lemma 4.g6. Lct M be o nonempty subset of {0'1'"''k closs Then the Honty inequality $'30) lor the oaerdetermineil .ACk-r(N*, M) it equiaolent to the weighted norm inequality

all f e Fu

=_ Mt:0-asfollows

:

Fu:Vi

ffiduality

Fy:ii.

Theorem 4.18. Letl-

that

llf/llc,, < "ll/llp,, lor

by-e.t}Z). Moreouer, if the weisht v ' '' --v

Remark 4'gr' (i) Thus, the investigation of the Hardy

Th9 main result reads

the "orthogonal complement" of Vv'a'

o@tt

sotisfies (4.10A), then

me€lns: To investigate (a.30) under the standard "p*u conditions ta.roo) is equivalent to investigating (4.104) on the whole spare tl(r\--

(t'zilo:o for j€M' 1l

functionv

(iii) lt M is empty, then we have no additional conditions on u and the subset Vnj coincides with the whole U@1. This

h e I?' (a)r

then assumptions (4.102) can be rewritten as

k,Pil, :

yh Fu c I/(u) d:r:yhd

(4.104)

1'1' concept is difierent from that introduced in Sec'

Then the k-th otder H:(y- inequality Be hotds for g from the oaerdetermined eloss z{C*-r(N*, M) and.only it ii noii io, g e ,,{Ce- I (N1, 0) (t. satisfyinj the standard f OOit.'

if

(

*niuo* f+. "., Pro?f.-The r/part is obvious since.AC&-r(Nr,M) is a subset

of .ACr-r(Ne,0). To prove the only if part, let us first suppose that M contains only one element, M : {jo]r. Then there is only one function g : p(t) : (1 - s;r-J'"- tu-r (t), 9 e .t/(u) due to (4.108), and the set V7a

is onedimensional.

226

Weighted Ineqrclities ol HonlV Tgpe

Iligher Oder Eardg

Defrne a function ry' bY

,bft)

:

Co(1

- t;(fr -io-t;1P' -l) ur-P' (t)

with a suitable constant co > 0. condition (4.105) guarantees that rb

e U(a) a"nd (tlt,P)" >

g:f+h where

I,'

"E

(1 lo'

-

t1(t-r"-r)P' ar-P' (t)dt

ZZT

furytf3ns pj(t): (1_- 1;*-r-ru-r(r), j e M,introduced above (see with rp1 € It'(u), there exist functions U(u) e such that $i l-,rrn),, dij, i,j e M, and we can write, in analory to (4.102), \rlt;,gj)u: every function g € U(u) in the form

0

slnce

{.P(t)u(t)dt:

ltqualities

(

b

/

belongs vrt and ft belongs to the linear hull ip of the to a finitedimensional subspace of U@).It can be shown that the functions t/t; can be expressed as linear combinations of the ry';'s, i.e.,

oo

IUnc[lons

and

(rl,,pl,

:

,tt!r(t\a(t)dt

:

Co

(1 to'

lot

If

we choose Cs such

that (r/, 9)a

:

-

(1-t;(t-i-r)@''\rr-p'(t), j

1;(t-i'-r)P'ur-f (qdt.

1' we can write every g e

U(a)

in the form

g:f+Cl), IeVfr.

(4.107)

if we put f : g - (g,p)o'h, we have (f ,pl" : (g'.rg1' :0, i'e', ! e Vi' and (a'107) follou's with c: : (g'9lo' k,pj,kl',9)o 'assertion

Condition (4.106) implies that Tg; € Lq(u), T: ![ -+ fcfu). The conclusion follows as in the

Ttb e Lq(u)

g(0)

If(u) =Vfi e {al'}

:

Fu

@

{a!},

maps the whole space Lp(u) into.Lq(u). But then, according to and Remark 4.37 (iii), inequality (4.30) holds for g € ACk-l(Nt,0), : this completes the proof for M {iol,' If M contains more than one element from the set {0, 1, ' ' ' , k- l}, (see' we can proceed similarly, using the concept of biorthogonality : ma:<{j T. KATO [1, Theorem 1.221)' We denote again io ".S., j-eM}andsupposethat(4.105)and(4.106)hold.Thenforthe

?

cr.use

:

g,(0)

: ... -

9(&-r)(0)

:0,

g(1)

We have the situation just described, with M: thus, according to Theorem 4.3g, inequality (aJ0)

if

of Zp(u) and thus ? maps the onedimensional subset {(:tlt!, " € R, continuously into Lq(u). Since ? maps Vn) continuously, into .Lq(u) if and only if (4.30) holds for .ACt-r(N*, M) (due to Leinma 4.36), and since according to (4.107)'

i e M, and thus M: fub] tr

Example 4'89' consider inequality (aJ') under the conditions

Indeed,

Condition (4.106) is nothing else then the

e M.

conditions (a.8) (t?,

:0.

ffo) : {0}, and

rrr-rar ir rrra oory

p < s) or fa.gl O, p > q) *" *rirn"a,

provided that, in addition,

I'

,, - t)Q-Dp'uL-n'(t)dt < x

and

I"'

(1""

O- r)t-r1r -

t)G-\(e'-t)ar-e,(t)dt)c,.,(r)d, <

Another example Consider inequality (4.80) under the conditions

g(0)

: g'(0) =... - 9(e-r)(0) : 0, r(r-r)1t1 :0.

*.

228

Higher Order Eatdy Ineqnlities 22g

Weighted IneEltalities oI Hordg lbpe

conditions (4.109) describe a closed subspace F(A, B) of lr(u) (with finite codimension) and we consider the weighted norm inequarity

holdsif andonly ( Nowwe have M: tk-l), andthusinequality '30) in addition' that' provided if conditions (4.8) or (a'ir) are satisfied' al

Jo

ur-e'Q)dt

(

llTfllq," 3 cllfllp,"

oo

for

and

\c fl / ft IJo (\Jo I (c-t)e-rur-P'(t)dt)/ u@)dr
pp' 2LC_i2l2 Compare the Exactly this case was investigated on sults with Theorem 4'31'

re

g(r):(r/X") : JO[' C1',t)l$)dt, f e Lp(u)'

Example 4.4O.

9(0) We start

(4'108)

:o

A,ryc Nt, Anfio :0' B nfrr:0- (so that in frr"'e,-Mt' = frru A), we simply use these conditions

(4108) and obtain additional conditions on J:

I

#rt,t)f (t)dt:0' I, #tt't)f

(t)dt =

0

'

(4'10e)

function a e A, P e B. Supposing additionally that the weight

: 0,

l,l#rr,ol"

d't


h#rt,Dl" "-un,

dt

g(1)

: 0,

9,(0)

=

0,

gr,(1)

:

0.

(4.111)

with the stondord conditions

0, 9'(0) :0, S(t) :0, i.".,-l1h fuo : {0,1}, fr, : {0}, and the necessary and sufficient conditions of the validity of inequality (4.110) for g i eCr(fro,rtr) have, according to Theorem 4.13, the form (dor I p . i C . ;j 9(0) =

,::1,

(l"c(r - t)cu(qdt)

r' (t - t)p'ur-e'(t)dt) "o' " (lr"

.*

,

(4.rr2) u

satisfies

,'-o,,

We consider the third order Hardy inequality

satisfying

with o e A, P e B,

;;:

a certain

/ ft. ..._ \r/q / F rl/p
is known' If our ouerde' where the kernel G - the Green function by ad'ditional conditions termined,couple M o, M r is determine d'

n(o)1r;

of,

example.

fio - Nt' fr'' = 0 " n The approach described for the P""El case Let us describe shortly how be used for any standard couple fro,fr'" to proceed: i function u e A?k-r(fro,fri can be expressed with help of an integral oPerator T:

:0,

from F(4, B) which is the orthogonal complement

the last inequality to the whole space.Lr(u). Without going into details, we explain our approach by a simple

A general standard couP'e

n(")1g;

/

subspace of I/'(u). Finally, we try to find conditions on u a.nd u which allow to extend


,::1,

(/'

t2tultyat)

" (1,' o -

t)2o'ur-o' (qdt)

< oo.

230

Weighted Inequalities ol

In this

Hody

fup

Higher Order Hodg Inepolities ?31

Remark 4.41.

ca{;e we have

s@)

:(rfX") :

rU

ttzr'

fo"

- r2t - 2r + t)f (t)dt

particular additional conditions on z and u. So, in Example 4.40, we hN Mo: {0,1} and

rtl

-;* l,G-q2ft)dt and the additional condition g"(1) tion on /: Io'

:0

Thus inequality (4.110) holds for

we have chosen

leads to the following condi-

,r, - t)f (t)dt:0.

(4.113)

||f/||o,"Sc||/llp,,holdsforall/€trp(u)satisfying(4.113).But

l"iiii.tr ta.lis;'"*, be rewritten in the form \f ,plo: 0 with p(t) : t(2 - t)u-:r(t) provided g belongs to IP'(u), which means that u

s@)

(

to'rr-r' dt

Then, similarly as in the proof of Theorem 4.38, we can construct a function E e U1"1, ,1,(t) = Cote'-r (2-t)p'-rar-p' (t), uttd decompose the space U(").If, moreover,

l'

(',t -,, I:

r'(L

- t'1u'-r'p)at

*r, [' G - t)2f'-tur-e'(r)dr)/ J,

(4.115)

u@)d.r( oo,

(Note that in (4.115) and (4.11a) we used the fact that

G(c,t) =

r(1-r)t(l-t)for0
I

2 for

t€

(0' 1))'

Ml = {0,2}

and

efl@)

:; Ir' 4t - zx)f (Dat - !,,

l,' Ig)dt

e-tt2)

for g e AC2(fro,frr,1 (for I . ; S; :;i

_

,::1,

(l

f u@dt)"' (1,' p'a,-p,(t)dt)"o'

,::?,

(/

*cue)dt)"' (1.' ,,-o,(qd)

. *,

< oo.

: 0 leads again to condition (4.113) on again choose ,p ana ry' as in Example 4.40. leads to condition (4.114) again, b,ri *"umption (4.llb), which guarantees that Tr! € Lq(u), The additional condition

9(l)

y9 consequently, we:an IThis

then ?ry' e Lq(u) and we can conclude that conditions (4.112) a,re necessary and sufficient for (4.110) with g satisfying t'oe ouerdetermined, conditions (4.111) provided u and u satisfy a.ssumptions (4.114) and (4.115).

(4.zg),and that L < 2 - t

:

-

(4.114)

oo.

:,{0,t},

and the necessary and sufficient conditions have now instead ot the form

satisfies the assumPtion

lot

fro

A: A, B : {2} in the notation from above (see pp.4r:-10},'i.u. ZZA_:iSj. But as a g"n"ri stand.od, couple it is also possible to-take fro: {0,1}, frr': tZl *fri"t i, again a standard couple, and now A:0, B : {0}. fne., *,e h.1r"

satisfying (4.111) if the inequality

I

If. an overdetermined

couple (Me,M{is given, we sometimes can choose differcnt standard couples Mo, Mt and obtain difierent necessary and sufficient conditions in dependence on the

now reads

F'at-p'(t)dt*,,

I'

0

In

Example 4.40

[

I,

sp,-tot-o,1r)dt)q u@)d,E

<6.

it is also possible to choose fis = {U,

^ M1= {0,2}. In this case, we have s@) =

Qfl@)

:

I"

te -

lf

(t)&

+;

f,' {r, -

x2

_

*)f

g)dt

Ineqtolities ol Hody TWe

232 Weightd

Higher Otder Eardy Ineptalities

and according to Theorem 4.12, there is only one necessary and sufficient condition for the Hardy inequality (4.110) with g e AC2(fro,fr):

(l',, - c)qu(r)dx)'^ (l'

tp'at-P'(t)dt)"o'

. *.

The additional condition 9(0) :0 leads again to assumption (4'113) about / while assumption (4.115) is replaced by

l'

(,t - ,,

tp'ut-p'(t)dt+ (r

I

: (l',t

- ", t

-a)qu(x)d,r)

(l'

r'ut-p'(t)dt)o up)a,

p'ut-p'(t)dt)'.

-.

In view of (4.114), the last two conditions can be replaced by the

If

we define

s@)

:

xp,"y@)(sr,,f[r)

*(-t)ex(,,r {x)(52,, f)(a) :: (7" f)(t), then g satisfies conditions (4.116) and we have 9(e) (0, z) u (z,l). lf / satisfies tf,, *"uropiiorr" 1r

Jo

ttf(t)dt

=0,

/ Jo

-

r)qu(r)dr <@

:

Mr

-

Nt

Now. let us consider the mosirnaL overdetermined class, namely, the functions g e ACk-r satisfYing

9(t(0) :n(t)1t;

=0 for i :0,1,...,ft -

1.

(4.116)

51,r, 52,, bY

:

(sz,'!)(s):

ffi tr

t

f'

J, @ -

rr

J,$ -

.L r--. D*-r f(t)dt,

t

e (o,z), (4'117)

x)E-t

t$)d't, t € (',1)'

on

(4.11e)

and we have immediately

equality

s

cll|llp,"

(4.120)

!j.f e I/(a) satisfyins unditions (4.il9) whereT, is giaen (4.118), z€(0,L) fwed.

by

So, similarly as in the foregoing pags, we have reduced the Ha,rdy inequality (4.80) under toliru""pe"ial weighted ,ror* ,;,ruf_ ity (a.120) on a subset ot!4:116) U(i). No*, rrlu try to extena4-io the whole space 'Lr(u), and for this purpose, ",iff we introduce the functions

p;(t):tia-r(t), i:0,1,...,rc

_ l.

The additional assumption

fo'

o'-o'tDa'

t*

guarantees that

1

:.f

Hard,y inequality (A;iJD) with oaetd,e_ l.! yd* tennined conditions (4.116) is equiuirent to th,e-we;ghted, oonn nn-

In this case we can modify the foregoing approaches' We choose z e (0,1) a^rbitrary but fixed and introduce operators

(sr,".fX')

1.,

t:.,

which together with (4.114) guarantees the validity of (4.110) for s e AC2(My,M). The case Mo

s i < /c -

l,

(4.r18)

Lemma 4.42. The

llT,'fllq,"

(t

i = 0,1,...,& _

then 9(d)(21 0) : g(i') (z- 0), 0 the following assertion:

single condition

1l

2gl

(4.tzr)

g; e Lp'(a), and conditions (a.11g) can be rewritten as ('f' pr)v : 0' Analogou:1y.* in the proof of Theorem 4.3g, we introduce functions th; e U(a) such that (ry'i ,9i). =dij,i; hrms of the functions

ri(t)

:

f@' -t) or-o' 1rr.

234

Weighted Inequolities of Hordg fupe

Higher Otder Hardy Inequatilies ZS5

Condition (4.L21) guarantees that rc;

e U(u),

(ii) $ k is odd,, then e is the difference of positive operators ,S1,, / € If (r), define

and the additional

and, 52,r. For

assumptions

t)k-r ri1n' -t) ur-P' (qdt) u(r)d,r < a, Ir" (lr" O -

|"' (1,' o -

t1k-rri{n'

-\

at-p'

(qdt) u(r)dr

<

(4.r22)

a

guarantee thatTrrpi e Lq(u). Consequently, Q maps the whole space .Lr(u) continuously into .Lq(u) provided the Hardy inequality (4.30)

n@:{ f(') for xe(o'z)'

( -/(r) for x € (z,t). h. lhen e^LP(u), llhllp, : ll|llp,, and llSr,,/ll q,u : llS;,2hf10,,, since *d 52,2 &te concentratea on ,ni (r, ii ,"rp""ii"lirl ii""" 1,: 10,

Trf = Xp,4S1,rh+

*Ty

/ rz

it

and

sup'<,<, U:A -c)(t-r)e,r1t; dt)rh sup0
(for

i:

$l

ug)at\r'o

1, i.e., on (0,

/ tx

,,

.

(ff @ - t)&-Dp'nr-o' $)at\r/n' . *

(1,, |(sz," f ) (,)lq u@d,)"

\rlc / rr \l/P' {\Jr I ut-p' (t)dt)'

=

"

(1,' V AX,1,ya,)'

/,

(4.127)

a'nd.consequentlS conditions (4.124) and (a.l2b) are also sufficient

tor (A.12}) to hold on lt(u). So, we have proved the following assertion.

Theorem 4.41. Let z € (0,1) De arbitmry but 'n"ii Let l oo. Then_the Hardy inequolity (a.eO) fi.xed,. n ge ACk-r(Ms,Mi with^Ms: Mr Jxr ffana only if conditions{4.r24) ond (aJ25) are satisfied' yyouided, thi weight functi,ons u antt a'sotisfy

*,

/ rx rl/ sup ( | ug)atl' ( [' f, -c;(e-r)n'rt-P'(t)dt)t'o' . * / / \J" "<'2t \J" (for i : 2, i.e., on (2,1)) have to be satisfied.

< cllfllp,,,

o

p< q<

z)) and

sup ( t, -l(k-r)o"1r1or1 / "
I

fff ur-n'1t1d't)r/v' ( 6, (4.124) .

(4.126)

follows from (4.125) that

llS,,,/llo,"

tions

/ f, rr/p V@)lp,(x)tu\" | ="(\ro /

s cllfllp,u and

< W" lf I llo,. S Cll lf I llp,, : cllfllp, (4'123) for every f e U@) and i : 1,2. Consequently, the necessary condi-

\l/q'

|\"/o | l6\"f)(t)lqu(ldx) /

.92,2, and we have

:f'lfl

"; obtain *u ugui' (4.123) and the nec_

conditions (4.124) and (4.12b). On the other hand, it follows from (4.124) that

holds for g satisfying (4.116) and the weight functions u,u satisfy (4.r2r) and (4.122). Now let us look for necessary conditions. Assume that (4.120) holds for all / € U(u). (t) If k is euen, then I is the sum of positive operators.Sl,2 and

lSi,"/l S S,,,1/l < (Sr,r +.92,,)lJl

Xp,gS2,"h,

assumptions (4.L2t) and, (aJ22). (4.125)

Obviously, an analogous assertion holds also for I < q < p < (4.1,24) and ( .l25)_replaced by the appropriate .orrdition, expressed in terms of .A and A, A,,ea.+ 1""" oo

with

could be replaced by

n"*"*

ilrr,ia.rzl

236

Weightel Inepolities ol Haily Type

A(0, 2;(t

-

A* (0, 2; u(t),

Higher Order Eotdy Ineqtalities 2Sz

c;(*-r)cu(t)' u(t))

I,' (I"

(r - t;(t-*)r,1t;;

:

and a,nalogously for (4.125).

Remark 4.44. Let us consider the special case k : 1. We have shon'n in Theorem 4.28 that for u satisfying (4.121), conditions (4.72) arc necsssary and sufficient for (a30) to hold with 9 € .AC&-I(N1,N1) (of course, ft : 1!). Since conditions (4.72) are exactly the conditions (4.124) and (4.125) for lc : 1, it seems that the second assumption on u and u, namely (4.122), is not tzleuont. But condition (4.L22) is satisfied for & : 1 this is not the case since To show

it, assume

without loss of generality that (for

z

€

(0,1) fixed)

/

fo" Ksr,,ur-e')@)lcu(n)tu s lo

ls 1,"(pr-n' +

c)

(t)lq u(n) dx

fz

Jo l(s+t)@)lqu(r)dr < Clllll$,, <

-,

which is the first condition in (4.122) (we_ have & while the second follows analogously t'rom ( .l2Z).

: l,

i.e.

-_

0)

4.9.

Again the Interval (0, oo)

lntroduction

I,' ,'-o'{!at S f, Define

u@)do

-

-

automatically.

=

u-,,(ilor)q

In contrast to the results of Chaps. I and 2 where the role of the interval (o,6) was not important and the results could

ar-e'G)dt.

be derived from the corresponding assertion. f* if," generic interval (0, m), the situation for higher order Hardy i.r.q,*ritio is substantially differ_ ent' we tried to illustrate it in sec. i.z; h.ru we will mention some interesting results concerning the inequality

by

f

(x): I

at-e':n)

'+: ur-e'(o)

t

t' r € (O,z), for

r e (z,I),

where the constant c is chosen so that

[' f @)*: Jz[' Jo

/ f*.

(/,

f @)0,,

and define h by

n@):

{

[ -/(") for

x € (2,1).

f, h@)da: 0, i.e., h satisfies (4.119) (note that ls : i :0).

Then

1, i.e.,

According to Lemma 4.42, we have llflhllo," < Cllhllp,r, and consequently, we have (4.126) and (4.127). But due to the positivity of ,S1,s, we have

from (4.126) that

\l/c" ls(x)lqu(x)dn)

/ ea

s" (I*

bG)(,)lo,(,,,0,)',,

(4.12s)

with parameters p,g such that I < p,Q various choices of the sets Ms,M1. These v' D' STE'ANOV and rra. nasvil.ovL; among resurts are due to other, the difierence between the well-detennined (Mol + lMrl: &) and overdetermined (luol + lMl > /c) cases in a certain ,urrr" di""pp"*r.

The results concern mainly the conditic oo. Therefor", lut u, recall a ,o,rtt *,rriJil";:tTr#l';:li,:T$ in Remark 4.4 and which is (in a *or" gnural setting) due to v. D. STEPANOV [U.

2gg

Theorem 4.46. Let L

lor

Higher Otder Hady Ine4nlities 2S9

Weighter! Inequolities ol Hordg Tgpe

I

p,q 1

a.

alt functions g satisfYing g(oo) : g'(oo) : "' = g(k-l)(oo) :

il

and onlY

(4.12e)

0

Theorem 4.4G. Let L 1 p,q 1. q. Then inequolity for oll functions g satisfying s(oo) =

if / ;:

ma:c(A*,0,A*,t)

(

(4.130)

oo,

:lE

(l'(t - r1(t-r)4,1 da')'l

:$ (l'

,r-o'1*yar)'/o'

for g sotisfuing

hokts

14.1s1)

(4.12g), i.e.

if arul only if

To prove the sufficiency of.( .130) assume that lp(r)1;o,t, and ,4 < oo. Denote 9(*) F and define by !

:0

,'-o'(r)or)t''

ror

p > q;

O(oltr) ff-"1 Hiilder's

:

vs,

F(c)

:

inequality that

g(k)(t), and since /

( -,

we have by lp

and consequently, due to Theorem 4.4b, inequality (a.fZA)

q,

(l"" (lo''roo*)''o *(0d) fot

P> q,

- tlP' Moreouer, the cnnstont C in (4'128) sotisfies CxA.

oo,

lo,, ( [* e - x1G-Dn'rt-p' r]* " b)dr\ ,'t"' (t{ - rrt|ltt'.' \"/" ) and fr(c) -+ 0 for o -+ oo, i.e. 9(m) : 0. But also 9(m) = 0, and hence, g : A. Moreover, g(m) = y'(*) = ... : |ti-il1*; :

u{da')'^

r;(k-r)n'rr -o'61ar)'/o'

(

:"+_ L)l J,I,* u - x)k-tF(z)d,2, r > o.

l,i(")l s

, (lr* oL/r = rlq

if it

i@)

r;(e-r)r'rr -o' p1a*)tto ro, p : ' (/-,' -

A&,rt:

< m.

(if)

(l- (l',' - r;(t-r)c,1 +a,)''o " (/*

.,4

g(oo)

ar-P'(x)d*] tor p9q, "[\JtI /

Ak,o :=

and, only

o

for g satisfying (4.129), and hence ,4 < m.

r l/P'

/ 16

if

(a.i.J2g) holds

P'oof'(i) The necessity of condition (4.180) folrows from Theo rem 4.45: If (4.128) holds for g satisfying (4.rg1), then it holds also

where

with

The key result reads as follows:

Then inequolity (a''28) hokls

i:

g.

O

ioia" fo, CI

Remark 4.47. (i)

It follows from the foregoing theorem that in the case of the interval (0, oo), only one zero condition at infinity for the least derivative is important. v. D. srEpANov calls this phenomenon rhe heuristi.c principre. Flom Theorem 4.46 it follows that necessary and sufficient conditions of the of inequality 'aridity (4.128) for functions g satisfying (a.181) [i.e. with the second line of the incidence matrix of the form 1,0, 0,. . . ,0, 0l are the same as for

functions satisfying

g(m) = g'(m)

: o

(2nd line

l,

1,0, . . . ,0, o)

240

Weighted Inequolities of Hordy fupe

Higher Order Hordy Ine$alities 241

or

S(m) : g'(-) :

9"(m) : 0

(2nd line

:

9(m)

(ii)

g(0)

(iii)

g(0):e,(0)=0

(ia)

g(m)

(")

g(0):g,(oo):0

(oi)

g'(0) = g(oo)

(uii)

s(0):y'(O):g(oo):0

:

etc. up to

g(m)

:

g'(m)

:...

:g(ft-2)(co)

: 0

(2nd line

1,1,...,1,0)

or finally (Theorem 4.46)

g(oo):g'(m) =... - 9(e-1)(*):0

(2"o line

1,1,1,...,1,1).

Instead of. starting with (4.131), we can also start with functions g satisfying go)(co)

: o

(2"d line

with afixed j, 0
1, and

0,...,0,1,0,...,0,0) addtheconditiongO+l)(m):0

- n0+2)(m) : 0 etc. up to the set of conditions g('3)(oo) : 0 for s: i,i * 1,.. .,k - L.

(l 3), (l 3),

(i)

1,1,1,0,...,0,0)

:

o

s(oo)

:0

(;

(l t),

:0

gr(oo)

(; i),

(l;),

:0

or the conditions 90+t),oo,

(ix)

g(0)

heuristic principle was studied and used by M. NASYROVA and V. D. STEPANOV [1] to the case k : 2 and p: g:2, and then extended by M. NASYROVA [1] (see also [2]) to the whole scale of parameters, ! I p,q ( oo. In this last paper' the case & : 2 is fully described and for /c > 2 some special choices of the "boundary conditions" a,re investigated. Here, we will shortly deal with the case

(*)

y'(0):g(m):e,(m):0

(ri)

g(0)

(ii) This

ls:2. The second order HardY inequalitY

:

g(m)

c,(0)

:

:

g,(m)

s(m)

:

(lt), (l i), (l t)

:0

g,(m)

:0

Tfrl_case (fff) is solved by Theorem 4.3, the case (fu) by Theo rem 4'4s and the case (u) bv Theorem 4.g. The case (i) t. r'"rih it is equivarent to (fu) due to Theorem a.ao. tt u heuristic "i""" principre

(Theorem 4.46) indicates that

Let us deal with the inequalitY llgllo," S Cllg"llo,,

(l ;), (; l),

(aiii) g(0):g'(0)=y'(m)=o

:

(ii) ("i) (uii)

(4.132)

on the interval (0, -). This inequality can be investigated under one of the following nontrivial bounda'ry conditions on g (we list them together with the corresponding incidence matrices):

; ),

and in M'

is equivalent to (ir), is equivalent to (a), is equivalenr to

(ri),

NASyRovA [2], these equivarences ( .$2) in ali cases *" ei";.

of rralidity of

are proved and criteria

242

Higher Otder

Weighted Inequolities ol HodV Tbpe

Example 4.48. In Example 4.?, the case (ui), i'e' g'(0) :9-(T] was considered for the special weights u(c)

: r", u(r):

:0

sc-zP

-tO forp_q.Thecriteriamentionedintheforegoingsubsectionshorl thai inequality (4.132), i.e. inequality ( '17) holds for a) 2p - l'

4.10. Comments and Remarks pa'rticular 4.1.0.1. The P6lya condition mentioned in Sec' 4'1 is a very of Birkhoff case of a general P6lya condition appearing in the theory

interpolaiion. For details, see R' A' LORENTZ [1]' 4.LO.2. Theorem 4.3 is in fact due to v. STEPANOV [1]. It is a consequence of his more general results concerning Riemann-Liouville operators.

4.10.3. The results summarized A. KUFNER and H. P. HEINIG

in

Theorem 4'8 ale due to

[ll'

4.10.4. Necessary and sufficient conditions for the validity of the lcth order Hardy inequality in the general (well-determined) case (see A' WANNEBO Sec. 4.3) have been investigated by A' KUFNER a'nd gave a proof for : 3. Then A' KUFNER [2] : [1] first for lc 2 and /c

g"rrur"t/ceN\f.MsnM1-0andformulatedaconjecturefor proved this !"n"rtl index sets Mo,Mt. Finally, G' SINNAMON [4] conjecture. In Sec. 4-3, we follow his approach'

4.10.5. The sPecial A. KUFNER [6].

cases described

in

Sec. 4.4 can be found in

4.1.0.6. The reduction of conditions (Sec. 4.5) was proposed by A. KUFNER in [3]. 4.LO.7. Condition (4.65) was derived by P' GURKA in an unpub lished paper for the case p S q. Then B' OPIC modifred his approach and extended the results also to the case p > q'For details, see

Hady Inequalities

249

[OK, Sec. 8]. The approach to overdetermined classes for the case when the weight function u satisfies (4.66) for k: 1 (Theorem 4.2g) as well as for the special overdetermined classes if & > 1 (Sec. a.Z) is essentially due to G. SINNAMON and can be found in A. KUFNER and G. SINNAMON [1]. The idea of sptitting the intentol (0,1) by sorne z was suggested by R. OINAROV.

4.10.8. The approach to general overdetermined problems described in Sec. 4.8 is due to A. KUFNER and H. LEINFELDER [1]. partial results can be found in A. KUFNER and C. G. SIMADER [1]. See also M. NASYROVA, V. D. STEPANOV {21 where the ma.:cimal overdetermined case (lMsl + lMl : 2k), k -- 2, p : q: 2, is

characterized.

4.10.9. The higher order Hardy inequality on dealt with in T. KILGORE [l].

(0,

m)

has been also

5 Fractional Order Hardy lnequatities

5.1. Introduction As mentioned in Chap. l, now we will investigate froctional order Hordy inequolities, i.e., inequalities of the form llgllc," <

cllg())llr,,, o< l < 1,

(5.1)

llg(^)llo,,

< cllc'llr,o, o < A < l,

(5.2)

and

where

llg(^)ll',. Here D

':

(1"'

Io

ffi.6,yyaray)'/'

ur(r,y) is aweight function defined in (a,0) x (a,b), _oo

(5.3)

( c( S +m, and g('\) denotes the (formal) fractional derivative of order

,\,0<)<1.

246

Hadg

Weighted Ineqtdities ol

IW

ftactional Otder Hardy Inequalities

The following result was mentioned in Chap- 3 (see Sec. 3.1):

Proposition 5.1. eaery g €

i-##dad")

Letl
Cff(0,oo),

ff- l9l"*)

"o

=

"

(1,*

fe#

**)''o'

:

(

oo, 0

<

m),

( [* [* \Jo lo

ls@)

lc

-

-

^

case

g(v)lP

< l.

Then

for

(lr* (1,* p -

y1-r-tell,"

eaery

fixed

c)

where C

"

I lt -

Jt

on)

*)'''

Yl-r-xP

n'avrlo

ll,'

I,

p-y1-r-xe+ob,(y)lody

ls@)

g€

drdy)'

f@/f@

(5.5)

b't

yl-r-)p+pls,(y)lody) dt

(lr"

)lo

f-

c)-r-ro*,

*) ou

f@

:21-' o

O-

ll,

Jo

:2\-p

is the best possible constont.

^-,

f* f* - ofu)le Jo Jo -i; - virilt *0, sz^-,

(lr* P'(r;r"tr-r"*)'

on.

(see chap. 11 the necessary and sufficient condition for its validity is satisfied since ,\ > 0). using this inequality and F\rbini's theorem in (5.6) we obtain

v1t+'rP

:2r/cs-r6(r - ,lr;;-tlr

roorl'

0:

f@

fo*

=

n,

Now we use the Hardy inequality for the interval (r, oo) with a

The following result is a counterpart of inequality (5.4) and a special case of inequality (5.2). It was derived in Chap. 3 (see Example 3.18) but we will give here a different proof.

Theorem 5.3. Let I < p

21/p

(b.6)

(u n)

of inequality (5.1): 1. It was derived inde pendently by G. N. JAKOVTEV [1] and P. GRISVARD [1]' but it might have been known ea,rlier (see 5.6.1 in Sec. 5.6, Comments and Remarks). Below, in Theorem 5.9, we generalize inequality (5.4).

Remark 5.2. Inequality (5.4) is a special We take e : p, u(c) : s-)p and a(s,y) :

.AC(0,

: ('I"* J,f*lg@)-gfu)lq, . \r/P

247

-E Jo ls'fu)l1/-^Pav.

( r, inequality (b.5) hords. The constant is the best possible because of the sharpness of the constant in the classical Hardy inequality. Thus, due to the fact that A

c

Proof. Using F\rbini's theorem and the symmetry of the integra,nd, we rewrite the left ha,nd side in (5.5) into the form

(1,*

J,

l"W

Remark 5.4. If boththe inequalities (b.a) and (b.b) hold,

a refinernent of the classical Hardy inequality (1.2b):

fP#a'au)'/

( f* f'lg(") - s@)lp d,ydn * =\/'

tr

llgllp,"

g(s)

Io*

-

g(v)lp

l,* lr- tir5'-

dytu

\ l/p

with (a,6) = (0,rc), u(o)

)

the assumptions on g

a,re

=

<

Cllg'llo,o

x-\p,

stronger.

we have

u(x):,(l-l)e.

Of course, here

248

Weighted.

Ineqnlities of Hodg

TVpe

ftu'ctional Oder Hardy Ineryotitiet 24g

Remark 5.5. This chapter is organized as follows: In

Sec. b.2 we present and prove some (unweighted) fractional order Hardy inequal-

ities of type (5.1) for the case o : 0, b : oo (see Proposition 5.1). Our key lemma (Lemma 5.6) is of independent interest and explains partly the restriction ,\ I l/p in Proposition b.1. Section E.B deals with some inequalities of types (5.1) a,nd (S.2) for the general (weighted) case, with b : oo as well as D < oo. In Sec. b.4, some relations between fractional order Hardy inequalities and the interpolation theory are discussed. F\rrther results and generalizations are dealt with in Sec. 5.5.

5.7, (i) In pa,rticular, Lemma b.6 implies that inequatity llT*k (5'9) holds for all t 0 pd ""v rocally integrable function 9 with compact support in" (0,oo). (ii) Inequality (b.g) does nof hold for a 0 even if we assume that . = the limit conditions in (b.z) or (b.g) are satisfied. As a counterenamgo(x): X1r,o1(c), f < o < *, *d

yle, we can take for g the function

let o

-l

oo.

Proof of Lernmof.d. Define

h(t):: 5.2.

An Elementary Approach. The Unweighted Case

Fbr 0 <

section we consider the standard case a : 0, b : oo, and the "unweighted" modification, i.e., r.r.'(r, y) L in (S.A) (cf. propo = sition 5.1). Nonetheless, it will be obvious that similar results can be derived also for b < oo. Our key lemma reads as follows:

In this

rs

I

a<0

and

and

ti* 1 z--+O r

/' s(t)dt :0 J6

ti* 1t /' s(t)dt: 0. JO

x+@

Then

- * l,'

=

:

(5.7)

Putting

tr :

nav' '

(5.10)

x1< oo, integration by parts yields

I:'ryt, : I:'+"-

Lemma 5.6. Letl < p < 6, o € R\{0}. Assume thot t{g(t)d,t edsk for eaery s > 0 ond that either

c>0

s@)

['i

[+"*[t, I fz.

lo'nG)0"0,

-f f"

l,"a,r"];

s(t)d,t-*1"^

^1"

s(t)dt.

(5.11)

s and letting es * 0, we obtain that in case (5.2)

[" 9or: t Jo

(5.8)

!a [' "'se1at, "/o

and substituting this into (b.10), we obtain that

ff-l9l'*)"'

g(x) = h(t) + Analogously, putting that in the case (5.8)

withC(a):t+l/l6j.

Ir'

ao: r in (5.1r)

Y" and letting o1

I [" [* 9or, J' tJo s4)dt

(5.r2)

-)

oo, we obtain

frvctionol Oder Hodg Ineryalities 251

25O Weighted hequolities of Hordv fupe and substituting this into (5'10), we obtain that

g(n)

: h(') - [* 9o'' Jx,

(5.13)

b

from Now, estimate (5.9) under assumption (5'7) or (5'8) follows Ha'rdy and (s.ti) and (5.13), respectively, by virtue of the Minkowski inequalities. Namely, in case (5'7), we have from (5'12) that

rlp

*l)

\lP

g\n)Ll ls(")

(1, 1"" ta

I

(r ff

.oo

\JO)

fiq

It t )"o

*

and

:-

ln(l *

r6 ,

Ia

I I

e)111..",-y(c)

e)x1s,11(r)

+ (r + hc

- ln(l + e))X(r,r+")(c),

Theorem 5.9. Let 1
I

i)( )t

*

Now we are ready to prove the following sharpened version of Proposition 5.1:

l

a

+ hc)111,r+e;(c) + ln(L

:

(l- l'-" !: ry,,|'*)'''

rl s@)-!ffs@at

:(f

respectively, inserting these functions gd into (5.9) and letting e -) 0, we find that the corrcponding "constants" C d(e) tend to infinity.

r/r h(")lP dc\ "l I s,(t + a;) ( I"* *l;) )t

:\= ('- + 1

gr(s)

g"(t)

r) lP d,r 't- dt llh(r) I

Remark 5.8. Inequality (5.9) does nof hold for 0 < p < 1 with any positive constant C.To see this we take h(r) = h"(r) = X(r,r+.)(r) with e > 0. After calculating the corresponding function g(a) : g.(t) by (5.12) for the case c > 0 and by (5.13) for the case a ( 0, i.e.

l'*)"'

,

1

. ,r . t

: a,nd gA 1 lo" n{r)a,

o

(5.14)

se (55.8 ), we have from (5.13) that while in case

il*l9l'+)"' (l-l9l' +)'''. (l- l*" !.* *"1'*)''' = = =

o<)<-lp

('. l:l) \/'

|

*

"lg

(5.15)

(/-l9l'*)"' Scxo(|,*

r'\ "o 17)

* fo' n{r)or: o.

Then

(,.l:l) ff-l9l'*)''' ( r*l'p) - ! !{

and

[*e#ddy)'

(5.16)

with

s1latlo

n

c^,p:24/p

(t. 1;fu)

(5.17)

262

Weighted Ineryolities of

Hody

7!p

Fluctional Otder Hatdy Ineryalitiet 2S3

Proof. Htilder's inequality yields

aoo,l' lor'r I I,'

: lI

then by inserting into (5.16) the function

s(tDdtl 1," ro, -

s :1," lg(r) - s(t)l?dt. (ana hence, ap*2:1*)p Therefore, putting a: l-| > 0) and using F\rbini's theorem and the symmetry of the integrand, we obtain that

(1,*1'o'

-

'r'

'n'"1'

*)"'

/ \ : I-e

9elt1

1-x1e,4(r) *

xer,t/zl(c) + 2(l _ r)yg1z,r;(o), e )

0,

and letting e -+ 0 we find (after some tedious but straightforward calculations) that the corresponding "constants" cxp(e)ierrd

to in-

finity.

(if) As already mentioned in Remark b.g, the basic inequality (5'9) does not hold for 0 < p < r. Therefore it is a surprising fact that inequality (b.r6) hords also for the case 0 < p < I provided that g and '\ satisfy the assumptions of rheorem b.9. ?his can be seen in the following way: Choose parameters pr and )r, pr ) 1, 0 ( .\1 ( l, such that Arpr =,\p, apply Theorem s.g with .\1 andpf instead of ,\ ilrd p, respectively to the function lg(r)1n/n, .; ;;;;;;;;"r*, inequality

-

I lo@)lo/o,

1n161n/n'l

s ls(r) _

s1r11o/o,

(cf. A. KUFNER and H. TRIEBEL [tJ).

=

(/-

!o'tg(4:glt)te

dtd,)'/o

. Inequality (5.a) (i.e., inequality (b.16)) appears in literature soms times in a slightly modified form (see ihe references mentioned in Remark 5.2). To cover also these cases, let us close this section with the following equivarent form of the assertion of Theorem b.g for the case

|fp <,\ < 1.

Theorem 5.11. Let I < p ( oo, l/p If s@at ez;,sl.s for eaery E > 0 and that

ti* 1

a-+O

The estimate now follorrs from Lemma 5.6 since

t

/" g(t)dt:p,

< A < 1. Assume

that

p€R.

Jg

Then

/-19[*:lo*Wf * Remark 5.10. (i) The restrictions on the parameters in Thee rem 5.9 are essential. Indeed: If either l > 1 or I ( 0, then the integral on the right hand side in (5.16) diverges e.g. for each nonzero function g from C6"(0,oo). Moreover, if A : llp, L ( p ( m,

I,*

n@ I

-l' a, a

cx,p

(1,*

l,*

with Cs,p fum (5.r7).

ffid,dv)'

Remark b'r2' If the right ha"nd side is finite, then g is equivalent to a continuous function

f

on [0, oo) and

IL

:

O$).

Weighted, Ineryolities ol Hadg

5.3.

7W

Fluctional Oder

The General Weighted Case

Now we consider the case when the two.variables weight u(c, y) is not identically equal to 1. Also the interval (o,b) ca'n be arbitrary finite or infinite. First, let us note that by adopting the methods used in sec. 5.2, fractional order Hardy inequalities can be obtained also for some weighted cases. For instance, if we choose w(try) : {,'f € R, we can easily see that by using the methods of the proof of Theorem 5.9, we obtain the following generalization'

Theorem 5.L3. Let 1 < p < oo, .\ ) -t/p onill * thot [{ g(t)dt exists for eaery a > O and thot either

I

<

\p-l

ti*

and

I /' s(t)dt :

r-+0 0 Jg

\p-l'

'y

>

^p-l

a,nd

(5.1e)

O.

,tg * fo' nt!o, : Morenuer, Iet

u(x) ond w(r)

ond" denote

B

:

Apf

(p

c^,p=r-vo

I,*

(5.21)

d-p'(qdt)''o'

oo) sotisfying

. *,

(5.22)

- \t-p. If a0

-r u(n) +

r?-r-P.(*),

then

lo*

s c x,p (1"*

:

o.

be weight functions on (0,

i,:::g (1,' "toor)"o (I*

Then

ff-191"'0,)''

e, I ) 0. Assume thot

Theorem 5.15. Let | < p <

u(a)

ri*1/ s(t)dt : u-+@ fr Jo

256

In the foregoing results the weights have been puuer functions. Now we deal also with more general weights. The next theorem is given without proof since the argument is essentially the same as in Theorem 5.17 below.

Assume

(5.r8)

o

Hoily Ineryolifies

b{dlou(x)d,x <

"

Io*

lr'

lg9)=g!r)lP u@)d,yd,x

(b.28)

ruuith

ffi

"'

Remark 5.16.

(r. *+=1)

Remark 5.14. Due to a certain symmetry of the right hand side of (5.20), we can replace there c" by y1 . ConsequentlS we obtain an inequality of the type (5.1) with the twovariables weight

u(s,y): at1 + 9Y6, "Y,6 + )P - 1, on the right hand side, provided a > 0, B > 0 and 1,6 both less than or both greater than )p

-

1'

C:lP-r

a*av)'/o (s.zo)

arc either

(i) Apply

ma:<(l,8). Theorem S.lS

(5.24)

for

pos,er

weights

(i'e. weights of the form ta, a € IR) to obtain various modifications of Theorems 5.9 and 5.13. Note that the e.stimate (b.23) is better than the corresponding estimate (5.20) since in the inner integral on the right hand side, the integration is taken over (0,c) instead of (0,oo). (if) Also notice that condition (5.22) is a (necessary and sufficieni)

condition for the validity of the conjugate Hardy inequality with weights u a,nd tr.r (see condition (0.12) with @ 0, D: oo). = (iif) Due to assumption (5.2r), Theorem b.lb corresponds to the second part of rheorem 5.18 (under assumption (b.19)). The counterpart corresponding to (b.18) reads as follows.

256

Wei.ghtet Ineqtolities of

Hady Tlpe

hvctional Oder Hody InegrcIities

Theorem 5.L7. Let | < p < @, B > 0. ri*-

267

/(1) : ry, which is satisfied due to condition (b.2b). Thus we l* have that

Assume thot

1 /" s(t)dt:0. JO

,-+u g,

Moreoaer, let

u(r)

ond

be weight

\r/P / r,

\Jr

,/

u(t)dt)

.*,

then inequality (5.23) hokls with C

Jo

:

[* Ilo(,) - |zJo [" " ro

g@aulo --l

*p10,

* r--out(z) : *-Fa(r). This implies inequatity (5.23) with C from (b.24) since by H6lder's inequality,

(5.24).

lrrrw

lg(,)-)lstu)dvl &Jo | I

Proof. As in the proof of Lemma 5.6, define

h(r) .

ls(x)leu(r)dt

with W(c) = u(x)

x9 -L-P w(r),

furn

l" Jo

(5.25)

: Apf (p - l)r-p. If a(r) = r0-r u(r) +

f@

functions on (0, oo) sotisfging

\ l/P' (/ ur-p'(qat} / \Jo

/ f@

/::sup{/ c>0 and denote B

w(z)

:: s@) - L s}at fiJo["

s

(: l,' b(') |

grilvv)'

1," ual - s(iltpdv

and find that

g(a)

-

:

h(r) +

,'^9)0,

Jo =,

due to the fact that for

see (5.10) and (5.12). Hence

lo*

zp-r

s2p-r

(/- nt'tt

r9lr-Vl-p > l

with 0
withl(p(6, p

r.

u(x)dx

* Io* I,' Y "lo |

where we have used the Hardy inequality

ilqdrl u@)tu

=

u

Io*

^

provided that either

t

< \p

- | and x_+O liq 1 /"ot t)dt: t Jg

s

or ff@)lPw(r)d'a

2

[* lt"l ln+l' ,1dr 1" [* sg) :#.vit#p - ylldndy --Jo[* Jo Jo lr-",

u@)a,)

(/* nt"ltpu(x)dr * u Io* l9l"u,(o)dc)

I,* ll,'

0,

tr

Remark 5.18. By applying Theorems S.tb and b.l? with g | = + Ap, u(t) - xl-lr and ru(c) _l/p - ,t-\n*p we find that for

lntdl'u{,)o*

s

p>

\p-r(?()p*l

a,nd rim] ['g4)at:0. tt JO

258 Weightd

Inequalities

d Hadg

Itvclional Oiler Hordg Inqulities 259

TVpe

In porticulor, for all g e U(0, B),

Hence,thisresultreflectsthepossibilitytoinvestigatethemore general inequalitY

(1," uax'ut ya,)'/' lo*

btdl'u(r)d'n 3

"

Io*

f-

ut"l - s@)lpw(l'-

vl)tudv (6'26)

(5.30)

I

=, (lr"

btd - s(iltpwln - yt)dsdy)

fo"

whichisaspecialcaseof(5.1)forthetwo.variablesweightfunction

u(r,y) of the tYPe

u(r,y):

u(lo

-

for the interval (0' b) with Inequality (5.26) and its modification and W' D' EvANs [1]' b S m was studied ; t' I' BUR"ENKOV Let us give one of their results'

( and'Ietw Theorem 5.L9. Letl3p <@' 0 B S oo

be

Suppose thot there erists

for

eaery

a

constont c

6(r) s

€ (1'2) since, due

r € (0' B/2)'

u(n)

Then

:: U(ar(x)),

0 € (0,b)

(5.27)

u(r\;: JtI w(t)dt
6(r)

with c the constant from (5.28). Then 1B

(0,

uous and strictly decreasing. Define the function d bv

aweight

that function on (0, B) such

for eaeryc € such thot

U be the inverse of u from (5.27) so that U{u(o)) : u(U(a)): s. Since u is a weight function, both u and U are continProof. Let

Yl)'

to (5.28), d(c)

I

y

(5.31)

: U(at(r)) < U(u@/2)) :

x/2

nd

(5.28)

=an(2t), b € (0,8) ond all g e U(0'b)'

u(d(c))

:

u(U

(a4x)))

:

ar(r).

Consequently,

(f'rrt'lr,

. (l' with C

indePendent of

loo

g

t

{(',, /'le(")F)h,., (b.2e) btl- g(v)lp'r'(lc - sl)tudv) I

@1a,)'to

=

onitb'

I

J6(t)

w@at

: J6(x) [* -e)at - Jr[* wg)dt :

u(d(r))

- u(s) = (c -

l)u(c)

.

(S.82)

260

Weighted Ine4ualities of Hotdy fupe

Let e € (0,b/2). Since le(c)l S inequality yields

r

::

&nctional Otder Eody Inqualities 261

lg(y)l+lg(r)-g(y)|,

the Minkowski

(1,''' lo b{')lo'tu -')drtu)

:

(1"' vror

tc

- L)u(y)dy)"'

.

Due to the symmetry of the integrand, we have

/ fb fu

Iz

rr/p (5.33)

*

/

\l,,' l'

:: It*

val -

stu)l''n(Y

-'laua-)

:

:

(1,"'

ls(x)lP(u(x)

< (c-

'

or(b) for c € (b/2,b),

-

u(b

- or4"'

t)'!/o,l(l:''

p1,11,,1,y*)

(1""'rg(")ro f,o -{r)oro,)"o

(1"'''

ls(r)l,(u(r)

-

u(b

while F\rbini's theorem and substitution y with (5.31) and (5.32), that

11

<

Iz

(notice that c e (e,b/2) implies 21 1 b, and y € (2x,0) implies U ) r,i.e. lc - yl : y- c). Substitution y - t :t a,nd (5.27) yield

r:

and hence finally, since u(o) < u(b/2)

-

\r/P

*([' 'Do')

- t : t yield, together

(l"rrrv I"o'' -ru -,)d*dv)

: (l-'r'' l,o-"'t'l''o')"'

)

\lolrls@lotu

Fo11e (0,b/2), we have bar(b) due to (5.28). Thus

r)

/ ro/z

\

(l

I

(-'(t))'/of

+ co.

b/2 andhence u(b_ x) < u(b/2) <

l/p

ts@)teu@)aa)

= (1,"'

ls(r)lp (u(n)

* (1,"'

ls(t)lpu(b

-

u(b

-,r*)

- o4""

262

fructional Order Eordy Inqluro.lihies 269

Weighted Inqtolities ol Hardy TUpe

s

(c

- ,r',o (l"u''

Now, (5.29) follows by letting e -r 0*, and inequality (b.30) D tend to B. tr

follows from (5.29) by letting

ls(,,)lo'(dd,)'/o

Now we shall present a fractional order Hardy inequality with a general two-variables weight ,(r,y). we shall work in the interval (0,6) with 6 finite or infinite.

- 1)u(b))1/P (l',,rOn'*)''' *Co

+(c(c

+ (cu,(e)r

/n

(!

"

1n

Theorem 6.20. Let | < p ( 6, 2 -l/p. Let u(a,il b" o y:iSlt function on (0,b) x (0, b) with ^0 < b I q and suppse thot ld ur-n'@,t)dt < a for eaery x < b and, I{ or-o'(r,ildi'< o for euery y < b.

1,11oo,)

""

and consequentlY

(1

(i.) Denote

w(r): (:

- (" - r)1&1 (1,,' 1r1'y1",1"p) s (*(b))r/p

* (;)""

(f'rnt,lr *)''o

(1,'[

ls(")

-

(5.34)

s(a)to-(t'

I,' ,L-o'(r,ilor)'-o

and

u(c)

- u11o'oo)

:

u(r)n-^n.

Moreover,

Ao (5.35)

(!,'tnr,tr,ov,)

I ra \ llP / rblz \ r/P s- I I ls|"")lp"(fldrl +(cu(b);t/r | 1.,^ln{dloa'1' / \'u,'

/ \J" and since 1 - (c - l)tlo > 0, it folloq's from (5'35) a'nd (5'34) that

(f

vov,1,1o,)''o = " "'

. (l'

loo

{(,*,

/' ts(dte ar)

bt ) - g(v)l'.,(l, - nn*on)

with C independent of

g, b

and e.

\

,:

olllo

(1,' ffior)''o (lo'

,,-o,rt)t^o,at) 'o

ond,

K then

:: -.-A8a (p - t)'tt

for g e I?(u) ue haue

(1,'van',@dr)

1

.*

1u.ru;

264

Weighted Inequolities of

(ii)

Hady Tlpe

frzctional Otder Hailg Ineryalities

Denote

Hiilder's inequality yields

n(il: (i l,' ,'-o'(",iltu)

ll,'tnal -

ntuDdrlo

u(il:6(ilv-b. <

rf A,

': oltlu (1"' ffio')''o

(I"

''-''t)t^o'at)"o'

' * 1''"1

ond,

l'

ual - stu)lp,(s,oo,

xP-l Ilx

= .@)

-

Jo

ls@)

(l'

ot-P'(,,u)or)o-'

- s(y)leu(x,y)d,y

and consequently

7. +p
d and K,

268

lt(u)

uith u ond

K

I1

reploced by

respectiuely.

Prool. First note that

f'

J,

fo ls@) - g@)lp

lo'

Io"Wu@'v)dvda

* Ioo I,''tfL

- nlrlvoulo o,

ri,9[

using now the Minkowski and Hardy inequalities (the latter due ro the fact that Ao < m) we obtain ttrat

(1,'waw,ov,)

u@,v)d'vdr

:

(1,'

=

(I'#l'o, -I l,' nrutorl'*)"'

. l,' Ir'ryu(r,v)dvdx

* Iou l,'ffi-u@,v)dvdx =:

oat

I' #l,r'r - I l,' n{oa,l' a,

l, i=Affia(x'Y)dxdY

:

I' Wll'

hI

Iz.

b @)lp w (x)o- ^,

.(f #11,"

o,)

nr,vulo

*)"'

266

Ftvr,tional Oder

Weighted Inequalities of Hadg TVpe

f

The casep

s and since

t* * * (1,' p(r)lPu(c)d ,)'''

K

,

11, inequality (5'37) follows'

the Part (ii) can be proved completely analogously' starting from o integral ,[2 instead of f1'

Haily Ine4taliti,es

267

q

Up to nonr, we have considered inequality (5.1) for the casre p : g. The case p f q is more complicated; in order to illustrate this fact we present a result for the case I

wzfu): Remark5.21.(i)TheproofofTheorem5.20showsthatalsothe

lou

wz@)ax

inequality

(1

- K) (/'ut'lr u@dr)''o * r, - fr) (1"' b@)leu@)ar)

(/, J,ffi'P'Y1a'aY)'/o

u(r)

holdswheneverconditions(5'36)and(5.3s)aresatisfiedsimultane. ously (and

K,R
Ao,o

Aok(p,p) according to formula (0'14)' in olljr case, k(p,p) - prlppttlp' : p(p - L)-rlp', which leads to the formula for the constant K'

inequality can be estimated

Example 5'22' Ifwe

take

,"ilt"u g(c) by g(x) -

by,

u(r'y):

17 and

b:

l)

-l/pand'"Y

,:l:,

(l' u@)wtq (t)o')''

x,=ffi
9(0), we obtain the inequality

b@)-g(Y)lP, t"' .\p*p-t-1 j \p-t-r (f\Jo J,f Effi{dxda) oo,

::

< oo (5.39)

ond,

oo in Theorem 5'20

ff- lg(') - e(o)flar-rr or)t'' 1(p(

Wo,o(r)r-^t

, (lr' o
Then Jor g

whenever

:

1( p
and, ossurne that

Theorem 5.20 where the Hardy 1i;j r" the step of the proof of in this in"quulity was applied, we used the fact that the consta'nt

urrd

(; l,' (ffi)"'-'d,)-"'' (; lou .z@)a,)o -,(u).

Theorem 5.23. Let0
rftfi

=

w,,o(y):

<^p-1(cf'

TheoremS'13)'

(5.40)

€ Lq(u) we hoae

/ rb \l/c (/o lo(')l'u(x)h) s

i

-; (1,' (|'tr#

.,67a,)ot' *,rn)do)

268

Weightet Ineqnlities of Hadg

IW

ftu,ctional etder Hordy lrlepatities 269

Proof. Hiilder's inequality yields

(foo t'uit,t',(ilor)"o

(1,'val - s@)twz",)*)

(I" b(,) - g(otpw,(x)*)o'o =

(1,'

(ffi),-' 1"'

Consequently

=

(I'"rrrls{0 -

* (I'

d6

fou

o{,),,tdo,l'

"rrr(wz(y))-qll' o@).x,)o*l'

ou)

q,

lo,,,2(v)dy

I' (I' W

-, 6y a,)o

** (I'b@to,(ildv)

.

foo

r-o,o-"(1,'

"l[ 2

fou

b@)

(m)'-'0,)-o'o'

a-oro-" (1,"

(m)'

rb .l -. Jo "tu)lo@

I

- W6

fv Jo-

The result now follows by subtracting and using (5.40).

Notation. Now we will deal with inequalities of type (S.2). First, Iet us introduce some notation.

- s(y)b'2(r)drlo .,rrro,

'o,)-o'''

'lr(v) fo' ,r@)a* - fu o@),r(ddxlq =

,

-2(y)d.y

For two weight functions ,(r,y) and ru1(c,y) defined on (0,6) x (0,b) with 0 < b ( oo and for parameters p > 1, ? ) 0, define a function Bo,r@) on (0,6) by

Bc*@)

::

,Z*r(I' lc - 11-r-rq .@,t)a)'/o

tq

o@),2(,'1a,1' au.

using this estimate together with the Minkowski and Hardy inequalities (the latter can be used due to (b.39)) we obtain that

" (1"' 'i-,'(,,0d) forl
270

Weishted Ineqnlities ol

Haily 7W

hoctionol Order Hardy Ineqnlities 271

(P/ rb Bo,,@),: (.l (/, - tl-t-ra', x,t)dt \"'o ,

wherc

( C:1

'"

( f w!-n'6,11or\''o' " \Jr// for 0 < q

)

1, Here we assume that 0

*: i < .\ <

,al-o'1x,r)rr)

I

"'

lr*f. Due to the symmetry of ur(c, y) we find Theorem 5.3, see (b.6) - that fo fo ls@) J

V(Y):: suppose

oo,

finite a.e. in (0,b),

(r | (

fou

"E,r{*1-orr'

denote

^,:

(loo ayltn-d6)a,)@-d/o

lo

f

Jo Jo

lou

as

in the proof of

Y)d'tdY

(1,'gl|'lroo,1'

or)*

ls@)-s@)lq

?fitru(r,Y)dtdY

(5.42)

p) q olso <

. *.

ag*{,) (1,' w,t ttrwr@,s)du)o'' or.

First, suppose that 1 < p < q. Estimating the right hand side in (5.42) by Minkowski's integral inequality if p < q (and thus, q/p > l) and by F\rbini's theorem if. p : g, we obtain

(I |ffi-P'v1a'av)'/o (I'b'@)lpv(x)0,)"o

2kq(q,p)

loo

Then

="

:r

for p)e,

thatV(y) is finite a.e. in (O,b) and that for

"

sfu)lq

i:ffii-ta(r,

J,

-

Thus, using for the function g' the Hardy inequality on the interval (c,6) as mentioned in Remark 5.24 we find after integrating with respect to y over the interval (0,0) that

P,o!a')o/o for p s q ,

{ ); I\JO I w1(r,g)dn

p>(r

ptb1o,1r1p' for p>q.

I

1.

0 < q < oo. Let w(r,y) ond (0, x (0, b) arul assurne thot u b) on be weight functions -{r,y) 'i.s symrnetri,c: w(t,y) : w(y,t). Assume that the function Bn,o@) defined aboae is

fm

x(q,p):I Q+fr)tlng+$1rt' for p
t"

substantially in the proof of the following assertion.

(

zrh*1q,p)4/o

uith

Remark 5.24. Notice that for c fixed, 0 < r < b, the condition Bo,o(r) < oo is necessary and sufficient for the Hardy inequality to hold on the interval (r, D) with weight functions U,(t) : u(z,t)la - tl-t-rq and %(t) : ut(s,t). This inequality will be used Theorem 5.25. Let L < p

zr/ck(q,p) for p1=q,

fu f'lg(x) - sfu)lq lo =i:ifffu(t,Y)dtdY

Jo (5.41)

<

Zkq(q,

p)

(I' ([

ax sx)wll e (*, v)tg'(v)tq o,1o'

o

o

or)o'

272

Weighted Itequalities of Hardy fupe

:2kq(q,p)

-

2kq(p,n)

/ rt

l/o

tn'trlto

Fncti,onal Order Hardy Ineqtatities 2Tg

/ fu

(see the

\p/c \c/e

-t-

\J, "x,r.iro@,v)tu)

ds)

1o/n / rb ls'fu)l'v(ildv) (/,

i.e., we have inequality (5.41). Now, suppose that p > q. Then we estimate the right hand side in (5.42) by Hiilder's inequality with parameters s : p/q ) 1 and s' : pl(p - g), and we find that, by F\rbini's theorem,

(0,c) (with c < 6 fixed) and modifying appropriately the definitions of. Bo*@) and V(y). (Now we are dealing with the conjugate Ha^rdy operator for { : g'(t)dt.) The formulation of the corresponding ff theorem is left to t-he reader. (ii) It is also possible to omit the assumption of the symmetry of u, but in that case, we have to replace w(c,A) by fr(r,y) :: u(r,y)* -(y,.r) in (5.a1) and in the definition of the functions Bo,o(o) and V(y). Indeed: as in the proof of Theorem b.3, see (S.O), we have that

b,@ [' f fl9)]q ta(r,v)dxd,v Jo Jo lr - y1t+'^n

f Jo f 4t"l lr-cr.-

--,?{v,)-l:

Jo (P-

s

2kq(s,

p) (1"'

"fJr-c)

first lines of the proof of rheorem 5.25). consequently we

can modify the proof using now the Hardy inequality on the interrral

(')dr)

il lP

:

" (lr' (l,u v'ot,wr(x,y)dy) o*)0"

*

:

2kq(q,p)q

(1"' (1,' wr@,v)dx) b'(v)l'dv)

:

2kq(q, p)co

(lo'

W,

tolt,v

This is again (5.41), and the proof is

Remark 5.26. (i) Notice that

tilda)

:

complete.

o

due to the symmetry of

ut(r,y),

can also write

- g(y)o w@,y)drdy fJo lo f E(") l, - yr "''

:'fo'(1," ff&11""

I"' (1,'

we

1,'

u(r,y)d.xdy

ffill,'

I,' (1,'

ffil!,'

n'

ruo'1' aa) a,

where fr is symmetric. Now, we proceed as rem 5.25.

in the proof of

Theo_

b.2S with b = oo, Q : p, yl-r-^p+p, we obtain inequality (S.5) lc but with a difierent constant. Notice that in this case, ao,olry i, ,

w(c,y): I and ,{r,y):

or)*'

roo'1' aa) a,

(I' ffill"'tul0,1'0,)*

Example 5.27. (i) If we apply Theorem

n'av'lo

n'

constant!

274

Weighted Inequalities of

(ii) If

Hady Tlpe

hactionol Ord,er Hardy Inequalities 275

we apply Theorem 5.25 with g

w(s,y) = lr we obtain that

- ylr+N-(lx -

:

5.4.

p,

yl) and -r(r,y) --

b(') - s(illpw{r lr' lr'

=o

|oo

-

.{lr -

yl)

vl)drd,s

ls'(r)lPu(x)dr

(5.43)

:

lo'

Some basic facts

First we recall some concepts and results from the real rnethod, of intetpolation (cf. e.g. J. BERGH and J. lOfSfnOU 14;. Let (As,.41) denote a compatible couple of Banach spaces (i.e., both are continuously imbedded into a Hausdorff topologicar vector space). A Banach space .A is called an interrnediote spore, between

if

As and.41

where

u(x)

Hardy-type Inequalities and Interpotation Theory

ug*{r)*1(fu

- tl)d,t

AofiATCACAo*Ar.

with

Bl'e(t):

,."r1?*,

\ p-l rb-t / fy-t -!-o'1rya") -(")0" Ju-,

\lo

Obviously (5.43) may be regarded as a counterpart to inequality (5.30) - see Theorem 5.19. We close this section by stating an elementary result. The proof is left to the reader.

Now, let (Ao,A) and (Bs,B1) be two compatible couples. Then spaces .4 and B arc said to be interpolation spou,s with respect to (,4o,Ar) and (86,^8l) if .4 and B are intermediate spaces to the respective couples and if, for any bounded linear operator ? such that ? : As -+.86 and T : At 1 Br,we have T : A +^B. In terms of inequalities this can be expressed as follows:

Theorem 5.28. Letl1p,9 ( m,0 < b !x, ondletv(r) ond w(r,y) be weight functions on (0, b) ond on (0, b) x (0,b), respectiuely.

If

Moreoaer. denote

v

(r)

:

fo'

,'-o'

o': Then

Ioo

lv(") - v(illsle' 'ffiw(a,s)d.ady
then llrflln S G(Mo,ut)llflla

tt)at

and suppose thot

where G(s,

t),

f)

,

(b.44)

0, is a positive bounded function. Nowadays there exist many methods of constructing interpolation spaces but the most frequent methods are the real method, (.,.)r,q, 0 < ) ( 1, I 1q(-oo, and the complexmethod [.,.]r,0 < l < 1.

In both

with C = Arlc

llrllla, < Mollflle" and ll"/lla, < Mrll/lln,

s,

cases we have

G(s,t)

: sl-)tl.

276

Weighted Inequolities ol Hardy Type

Fraclionol Order Hardg Ineqtalities 277

For our purposes, it is sufficient to consider the real method defined in the following way: For a compatible Banach couple (.40, Ar), the PEETRE K-functional is defined for any I € Ao I Ar

The technique just described was first used by A. KUFNER and H. TRJEBEL [l]; some inequalities of the type

andt)0as

K(t,d : K(t,g;Ao,Ai : inf{ll96lla" + tllgrlla, : I : The interpolation space (Ao,Ar)e,q, 0

fo* 9o

* gugo € Ao,grE Ar}.

< 0 ( 1' 1 ( g (

for

g:

: (l-

r-eoYo

(t,s)*)t't,

was mentioned

X.+

ca,n be

Lq(u)

llgllx

:

lls'llp,'. Therefore, if we use the space

X

forevery g€Ao, llglla"

s

then we have the following general fractional order Hardy lloll1r",r,"1,,11

^.,

(i) rf

olpr(b -

inqnlityt

s M;-^ M|llgll(e",x)^'"

Notice that in this case, the operator T from (5.44) is the imbedding operator, i.e. the identity.

r)t/0

/ fx

\Jo

l/P'

\ -. -ttttlat) ( oo, "o','<

thenforanyd,0
Mollglle",

<{ (

oo, 0 <,\ < l, } : f + }.t et g be a differentiable function on (0,b), 0 ( 6 ( oo, such that g(0): O.

< Mrllgllx

as A1and the spane Lq(u) as 81 and if we can find Banach spaces -As and Bs which are compatible with X and Lq(u), respectively, and if Ao'-+ Bs, i.e.,

where

weight functions on (0, m).

further investigated and applied. The key problem is to determine Bs so that the norms of the interpolation spaces (Bs,Lq(u))s,r ancl (Ao,wl,p@)).r," o, (Ao,Wkp@))r,, be idenri"* results fied with the norms of 9('t) required in (5.1) and (b.2). The obtained there are not completely satisfactory and here we only present two examples where we consider Bo : Wl,p(u) with u(r) : 1 and ,qo : Wi'p (u) in the first case, while in the other case, we choose Bo: Lq(u) and .46 : t?(a) with u(c) : 1.

Example 5.29. Let |
where X is either Wl'p{u) or WL;p(u) (see Sec. 1.2, formulas (1.29) or (1.30), respectively), i.e. we have for evety g € X llgllc,"

tl

spaces ,46 and

that the llardy inequality (1.25) imbedding treated as a continuous

it

pp'11'.p1a,

In A. KUFNER and L. E. PERSSON [1], this technique was

lnterpolation and the Hardy inequality 1

[*

JO

have been derived, with u, u and

oo, we apply the usual supremum norm.

In Chap.

+

oo, is

normed by llell(n",a,),,o

btdlou{,).,

r-^'-'

(I'ro+t)-

o

s(r)tcdn) n sc

ln particular, if. p - g, then r

-

Iob'@)l'a(x)dn.

p and we have that

fi ls@ + t) - e(r\rr f f =-ffi:Ldxd,tsc ls'@)l,a(r)dx. J, J, Jo Thus the last inequality is a modification of the fractionar order inequality (5.2) with the special weight u(x,t) = 1..

278

F\ucti,onol Order Hotdy

Weighted Ineqtolities ol Hordy fupe

(ii)

Assume additionally

that

)*

L/p and extend g by zero to

(-b, o). rf

.',0 (1, u$)dt)'/n 0(z(b suP

for

eb

I

Jo

ls@)l'"^'h1x)dr

=

"

1-lr-r lo'

In particular , if p : g, then r

btdlu^(c)dc loo

p>p-

u*)"o

in this form, we

see

1u.n

-

(l',ro+

t)

-

(H"l)@)::tJoI ffu)dv for 0
at.

s@)toa,)

1

that it is

p-L that

1f,

/'

(H"l)@)--:fi1f6I I@)av for 0>p-r. Jz

p and we have

(5.48)

(We call I/o the Hardy aueraging operator.) In order that these two operators coincide, it is necessary that

t" I"' (l_,rywa,)at.

1fr116 :rJo ffu)dv: frJz ffu)dv,

I

This is a modified version of the fractional order Hardy inequality (5.1) with the special weight u(r) : 1. There are many examples of inequalities which hold except for one or more values of the parameters involved. Sometimes, this phe' nomenon can be explained via interpolation' Let us give two simple

-: I

l.e.

fo*

f{ilor:

o'

N the set of locally integrable functions satisfying this condition. In order to interpolate the Hardy operator on weighted Lebesgue spa€es, we have to consider not the spaces themselves but their intersection with /V. Denote by

examples.

Example 5.30. The (classical) Hardy inequality

(l- l*ro \" ,ua.)''o ' c (/- u't"l Yr'dn) holds, with p

2

1 and 9 € Cff(O,oo), for every

p

* p-

1

(5'45)

but

does

nothoklfor P:p-L. In order to be able to understand this phenomenon we first note that inequality (5.45) ca,n be rewritten as

(/- l:

1. When (5.a5) is written

ut'lr

impossible to interpolate between 0 < p- 1 (see (5.46)) and 0 > (see (5.42)) to obtain the inequality for p: p-1. The reason is we have in fact two d,ifferent operotors involved:

thenforany6,0
279

forBcp-landas

(/- l* l,* r{ilool" *0,)"o . r(/-

(@t

Ineqnlities

fo"

trovul'

*o-)'/'

' c(/- ut'lr',u*)''

(s'46)

Remark 5.31. Notice that for

weighted Lebesgue spaces, the fol-

lowing interpolation result holds:

(If

(or),

LP

{az)) e,p

:

7n @l-o uo),

i.e., the interpolation with two different weight functions arrt;2 pr\oduees a new weight function ae = al-eaf. tn the case mentioned in Example 5.30, we need to interpolate between U(ui ON and U(w) flN with u1(r): 81,.t 1A -\ and u2(r) : 9.6,6 > B - L.

280

hoctionol Order Hordy Inegtalitties

Weighted Inequolities of Hardy fupe

281

Example 5.32. Return to the fractional order Hardy inequality

("-tgXt) :xg(x)and recall that this inequality holds for I < p < o with l € (0'1) but ) I \/p (see Theorem 5.9 and Remark 5.10 (i)), and e.g. for every 9 € Cff(O,m). As we have seen

in the proof of Theorem 5.9' (5.49) is only

an

ciLsy (:onscquence of the inequality

|

g@avl'

I

""

le c

g(t)

":

{n We note that ?g € N

a + O.If we consider g € lP(r-o,rn+t) n N, ?-l coincide and interpolation is possible between such special subspaces also across the critical point c :0. boundedly for

all

- I li

d, Io

These observations lead the type

g{ilav .

(N n

?, inequality (5.50) reads

C llT gllp,,-dP+P-

(5.51)

r

a # 0. If T has an inverse ?-1, then (5.51) can be rewritten

a+0.

as

(5.52)

Moreover, if the assumptions of Lemma 5.6 a,re satisfied - see formulas (5.12) and (5.13), then ?-r exists and from the proof of the lemma mentioned we see that it ca,n be given explicitly in the form

("-lgx") :zg(x)+

[

JO

gfu)ay,

c(0,

investigation

of interpolation

@P),

N n ^Le(c?));,o.

(N Nf

^+

q

and 1 <

nE@\,N iLp(f))r,p:

(0+t -p)/(P -1) (N n

Ln @e),

p- | < B. Then

ry

nl,r(r(r-))B+)'v)

:

Cp

(5.54)

and

N n Lp (rl))x,p

(ae-r) n U(gJ,-')

if^:(0+t-p)/(0-t).

where

llT-rgllr,,--,-,
Il

Theorem 5.33. Let 7 1 p <

s@)dr: o}.

compact support. In terms of the operator

to the

The following result has been proved recently by N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [1] and is particularly relevant to the previous examples, in particular for the case from Example 5.30 where the exceptional value was p - 1.

e.g. for every locally integrable function with

llgllo,,--"-, <

(5.53)

spaces of

In Example 5.30 we have introduced the set € ,Lc(o,

follows from the Hardy inequality that

T-r : Lp(t-ap+p-r) + I?(r-w-r1

(b.bo)

with c I 0 (see Lemma 5.6). For o : 0, it fails (see Remark 5.7 (ii)). To become acquainted with this somewhat curious phenomenon, let us investigate the operator appearing in (5.50), namely

-

sfu)dv,a)o'

the two expressions for

[* ln142l' 12 s c [*lgi') - i Ii Jo l"'l r--Jo

(rd@)

it

F\rrther,

[*

Jz

Here Cp(u) denotes the weighted Cesiro function space of nonabsolute type:

C,(o)

:

llgllc"t,l ,=

{r("1,r

€ (0,m)

:

(1,*l* l,' s(fiavl"@)o')

'*)'

282

Weighted Inequolities of

Hadg

TWe

hoctional Order Hody Inequalities

Remark 5.34. If we use Theorem 5.33 for the Ha.rdy averaging operator I/o considered in Example 5.30, we obtain for the excep tional value p - 1 the following (trivial) estimate:

llH"flln,@"-'prvS and for the operator ca^se

c

:0

Cllfllpp,-,)ncp('p-,)nN'

(5.55)

2gg

Remark 5.36. Inequgtity (5.57) means that the Hardy averaging operator (H"t)@) : * I; Ifu)dy is bounded from N n rlfllogcl) into trr. Let us recall that ,F/o is not bound,ed from 11( log cl) i"to Zi'. By using the estimates from [OK, Example 8.6 (v) and Rema.rk in the same vray prove the following more general result: < p 1 a and, f e N n LP(rr-r1tog*lr), then

8-7], we can

Il |

?-l from (5.53) in Example 5.32 we have in the

the inequality

t llT- gllL, @-' ;nrv S C llgllp 6, -')nce (c'-, )nN .

(/-|*

(5.56)

Sometimes, we can obtain estimates for the exceptional values directly without using interpolation. This is illustrated by the following proposition.

Proposition 5.35. Il

f

<

Proof. Using the assumption find that

I,*l: :

c/*

/ €N

lrogzl& '

(5.57)

and F\bini's theorem, we

6,-'l

S Clll llrnr,p(cp-, I rogzlr).

Finally let us note that Theorem b.3B is only a special case of a more general result from the paper mentioned on p. 2g1 (with power weights replaced by general weights) which allows to describe situations where some Hardy inequality fails not just for one value of the parameter but even for an interual of parameters. For illustration.

h/h.

f

Then, for

,\e(0,1)\[a%;,#]

volaslat and

vtu)tava,

/

€ N, we have both the Hardy inequalities

Ir- E

I,' (1,'!0,)vorar* I,* (I'l;*)vtu)tdy

_f* -lo lrogyl lf @)lay.

This estimate states that the Hardy averaging operator .EIo is nI?(xn-lllogcp) into If (tp-t), i.u., that we have

Example 5.37. Let us(c): max(ooo,co,) with 0 S ao ( ar and ,r(") - min(c-9o,r-9') with 0 < go S B1. Assume that crsfcrl I

lo' rr'to'lo*

l,'l: l,' rrovrla,+ l,*l:

.

let us give an example.

v{dtooa, + l,* : l,* = l,' : fo' =

togxf,o-16r)"' =" (1,* l/(")lol

llH " t ll p

trt'lt

looul"o-'d*)

bounded from N

e N n rr(l logrl), then

I,- l: l,' rutouldt

fo"

fr' xovrluf-^1cy,f

s

"

Io*

x)d,x <

"

Io*

1c

)do

and

tr

Ir-

r{iloul,;-^t"l,it l: l,*

l/(r)luf-r(c )u!14aa

r/(r)ruj-r(o )ul@)ax,

284

hactional Order lIatdy Ineqtalities

Weighted Ineqtalities of Hardy Tgpe

Proposition 5.38. Letg e LP(n-ao-r! a / 0. Then

and therefore

(fv n.fr(us), N ff f,t(rr))^,,

:

N

l-l

rr(u;-^ul)

(see also Remark 5.31).

For ,\ e [Afr;, and we only have that 'ft'],

(N n.lr(u6),

5.5.

ff fl .Lt(rr))^,, : N nCr(ul-^ui)

n tr(?,}-rui).

Further Results

A comptement to Lemma 5.6 If 1 < p < @ and o ) -1, then we easily obtain the following inequality which is reverse to inequality (5.9):

( r*ls,) - I Ii s@aalo'"\

*

Ul s

(r +

l;)

-'J;

(1,*lgat-

none of these Hardy inequalities is true

s@taYl"*)"'

ruith the equiaolenu, constonts

1+ Lllal

=

withp2l and,a)

(lo*lgl'

and (a

*)'''

+ l)/(a + 2).

Proof. The upper estimate follows from (b.58), the lower from Lemma 5.6 since by Hiilder's inequality we can show that I I; g(t)dt tends to zero for r -r 0 (ifc > 0) and for c -+ m (ifa < 0) provided

I e U(t-ar-r1.

tr

Remark 5.39. For the case that g is decreasing and -l < o < 0, this is essentially the result of C. BENNETT, R. DE VORE and R. SHARPLEY [1]. They investigated the functional l*. -.f. where /* is the decreasing rearrangement of the measurable function / in the totally o-finite mea.sure space (O,p), i.e.

"o

#) (/- l9l' +)"'

lndeed: The Minkowski a^nd Hardy inequalities (for F\rbini's theorem (for p - 1) Yield

(5bs)

f*(t):

inf{y : p{r e (0,oo) l/(c)l > y} < r}, '

and

p > 1) and

f..(t),:+LJO['

I'(ildy.

More exactly they proved that the norm in the Lorentz Lp,,I-spwe,

( r*lso) - I It

Ul * l;) =

i.e.

stu)aYlo'"\ "o

(/-lgl" *)'''.

ff-l#

[

llflln nrildnlo

*)'''

s(r+*) ff-l9/l'+)"' Hence we have the following result about equivalent norms in

I?(t-an-r'1

.': (/-,r'

ro

y. 1tyo4)'/0,

(5.5e)

is equivalent to

lll/lllp,c

that l liml-a6/*' (t) = 0. provided

,: (lr* Gtle U,. e)- /.(,)))o+)'/o t/ @,

lSq<mand

.f'-(oo) =

286

fractional Order Hordg Ineqtalities 287

Weighted Inequalities of Hordy fupe

Using the estimate (5.58) a,nd Lemma 5.6 with g - .f' and a: -Ilp and p replaced by q, we obtain the following somewhat more precise result.

f0

t7

lr'vt xou+ l,r(,)l =+ Ir' f.(v)dv+ l/(41,

Let

lll/lllp,c

(ii)

Since

l/+(t)l <

f' ond l'* be defined os oboae. (i)Ul
Corollary 5.4O.

Ptwf. (i)

s (r + h)

< p < oo, 1 ( q < oo ond

we obtain that

f+(t)

ll/11,"-.

f'*(x) :0,

inequalities.

(ii) Similarly as in the proof of Lemma 5.6 (see formulas (5.10) and (5.13)) we obtain from (5.60) that

us denote

f+@:l l,

ttu)dv

- I@'

We have shown in Proposition 5.38 that the functions

f

(5.60)

f

and'

f4

are

equivalent in the (norm of the) If (t-an-r1-spaces' M. MILMAN and y. SAGHER [1] have shown that they are equivalent in the Lorentz Lp,q-spaces. Their proof is based on interpolation techniquesl here, we will give a completely different proof based on the idea of the

l,* 1


(ii) If L < p <m,

0

ll

x,

t +ll r',' S

(t): -!*(t) * I,* f#@)+

Now, the result follows again by the Minkoq'ski and Hardy inequalities, but we must consider separately the cases I S q < p, L 1p S q and 0 < g < l. < p < oo. In the second case, we must apply duality, in the third case we must first do the following calculation:

proof of Lemma 5.6.

Proposition 5.42. (i) If 1 1p 1 6,

t' (;)

(where I# : $+).). The desired inequality now follows by using the definition of the norm in 10,e, the Minkowski and the Hardy

then

ll/llr,." s @+ t)lll/lllp,s. Remark 5.4L. Let

s', lr"' f.(y)dy.

*a! :

lo*

ru{r)

(}"o-,,r,) ,,

then Cp,cll t

ll

L,''

( s < oo andlimsa*I It t$av : Jllr.'" ! Dp,qllfiallu'.

0, then

: I,* fiwhdv ' tJo fi(r)ou *

I,- f+@+

tr

288

Froctionol Order Hordg Inery,alikes 289

Weighted Ineqnlities of Hordy Type

5.38withp:2and

Example 5.43. If we apply Proposition a: -Lf2 we obtain that

(l- (n", -: I

s@)dv)'

The following theorem is due to N. KAIBLINGER, L. MALIGRANDA and L. E. PERSSON [U, and their improvement of formula (5.61) reads as follows:

o')"'- (l-

Theorem 5.45. Let g e Lz(rD)

s2(r)d.-)

q.

thatW(a):fr(O):

Then

for every g e L2(0,m) with equivalence constants 1/3 a'nd 3' (Note that for such functions, the condition lim"-a- I Ii S@aV : 0 is satisfied.)

and. suppose

llgllr,,

:

- H.gllz,.

(5.62)

llgllz,.

: lls - fr-sllz.-.

(5.63)

llg

ond

Remark 5.44. The last equivalence relation can be rewritten 1

itto -

H"sllz<

llgllz S sllg

-

H"sllz

(5'a8))' Moreover, signs: by equality inequalities the replace possible to

with H" the Hardy averaging operator

as

(see

it

is

Proof. The function grr is integrable on (0, c) for every r ( oo, since due to g e L2(rn) and W(r) < oo, Htilder's inequality yields

(5.61)

llsllz:lls-H"sllz

for every g e L2(0,oo), and in the next theorem' we will generalize this result to weighted sPaces. For this purpose, let us introduce the following generolized Hordy

(1,'s(il.tildv) s

oaeraging oPerator

(H.g)(r),:

#,

where ur is a weight function on (0,

W(n):

lo"

w(a)du <

: lo

stu).@)aa For a.e. o

m) and

a

for everY s

)

o.

Similarly,

(H.s)(a) :=

1

w(r)

l,*

o@)*@)aa

m

for every c > 0.

with

W(x):

too

t 1

Jt

ut(y)dy <

)

lo' o'til.fu)dy, Io fc^ w(r) s'{il.tildv Sw(r)llsll|,. < @. w(ilda

J,

0, we have

*"lA([ s(u)-r,,*)l : ry# s@)-(ildv fo' ffi :

(5.64)

s21r)w(rr

- (ef'l - #6

fo'

(lo" s(at'todv)

s(il.(ilon)'

,a(x).

290

hactional Order Hordy Inequalifiies 2gl

Weighted Inequalifies ol Honlg fupe

Integrating this identity from 0 to b with b fixed, we obtain

<m

arbitra,ry but

Consequently, for large b,

ftfi

h(|,"til,(ildy)

I s'1*1.14a*- JoI (g(')- (H-s)(r))2w(r)dr Jo :

h(!"u -,rlr*

:

ntol'tildv)

[*

(1,"

o(ut-t

s

*)')

rbP o'{r)-{r)d" - Jo(g(') - (H,s)(r))2u(r)dr Jo

and

-

< Ine' similarly as in (5.64)

-

that

P 1-. /rb \2 P^ ( I s@,(ilayl s J.I s'@).(y)dv. Jc I -fu)dy<)ew(b).r - --/

Moreover, since lIl(oo)

:

oo' we have that for large

h(1,"ril,(ildv)'

.I''

(!.0 nt

t-@oo)l

/

|fJe

s21x)w@)ax

I ^ : - Jo | b@) - (H-d@Dzu@)a^xl j s,

i.e.

llslll,-:

llg

- H-sl|17,..

But this is (5.62), and the proof of (b.68) is similar.

fora,nyb>c,

\"/"

,,,,.rn).,)' *

s(il.tilds)'

p

rrb ^lqa

> 0 arbitrary. Since g € L2 (w), there exists a consta.nt c(e) > 0 such that

Using Hiilder's inequality, we obtain

f"o

follows from (5.65) that

o_+oo

Choose e

J"

it

(5.65)

: i6(1,'otowtv)do) ' f@ ^. s"(Y)u(Y)dY

ft,f([

s(ul-fildy +

., (i,* |,) : ,,

The last limit is zero due to (5.64), and hence

c:

h(1,'

b

s

Remark 5.46. (i) The assumption tV(m) : Jf w@)dy: oo in Theorem 5.45 is essential. Indeed: if 0 < [f, rd)aU < oo, then for the constant function go(r) c O_we have H-g6 : F_go : go = I and_h_ence llso -.H.Sollz,- : llgo : Owhile l|gollr,, : fi."gollz,, u(y)dy)Uz s s. l"lffo(ii) It follows from the proof of rheorem s.4b that for the case

0<

f-

u(y)dy <

oo we ca,n replace formulas (5.62), (5.63) by

llgll?,.= llg -

' Uf, sfu)'fu)dY)' ,---jW

n.gil1,

292

fructionol Order Hatly Ine4ualities

Weighted Inequalities of Hatdy T\pc a^nd

and t f* ililw(y)dy)z : llg - fr .slll,. * *-ffit,

llslltr,.

thus by Theorem 5.45,

llsll?,.

:

:

respectively. ha-s some

Theorem 5.45 some of them.

sotisfy

the assumptions of

Theo'

:

rern 5.45 and' d,enote

(S.,gXr)

::

(S,gX")

:: lr" oUlffiar.

llhlll,,:

_ fi,nll?,,

r-l

I"*lnat*o', -

/@

J,

@)

1

h

roo

l,*

otu)w{rt,(o)dnl u(r)dr

- w(r) l,* ootffiorl'

ls(r) - (S-s)(r)l2u](a)dr

ffi*

: lls - S-gll1,.,

which is the first part of (5.66). The second equality follows analo-

l,* nUlffiao,

gously.

Example 5.48. (i) If lls

Then

we choose

u(x)

:1,

then we have for g e L2

_ H"sllz: llsllz: lls _ sgllz

where

: llg - S.ellz,-: lls - 5',gll '

llsllz,-

(5.66)

Proof. The weight function u(r) : u(x)lWz(t): (-1/W(c))' satisfies the assumptions of Theorem 5.46 and

r,.. f* u(ildv:- t*/ u(,):

J,

If

llh

: l"* lnaw

interesting consequences. Let us mention

Corollary 5.47. Let g and ru

2gg

we denote h(a)

llhllr," :

Io*

z:g(c)W(r)'

s2

1 1 \" dv:w6.

J, \ria/ =

Io*

s2

1r7.1r1a,

=

llsll|,.

Ir" s(ilda,

(ii) fiwe choose u(r) :c-2, lls

-

fr.sllr,t-2

:

^ee(c)

:

I,*

*or.

then we have for g € L2@-2)

llgllz,,-":

lls

-

S.sllz.,-"

where

(fi.d@)

then

1r1w2 1r]u(Qdr

(H"s)@)::

- , I,*

$r,

and (3,,g)(") = Ir'

*ou.

For the case p: 2, we also have the following sharp version of Proposition 5.38.

294 Weightd Inequoliti.es ol Hardg 7W

ftactional Order Hardy Inequalities

Proposition 5.49. If g e L2@9), B <

| oruI B f

Hence

then

-L,

. (.lr+pl\

min

lr,74

where (Hog)@)

:

*

l+#l

)ilgtlz,"u

[i

llgllz.'u

l0

:

since, using Theorem 5.45

shol.

- $ - B)H"sll2,,e

with w(x)

:

llflll,,-':

-

Hognz,,u

=ffiilnn

,,,

: r-9

and

f (")

:

g(r)rp , we

ll/ - H.flll,,-u

[*ln"1rr-np-r(r . [' - il Jo Jo 1". llg - (t - g)H"sllZ'u

g(ilavl'x-edr -'-. |

::

llg,

-

Using (5.68), the Minkowski and the reversed Minkowski inequalities,

Hogrllz,re

/llsrllr,ro

\\"*t,t

as r -| I-B;rl* md e" -+ 1 as r -+ oo, rye see Q, + that both the inequalities in (b.62) are sharp. tr

ryi

Remark 5.50. (i) The cruciar formula (b.6g) hords trivially for 0: I but it can never hold for any B > 1. (ii) Proposition 5.49 implies that the following sharper version of

Corollary 5.47 holds for q

.

"tX[o,U(t)

z'+B+t \'/' :((-:_1'*i@@)

(5.68)

S-ince

: :

ils

belong to L2(xF) and

Q,

obtain that

llgll3,"'

s

9r(r)::

we have

llg

ilsnz,,o

and (5.67) follows. For r ) (-B - l)/2, the functions

g@)ay. Both the ine4uolities are

Proof. For p < 1 and B

2gs

If.

f

:21

e LP,2,1 < p < oo, and .f-.(m)

:0,

then

we obtain that

(t

-

f)llg

-

H.,gllz,,u

: s :

ll

- gs+e -

l9l

llsllz,"u

(1

+ llg -

(L

-L) vp-r/

-

\

ItrVt/ef.(t)lr+-)

P)H"sll2,,o

YP-L/

(01+ r)llgllz,,,

The lower estimate is the best one for 1

and

(t-0llg-H"sllz,,o

*in (t,

p)H"sllz'u

-

estimatefor2(p
-

0g+ e -

I l9l llgllz,"

l(lBl- l)l

(1

-

p)H"gllz,,o

-

llg

- (7 - p)H"sll2,"al

llgllz,"u.

5.6.

( p < 2 and the upper

Comments and Remarks

5.6.1. Inequality (5.4) in Proposition b.l was established independently by P. GRJSVARD [t] and c. N. JAKOVLEV (see atso

[lf

296

Froctionol Order Hordg lrn,egl;a,lities 297

Weighted, Inequalities ol Hodg TVpe

G' N' JAKovLEv [2])' For the case P : 2' see also N' ARONSZAJN and K. T. SMITH [t] (cf. also R. ADAMS, N' ARONSZAJN and K. T. SMITH [1]). Similar ideas can be found in the book by J. L. LIONS and E. MAGENES [1]. 5.6.2. The approach in Sec. 5.2 is taken over from N' KRUGLJAK' L. MALIGRANDA and L. E. PERSSON [2]. One motivation for this approach is to supplement the theory so that ttle disturbing condition * l/p in Proposition 5.1. could be understood also from ^ point of view. This theory was further developed in an interpolation N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [U'

5.6.3. The results, proofs in Sec. 5.3 are mostly taken from H. HEINIG, A. KUFNER and L. E. PERSSON [1, 2] (cf' also A. KUFNER and L. E. PERSSON [2] and L. E' PERSSON and A. KUFNER [t]); Theorem 5'19 is due to V. BURENKOV and W. D. EVANS [r] (cf. also V- BURENKOV, W' D' EVANS and M. L. GOL'DMAN [l]). 5.6.4. The results mentioned in Secs. 5.2 and 5.3 contain only suffcient conditions for the validity of the corresponding fractional order Hardy inequalities (5.1) and (5.2). The problem to find necessory ond, sufficient characterizations of the weights seems to be still not solvecl in general (some exceptions will be mentioned below). Hence, we propose here the following open questions. open Problem L. Find necessory and sufficient cond,itions on the ( c, Y < b, so that weights u : u(r),0 ( c ( b, and u : u(n,a), 0 (5.t) hokts (for the c&sep- q), i.e., characterizeu andu so thatfor

P)1,0<,\<1,\lllp, / fi

(I

\Jo

for

0
i l

I

([ [ffiup,v1a'av)'lo I

=

:

I I

I

* (lr' tu'(r)leu(x)d.x)

(5.70)

tor some finite K > 0 (cf. Theorem 5.28). 5.6.5. Let us formulate a proposition which contains holds

some necessary and sufficient conditions for the validity of inequality (S.?0) (or, more precisely, of its modification (5.72) below) provided the weight function u on the left hand side satisfies an add,itional assumption.

v:

u(o)


oo. Let u : u(r,g) and (a,b) x (a, b) anit on (a,b), rcspecu(r,V)+u(A,t). Suppose that the weight

Proposition 5.51. Let L < p be weight lunctions on

tiuely, and, d,enote {r(r,y)

:

function u satisfies

"Zy,l" I,'

,r,,, ([

I,u

,r,,)a,ao)-'/n

" (1"' Ioo *ro,nor*)-"0' oro, .

*.

(b.21)

\l/p ls@)lvu(x)dr

= hokls

I '.

Open Problem 2. Find necessary and sufficient cond,itions on the weightsa = a(r),0 ( c !b, ond,u: u(r,y),0 S s,g 1! b, co that (5.2) hold.s, i.e., choracterize u and u so that for I < p I @,

/I

* (1,' I"' ffiup,s1a,av)'lo

some f,nite

K > 0 (cf. Theorem5.20).

Then the inequality (s.oe)

(1,' l"' b@) - s(ilrqu(r,y)dydt)''o

. * (l"o v'avu@1a,)'/o (5.72)

298

Weighted Ineqtolities of Hordy fupe

hokls

if

onrl only

Froctionol Otder Hardg Inequalities 2gg

it

Hardy-type inequality

in Example

b.3?) can be found

in

this

paper.

r",j[?.,,r

(1""

t

{, @,y) dy

d*)'''

(l:

o,-o'1,1a,)''o'

. *. (5.73)

This result is due to H. P. HEINIG (for

p:

g) and A. KUFNER

(forp < q). Condition (5.73) is necessory for any choice of p,q,i.e., also for I < g < p < oo, and for the proof of its necessity (which follows from (5.72) by the special choice g'(r) : ur-p'(t)X@,il(r), a < a < p
Let us also emphasize that condition (5.73) is a certain modifica' = A(a,b;u,u) 1m with .4 from (0.8) (cf. also

tion of the condition A Remark 1.4).

5.6.6. As far as it concerns inequality (5.1) (i.e., the analogue of inequality (5.69), but with an .Lq-norm on the left hand side), the case p f g seems to be still not solued in general. As mentioned on pp.267-268, only for the case p ( g there are some results, but with a m:ined nor"rn on the right hand side (cf. Theorem 5.23) while the case p ) q seems to be completely open. 5.6.7. Interpolation theory is widely described e.g. in the books by J. BERGH and J. LOFSTROvI 1t1; H. TRIEBEL [1]; C. BENNETT and R. SHARPLEY [1];Y. A. BRUDNYI and N. Y. KRUGLJAK [1j. The idea of using interpolation theory to prove fractional order Hardy inequalities was initiated in A. KUFNER and H. TRJEBEL [1] (but, of course, this idea had appeared much earlier in the paper by P. GRJSVARD [1] mentioned in 5.6.1 in connection with inequality (5.4)) and further developed in A. KUFNER and L. E. PERSSON [1] (see also L. E. PERSSON and A. KUFNER [1]). Attempts to understand in full the excep tion ) * llp in (5.a) has led to some problems, but by using interpolation between special subspaces also this phenomenon can be cnmpletely understood, see N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [11. In particular, (a generalized form of) Theorem 5.33 and its consequences (e.9., the fairly surprising

5-6.8. The equivalence formula for weighted tp-spaces pointed out in Proposition 5.38 was proved in N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [2]. For decreasing functions, this was discovered by C. BENNETT, R. DE VORE and R. SHARpLEy [l] in connection with the definition a,nd investigation of weak .L6-spaces (see also C. BENNETT and R. SHARPLEY [1]). proposition b.42 is due to M. MILMAN and Y. SAGHER but the proof here is taken from N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [2]. our proof of (a weighted version of) the remarkable identity (s.6r) (see Theorem 5.45), i.e.,

llgllr: ll(I - H")sllz, s€ Lz = f,2(0,m), is taken from N.

KAIBLINGER, L. MALIGRANDA and L. E. PERS-

soN [1] where also another proof, based on isometries in Hilbert spaces, can be found.

In this connection (cf. Example 5.48 (i)) we also propose another open problem:

Open Problem 3. Find

T

(aueraging) operator

necessary ond, suffi,e,i.ent conditions on the such that

llgllz: ll? -r)sllz holds

for

att

g e L2

:

L2(0,a).

5.6.9. some results mentioned in this chapter can be generalized also in other directions. For example, in H. p. HEINIG, A. KUFNER and L. E. PERSSON [2J some multidimensional fractional order Hardy inequalities and also some extensions to orlicz norms are proved. A new related result was proved by H. TRJEBEL [2].

6 Integral Operators on the Cone of Monotone Functions

6.1.

Introduction

In the

previous chapters, we discussed the mapping properties of the Hardy, Hardy-type, Hardy-Steklov and other integral operators in weighted Lebesgue spaces. It is of interest - and sometimes even inevitable - to consider weight characterization of such operators on some subsets of the Lebesgue spaces, e.g. in the case when the operator is defined on the cone of decreasing functions.

Motivating example

If

we want to obtain a weight characterization of operators in weighted Lorentz spaces Ap(r), | < p ( oo, then it is necessary to consider operators defined on decreasing functions.

Recallthat,for0(p(m,

rr(-),= where

/

{t:

ll.f*llp,',

= (/*,r'(r))eu,(r)dr)"o

is a measurable function on a measure space

. -} X

(6.1)

(for example

302

Weighteil Inequolities ol Hodg Typc

R"), .f* being the

Integrol Operators on the Cone ol Monotone

equimeasurable dmreasing rearrangement

of lJl

defined by

f. (t) ::inf{y > 0 :,\1(y) < t}.

(6.2)

ftnctions

g0g

and considered on non-negative decreasing functions (notation: 0 ( ir bounded from ,[r(u) to Lt(u),1 <-p,q < m. This means that "f 1), one desires to characterize the weight functions u and u for which

the (Hardy) inequality

Here .\1 is the dzstrdbution functiondefined by

)y(y) Note that

ll/.llo,.

::

meas{r e X :

l/(r)l > y}.

is a norm on Ap(u) if and only

if u is decreasing.

But the expression ll/-.llp,., with

f,.(,)::1""

(/- (i l,' rav")o '$)0,)'/o . c (1,* r,aw@dr) holds whenever

(6.3)

f,(t)dt

/

(6.6)

2 0 is d,ecreasing.

Remark 6.1. (i) Although problems of the type just mentioned were studied by B. STECKIN

is a norm which is equivalent to ll/'llp,- (see, e.9., C. BENNETT and R. SHARPLEY [U). In what follows, we take the measure in ,\y(y) to be the Lebesgue rneasure.

Recall that the rearrangement of the Hardy-Littlewood ma;
(uf)(r):sup * t ffQ)ldz, r€lR', zeQ lVl JQ

(6.4)

where Q is a cube in IRn with sides parallel to the coordinate axqs and lQl is its Lebesgue measure, then (M

f).(t) *+ Irf*(s)ds, r > o.

( 6,

is a bounded mapping, or, in other words, to characterize the weight functions u and u for which M is bounded between Lorentz spaces'

it is equivalent to prove that the Hardy (averaging)

p-,.

lo..{9.1^'

and other mappings arising in harmonic analysis. (fi) The main purpose of this chapter is to provide the duality principle of Sawyer, given in Sec. 6.2, together with a corresponding

duality principle in the case 0 < p S 1. In Sec. 6.3, applications involving Hardy-type operators are given, while Sec. 6.4 is devoted to the case of integral operators with more general (positive) kernels.

Hence, to prove that

M : ItP(v) -+ Aq(u), 7 1p,{

in the early 60's for the discrete case, the first characterization of weighls for which (6.6) is satisfied (with 0 < t J!) was given by M. AzuNO and B. MUCKENHOUPT [U in 1990 when p : g and u: u. Shortly afterwards E. SAWYER [3] proved the general case in the index (exponent) range I 1 p,Q I @ thereby characterizing the weight functions for which the operator M from (6.4) is bounded as a mapping from Al(u) into nq(u). Simila^rly he proved such results for the Hilbert transform

operator

6.2.

The Duality Principle of Sawyer

.E[o,

The duality principte in I?(p,)-spaces

defined bv

(H"txt),: I ,{")0", lo'

r ) 0,

(6.5)

For (X,p) a measure space with positive Borel measurc p, ! < p < oo, and / a measurable function on X, the stondotd, duolity principle

304

Weighted Inequalities of Hodg

Integrcl Opemlors on the Cone of Monotone Fvnctiotug

7W

Then )1(t) is the Lebesgue measiure of. E(f ,t). Now, by F\rbini's

in ^Lr(p)-spaces is expressed by the formula

I -lJx - = suPl

llgllp',r,

theorem. (6.7)

XqnA)dP(')l I

1' where the supremum is taken over all f e l?(p) such that ll/llp,p.= U(p)' Here as usual p = + and' LP'(p) is the space dual to pp' and 224' 13 on already Recall that duality'was considered here given The duality in the l.it"r cate coincides with the concept will we in the ,p""itt case dp(c) - u(r)dt and X : (0,1)' Here' RD and be interested in the cases of (6'7) where X is R+, IR or weight : some with p is absolutely continuous, i.e', d'p'(r) a(t)dn function u. The other form of the duality principle (already introduced in Sec. 1.1) is given bY

ff f(x)g(r)dr sup #-+i:: \ r/, nb

(.[o- fn

(r)a(r)h)'' "

:

ll9llp,,,'-"'

,

(6.8)

where u is a measurable locally integrable weight function'

Theaimofthissectionistoobtainanexpressionforthelefb handsideof(6.8)whenthesupremumistakenoverallnon-negative decreasing functions.

wellBefore stating the duality principle, we recall the folloq'ing knontn result.

Lemma 6.2. If f 2 0 is function w we haue f@

I

JO

ilecreasing, then

for anY rneosunoble weight

fo*

f'{*)r{*)*

l,* (l,to' nt'-'\at) w@)dt

: :

(lr*

lr* fo*

or-txa(r,r)(x'1at) w61a

,ro-' (lo* *"u,rt@)-@)ar)

: plr0r@ r-'Il/,

at

\

u@)dxldt.

\rE(t,r)

/

But, since / > 0 is decreasing, the set E(f ,t) is the interval (0,,\r(t)). Hence (6.9) holds. tr

The main result of this chapter is the following (SAWYER's) duality principle for decreasing functions /:

Theorem 6.3.

Suppose 1 < p < oo. Let g,a be non-negatiae measurable functions on (0,a) with u locally integrable. Then

[f, f(x)s(x)dn

uqP

o-in (Jo fn(r)u(x)dx)t/e

fo(r)r@)a,

\/o

\1 is the distribution

/

functi'on defineil oboue'

Proof. Denote

E(l,t)

:

* (l- (1,' nr,.to,)'' (1," u(t)dt)-o' ,t )a,)'/o'

./rtrfu\.\ (p( oo, :p'JorF l-- {'-t - I l'- u(fldrldy, 0 where

308

- {'€ (0,-) f (') > t}'

*' If e@)a' (6.e)

(ff

a@)ax1l/n'

(6.10)

-

Remark 6.4. (i) Note that integration by parts shows that the first term on the right hand side of (6.10) can be replaced by

(1,*

{[ s*dt) (1,",av,1'-'' n@)*)

306

Integral Opemtors on the Cone o! Monotone Frncliotts

Weightetl Ineryalities ol HordA fupe

gO?

oo, then by convention the second term on the right hand side of (6.10) is taken to be zero.

But, since (t - pXt - d) : I and p(l - d) = -y', integration by parts shows that the last supremum is finite. Consequently,

Proof of Theorcm 6'3' First choose f (r) : C a positive constant' and denote the left hand side of (6.10) bv tk). Then J is decreasing

L(g)

(ii) If ff, u(r)da:

I ;c suP ;tb

and

f? f(r)g(r)dr

supL L(o\>=;:"" u\s) Ur- yn(x)u(x)dr)r/P

[f, s(r)dr

(6.11)

:

1

c

=SUD nio

ff q(,) E h(t)dtdt (Itr tr (s)Iv (n) / u(*)ln u14ax)ue /o- h(t)G(t)dt ([tr tr @)lv (a) / a(t)ln p(r)dr)' to

Next. let

f where h

(r): Jz[*

> 0 is arbitrary. Then

/

(6.r2)

n1t1a,

is again decreasing and

L(d

: ,uo

Jo g(") '['- h(t)dtdn

: *(1,- [ffi]",,v,)'o : I (1,- (/'nt"u";'' (|"'u(s)ds)-o' ot'a,)'to'

,;ilM

Denote

G(')

7t

= JO[' s@)a" v(t): /o o(s)ds.

The conjugate HardY inequalitY (cf. Introduction, formula (0.10)) shows that

(l-,,',

h(t)

(1.* nav,)' o*)'''

t, (I*

hp

(x)vp (x)o'

-o

p1a*)"

Here we have applied (6.7) with / : Y, g : g a,nd dp(r) : a(t)dt. The lower bound for (6.10) follows now from (6.11) and (6.12). To prove the upper bound of L(g) in (6.10), let

:

-'

/,*

{[e1,1a,)o

(lo' ,uro")

r |.fi'g(")e 1n'-r1r/n' *-'LF;(FJ

o

J

holds

if and only if with C >

-]-. - 1-r

o

,61*

30E

Weighted Inequalities of

Hody

Integrol Operators on the Cone of Monotone Flrnctioru 309

Type

But this expression is the right

By Hiilder's inequalitY we have

ha,nd side

of (6.10). Hence if

we

show that f@ : J, fu)su)h(t)h-t{,)dt

foo

t{ilnl)a, Jo'

(lr* r,aln-p(u)s(ed,a)"' =, (lr* yo1,y,1,)d,)'/' (6.14)

\l/p

/ f@

s (/. !n?)h-o(t)s(t)dt)

(6'13)

for 0

( / J, then the upper bound for L(g) follows from (6.18) by

dividins

. (l-

n''

1r1n1r',or)

''

integration yields

n-p(t)

Now, by F\rbini's theorem,

/ r@

(/,

w' (t)s@at

\

yn'-r 1 1x f roo / fE \ -p' : lrl, (/, ot')a") (/o ,t"lo")' ,1,1a,

1/P'

*

) p

=

(f'nt,la")''-' (1,",(,)a")

(l-r(,)

[/ ."[##]"-']

: (l-

o

(l- (1"'

' ff e(s)ds +(J;@"o

'

{

(|,"t'p')

"

(1,*01";a")o

(/'o1,va")o

-'

(f*,,",r")'*']'" o

[*

((/',1";a")'

- (f't"r'")'*')]

(1,' noo')'-' (1," '(s)as)

nuro,1''

'@)dr

o,)''' u@)

(1,'

."LffiF])'" =

fp(r)u(t)dx)t". to prove (6.14) observe first that

* (/-

sr'sa')

a'

*

: -o'

'@)a')'/o

" (1,'n1"10") (/-,,",*)'*')'*

{(/',1"v,")o

[*

- h(1,*,uro")'-''

([',u,0,1'-o

*" (1,*'t"8")'-']

)'" (6.15)

310

Weighted Ine4nlities of 1

since C

Integrol Operators on the Cone ol Monotone Frnctiorts

Hady fupe

0. Now Lemma 6.2 with

- F2

:

w: h-pg yields

J, f

Thecase0
Integrating by parts, taking into account that h-p is increasing and applying (6.15) we obtain

In order to prove an analogue of Theorem 6.3 in the case 0 < p < the following result is required: II h > O is decreosing, then for O < p I I and.0 < b ( oo,

n-n@)s(r)dr

lo s(ia,li'r,

h-p(^ t

s@

(yy "" Io^

-rt-'

" (1"'' = (p' -

(l

Io^"u'

(lo" n@0") oo-o1'1

nr,)a")

(l^'''

' since

: pp-'\he4)dt -, (l:r1"ya")o

1t

/ h(r)dr JO

Theorem 6.6.

:

roo

\P ,

'h(,) > 0

u that

ay

>

0 < p l- Let g,u be non-negatiae = loully integroble. Then (0,m) withu

Suppose

swoble lunctions on

mea-

ff, f(r)s(r)dr

.lPr

fn@)h-v(t)s@)tu

f-, (\J0l'",(,p")/

/ rt

th(t) and p - | <0. Hence F(t) > F'(0) : g n and (6.17) follows for t: b.

r(")'")

r(r)dr.

sp(/- rf-r J0 [-

(6.12)

that F'(r1

Substituting into (6.16), we conclude by Lemma 6.2 with tr

J"

7t

we have

(')

lo^t{o

xo-tnnqyac

F(t)::p I rn-r6n@)d"-( | n1*1a"1 Jo \"ro /

e(s)ds

,(")0")

Do;

-

P

1

is satisfied. The proof is easy: If we define

Jo

= t;n@)

1P

/ rb

I\"ro I h(a)dr) / sp JoI

,.rrfu\

3

L(g). tr

@)h-e(r)g(r)ds

\ / t^tfu) :p f* ,"-' (/" "" n-p@)s@tu)da' Jo

I

- l)r-r JO/@ !p(x)u(x)dn.

W

Hence (6.14) is established a,nd so is the upper bound for

foo

3ll

g;*Ft'l,t'lr'lm ::lB

(1,'

sau,)

(l'

,6)a*)-'to

(6.18)

3L2

Weighted Ineqnlities of Hadg

Proof.

If.

l(*) = X(0,')(o), r )

[f,

f

Integnul

TW

Remark 6.6. If f (r)

0, then

@il.\n- -

II

g@)a'

{li,{dd'o

fo*

bound for (6'18) follon's' p To establish the upper bound, we apply (6'9) with = P = 1. Then

lo*

r > 0, hence the lower

I

:

Jo U'

:/,f* s

sup

il'@

\

:::E

qp

= XE

o@)d'r)aa

0, then h


1q it

.L

if

and only

: u(t) we obtain that

if

/

n*oro(dd,)ot'od

'1.'

-

follows that

(1"

'op)''' (1"' ,{oo,)-"" (l*

n"1'1,1';a')'/"

or

g@)a, (

rrP

\ dY flvt u(r)dt)

llhllo,"
ffi4;*;-1/,

d#p[l-

ffip

)

/ r@ lr/q hq(r)u(r)dxl /

(l^''''t"r")'''

o4

"::lB

o4''' I,- n*' (lo^"')'t')a")

ffi'll'*

0("
0
(6'19)

with

I{4ntdo'

S sup

ho(r), g

I\J0I

and with Applying inequality (6.17) with h(v) = (f''l u@)dt)Un b: *, and then Lemma 6'2 with 7r: at we obtain that

fo*

no{,)u{,)h

With s :

and

ftdntdo' ra / it@)

:

Hence, from Theorem 6.5 with 9(c)

'?i, m;,(")d")iE' holds for any

Opentors on the Cone of Monotone F.nctiorug 313

(1,''av')'^

(1" 'P)a')-'/"

'

h(n): X(0,")(c), r ) 0, in (6.19), we see that the condition C < a is also necsssary. Let us point out that inequality (6'19)

Taking

is a weighted norm inequality for the identity operator defined on monotone functions. Such characterizations appear quite frequently in literature. See also Corollary 6.15 (iv) below-

6.3.

Applications of the Duality Principle

An integral operator Let us consider

raw@04

The upper bound follows after dividine bv

U0

fn1r)a(t)dx)Llp'

a,n

operator T defined by

(r/X') : Et

to*

k@,y)f(y)d,y

(6.20)

314

Wei'ghted

Integrol Operotors on the Cone o! Monotone Frnctiotts

Ineqnlities ol Hody fupc

where lc is a non-negative kernel. In order to characterize the weight functions u and u for which the inequality

\l/c / f@ .-\l/P / f@ gf)q(r)u(n)drl scll fp@)u(r)dr) (l / \ro / \/o

I 1p,e < m holds for all /, 0 < .f l,

(l=

ff",ts)@at)e'

"

(l'

,av,1-o'

Ha is the Hardy averaging operator,

(H"f)(,):!sJof

(6.21)

wu can use the duality principles (6.8) and (6.10). They show that (6.21) is equivalent to the inequality

with

:

Example 6.7. If.T

u(r)d,r:

(fr"s)@):

,{,)u)'/' (6.22)

,

lo"

@"dfu)dv

:

Iu*

I,', U,*

*or.

a

I #9l@-(4 (Ji fn(r)a(z)dx)'t' -

*")*

l,'(1"'*0,* I,* *")*

(6.23)

I,'+

and use (6.10) with g replaced by frg, we get from (6.22) that

",,0 d'iir

(6.24)

then its conjugate is given by

where f i" th" conjugate of T and I is an arbitrory non-negative measurable function. Indeed: If we assume for simplicity that

J,

f4)dt,

Now

(lr* no'1x|ur-t'140,)

foo

315

(1,'*)di*z I,*

*, lo" nl!o,

cllsllo,,u,-",,

I,*

*o'

n*or,

that Ii@"dfu)dy is essentially the sum of the Hardy operator and the adjoint of the Hardy averaging operator. Therefore, if we assume that condition (6.23) is satisfied, then (6.22) is equivalent for our special operator I/o from (6.24) - to proving weight characterizations for which both so

i.e.,

"uo o"iir

1 cllsllo,,u,-", #N*= (J. !n(r)u(r)d.r\'ro -

.

Butthenfor0(/jwehave

sup Jo[* {riltdn@)dr < cllfllp,o, ll?/llc,,: ll9llo,'.r-"' =l which is (6.21). Similarly we can show that (6.21) implies (6.22). In some special cases, weight characterizations for which (6.22) is satisfied are known. Let us consider some examples.

(1,*

t[

nsa') v

sc

n'@)a(x)0,)"o'

(lr* ot'(r)ut-e'(x)or)"

(6.25)

Integrol Operotors on the Cone of Monotone Functiotu Afz

Weighted Inequolities of Hodg Type

316

then 1 < p' < q' 1cr, and hence, by formula (0.9), if and only if

and l/p',

\

(l-u- *o')'' ,o'y-n' (r)u(")tu /

/ f*/ f* (t (/ \Jc

|

\J0

t (lr* "

st'

(r)ut-t'14ar)'/

(6.26)

are satisfied with

f,

v(z):

Jo

/ t' \"/P \l/' r (/, u1t1at) u(r)h)

and hence by formula (0'8)' (6'25) is satisfied

In the

if and only if

I

\r/P' / tt

/ te

y-n'p)u(r)dr) (/o "t'-art.d@)dx) :$ (/,*

81

r/q

:

-llq

\ \r/P / f, <m' ,olo") lJ, "@)o*) :lB lJ, / fr

(6'28)

similarly, formula (0.12) shows that (6.26) is satisfied if and only

Ar := :$

(l'

,o'Y-n'(r)u(dd,)

/ r@

' (/ If

\l/q

{qu(r)dn)

we suppose

1
if

<

oo.

(0.80)

sarne rvay, by formula (0.13), (6.26) is equivalent to

/ f* / f, \r/p' :: (,/, o'r*'(t)u(t)dt) \J"

is finite. But since (1 - s1(1 - q) : 1, we obtain - using (6'27) with V(m) : oo and integrating - that (6'25) is equivalent to

,o

.-

Bs:: ll/ t* ll/ r, ultyatl\-r/p / \ro \Jo

lcp
\J0

\l/"

u@)arl "1t1atl / /

l: i - i:i-i

Now suppose

Then 1 < q'

,/

\r/p

where But again, calculation of thefi.rst inner integral shows that (6.25) is equivalent to

(6.27)

a(t)dt.

\r/p'/ f, v-e'(t)u(qdt) ( l-

(0.2b) is satisfied

/ f@

,. (\/" t-su(t)dt

\r/p

)

o-qu(a)dr

\l/"

)

<

oo

.

(6.s1)

Hence, we have proved the following assertion.

Theorem 6.8. Let L I p,g < oo and, let u,u be uei,ght functions such that V(x) : ff u?)at sotisfies y(*) = a. Then the inequality

/

(1,* "o, <

m.

(6.2e)

is

satisfi,ed,

(i) for

(| l,' y1qo,)o o,)"0 = (I* "

,p1yo1,yo,)"o

for all non-negotiae decreosing functions if onil only i,f < p < { ( oo, max(As, A) < a where Ao,Ar ore giaen

1

by (6.28), (6.29), rcspectiuely; (ii) forl < g < p I 6,ma:c(Bs,Br) by (6.30), (6.31), respectiaely.

<

a

where Bo,Bt ore giaen

318

Weighted Ineqnlities ol HodV

Integnl Operctors on the Cone ol Monotone l\nctiotts 3L9

TW

M The Hardy-Litttewood maximat function in the motivating Let M be defined by (6'4)' Since, as mentioned example (Sec. 6.1),

(M il'Q) = + /.(s)ds' J,

But then

lo'{rdtt)at:

we obtain from Theo-

6-9. Let I 1 p,q 1 q and l.et u'u Corollary "rJiin*"vt"l: V(oo) : a' JJ u(t)di iatisfies M : N(u) -r Aq(u)

be weight functions

Then the moppins

derive the following result' Analogously as in Example 6'7, we can

thot v(m) = m' Theorem 6.10. Let u,v be weight functions such If I < P S S < oo, then the inequalitY

holds

for

::t t

otl

f

>-0,

="(1,*

!p(r)a(r)dr)"o {u'")

f L if orul only if the erpressions

l- 'u'll,

'o/r'

1o1v-ar'lr"1a']' o'\'''

t

f 't'w1'"

and

:g

tl-'u' [/ '{ l-

uetr 1o1v-t1v7nt'taa]'

v-P' (t)u(t)t^'

nlat

o'\'''

\ r/p'

y-o'1x)u(x)dx)

(1,* ll,' s(t)t^id4"

6'8 is sotisfied' is bouniled if orut onty if (i) or (ii) o! Theorern

(/-,,", U,* +*f'

o$)

and by Theorem 6.3 (cf. (6.22)), (6.32) is equivalent to

rem 6.8 immediately the following result'

0,)'''

to"

=

(lr*

"

so'(n)ut-o'(da,)t/

However, we observed in Chap. 2 (see Example 2.7 (iii)) that ln(f) is an Oinarov kernel, i.e. a kernel satisfying conditions (2.25\' (2.26). Hence, applying Theorem 2.10 with p replaced by q', q by t', u by

V-P'a and u by ul-c', it follows that (6.33) holds conditions ofour theorem are satisfied.

if

a"nd

satisfied.

(ii) In Theorems 6.8 and 6.10 we gave weight characterizations for certain Hardy-type operators defined on non-negative decreasing functions. This indicates that also more general Hardy-type operators can be considered, which is indeed the case provided we impose additional conditions on the corresponding kernel. Thus if

(rfl@):

Io',

k(r,y)f(y)d,y, 0StJ,

are finite.

then

Proof'

lf'(r/Xt) : ff *at'

(rg)(si:; lr' s(t)d't'

?

only if the tr

Remark 6.11. (i) An analogous result holds also in the case 1 < q < p ( m. The argument is as above only now we apply Theorem 2.15 to obtain equivalent weight conditions for which (6.33) is

10'\'''

then the adjoint of

(6.33)

has the form

gs)(il:

foo

lo

k(r,s)g(x)dn,

0Sg

320

Integrcl Operators on the Cone o! Monotone Frnctiorts g?.l

Weighted IneEnlities of Hotdg TVpe

are satisfied for any g, g )-0. But weight characterizations for which these estimates are satisfied follow from the Hardy and conjugate Hardy inequalities given in the Introduction. (iii) Up to now we have considered operators on the cone of decreasing functions. However, the main duality theorem Theorem 6.3 - can be applied to obtain an analogous result for non-negative increasing functions (notation: 0 s / t). Flor simplicity, let us consider

and

ft_ Qo)(u)aa Jo

: [" [* Jo Jy

: : lf

ke,fis(t)dtdy

fr r, f' f* k(t,y)s(t)dto, * Jo J, J, J"

!o'

stl

(1,' uu,il*)

o,

*

k(t,y)s(t)dtdy

l,* '$\

(1, *g,fiav) at '

we denote

K(t):

lo'

only the case V(m) * m. A change of variables shows that

-/(")g(r)a"_.

and by Theorem 6.3 we arrive at

u1r,n1or,

If,

f @)s@)a"

o.ir (Jfl fi

@)a(z)d.r)t

JUlr-

then, by (6.21), the inequalitY

,t =" (!,* ttop1a,)''

(1,* ,orrrryd.)''o is satisfied for 0

(l- U'

s / | if and only if (cf. (6.22))

*p1n1qo')'

(|,"t'u')

, (lr* "

"

(!r*

7a1ur-t'

..f,

If, (J;-

?@)o@)a"

fo 1r,1t 1,1a,),

/, )0,)''o'

'{*)a')'/o

st'(r)vt-a'1rr*)''

no'

r-1,

* (1,* (1"" oav,)'' (lo'o{r)0,)-'' ,t

-o'

x'lnlqo'), (1,"'t';") s

-

lp

the inequalities

and

(l- U-

ffr(ilgH#

=suD ",ro o.it (,ff !n(x)a(x)dn)r/p oiir (Jfl f, (il u (!) Slrrn' Now write f@): fG),i@): s(IWz,O(r): ,(*)r-".Then f1.

-o'

'{')*)'''

1r)or)"

(1,- (ff,n@',)

(t,,u,*)-''

(f

(1,*,u',0,)-''

(1,-

n1,vo,)o'

.$.0.)''''

,or*)"'

consequently the analogue of rheorem 6.3 for increasing functions consists in changing the integrals on the right hand side of (6.10) from (0,c) to (a,m). The corresponding analogue of Theorem 6.b holds, too.

IW

Weighted Ineqnlities oI HadY

6.4.

Integml Operators on the Cone ol Monotone

lntroduction The results mentioned in the foregoing section indicate that it is of interest to investigate more general integral inequalities of the type

e)u(t)d&)''o

"

=

(!,*

raw@d,)

(6'34)

:

Io*

k(r,t)f (t)dt

ca'n also be interesting to study inequalities reversed to (6.84), in particular on the cone ofnon-negative is to prodecreasing (or increasing) functions. The aim of this section

with lc(c,y) > 0'Moreover, it

vide such an investigation in the following more general frame.

Let(Mi,tt),i:1,2,denotetwoa-finitemeasurespaces'F\rrt e Mi tet doi(il denote a positive measure on (0, *) and define Q bY

ther, for every

(rif)@),:

Io*

(Ti

il@)

:

Io*

ki@,v) ! (v)d,v

If ?r is of the form (6.36) and T2 is the identity operator, then we obtain (6.34). If ?2 has the form (6.36) a,nd Tr: I, we obtain the reverse to (6.34).

f

fu)do;(il, i:1,2.

t g1f)q(x)dtt,('))"n s c (\"fru, [ e2f)e(r)d'1't'r("))"o / \Jfi,'-Lr'\"^"/

(

A crucial role will be played by the following constant connected with the framework mentioned above: / ^ rl/c"

cr:

\l*t'QtL.')q(')dp(')) . "o (l @)dpr@))"'

cr(p,q,Tr,Tz):: sup

^"{rrx(o,i)P

V. BURENKOV and L. E. PERSSON

[1,2]. decreasing

(1,*a"rw'*) .

(0.35)

6.12. Let Mr: Mz: (0,*) and dprl(c) : u(r)dt'

Example dpr2(r):a(r)dr,whereutuaxeweightfunctions'Theninequality

< pr/Pq - | t q rt / p-t/' and, the constant

is

(lr*

Prool. Applying inequality (6.1?) with

. c (1,* e,rrplopla')'lo

,

O", @,

*)t''

D

:

oo and with p replaced

by p/q we obtain that

/ f@

(6.38)

sharp.

(6.35) takes the form

i.e. we have a general (two operators) weight inequality'

(6.37)

Before we formulate the main result of this section, we shall prove a useful lemma. The proof given here is elementary and the lemma is in fact a special case of some more general results from J. BERGII,

Lemma 6.13. LetU 0. Then every function f on (0,m) satis.fes

We investigate inequalities of the type

(1,* v,rr (c)u(r)ar)'/o

(6.36)

.

An important constant

where now

(r/x,)

823

f. dof (il is the Dirac delta function dr(y) then ?i is the identity operator t. lt doifu) : kj(r,y)dy, then

More General Integral OPerators

(/*t"ll,t

frnctioru

1r/c

I\./o I h(x)dnl /

S

|, [* Q Jo

ro/t-r6n/o@'1dz

324 Weightel Ine4talities oI Hodg

Integml Operctors on the Cone of Monotone Rnctiotts 826

TVpe

(iu) Let 0

holds for any function h, 0 < h J' Hence also

(1,* oor*)o'n

/ f* \l/q / r@ rr/p fq(r)dpl@)l 0 independent of I rf ond only if C1 < a. (u) In oll mses, C : Ct is the sharp unstant-

fo*

and

I hPlc('.s)drq. n1".o1ar"ql s / -Jo

l* \Jo

The choice h(r)

t€

\ P/q

/ roo

(

:

fc$\/(dd)

Proof- our main tool will be the following identity for the operator Q which can be easily proved by using F\rbini,s theorem:

Yields

\ l/c

/ r@

/ f6

sl\"ro l^

( l- 1np1a""ol /

\/o

/

Qil)@)

,

: :

Now we are ready to formulate the main result'

Theorem 6.14- Let C1 be d'efinetl by (6'37) ond let

(

(i)

(

<

a'

t \,lrur"

f

be non'

,,tf)q(r)dtt,("))"0 '"'/

sc(\J''u,' [..Qrf)'t )orr('))"0 /

I\Jo I

\t/c"

fqo)d'p1l""\)

/

rr/P / f ._ \rM: = "(l..Q2f)p(t)d'p'2@))

f 'f ond only if C1 < a' (iii) Let0
a'

:: E(f,t): {y : f(y) > t}.

(i) In this case we just inequality for g )

I

a,nd

use (6.39) and twice Minkowski,s integral

1/p >

1:

(

t (r1f)q(x)d.t',("))"0 '-' '/ \./,,vr,'

: (l-,(lo* t ,*"u,11'1rr)o or,@))'

hotrls with C > 0 independent of

(l.,rr,tr hokk uith C >

(c)apr(r)) ''o

O ind'epenilent of

=

I

"(/-

$ and

r't"l o"*'')''o

only

if Cr <

a'

(6.3e)

{rrruut)@)at

fo*

E(t)

C > 0 independ'ent "f f 'f and only it C1 < a' the inequalitv Leto <ma:c(l,p) S g ( oo anilT11: I' Then

/ r@

I,* (lr,'rut,,tdoi(il)

where

Then the inequolity

hokts with

iirl

foo foo : Jo-Jo I t@)aoifu): I oi(v : !fu)>tl)dt

-

l/P

--\ ynP)dn"e)

whichis(6.38).Thesharpnessoftheestimatefollon,sbyinserting trI I@):x10,"1(r),0< o < oo.

negatiae ond decreosing' I'et0 < p S 1 q

Then the inquatity

ll\ro

norng)a,olo

=

m anilTl-?2: I.

=

Io*

sc,

(l-,t Io*

r,"1t)q@)dur(*r)"

(l-,t

r,urt)p@)duz(d)'

326

Weighted Inequalities of

Haily

Integml Operotors on the Cone o! Monotone Rnct:ioru

Type

, r, (l^"(lo* v,*uorX")at)"

:

:

or,@))''o

:

rl/P

I f

"' lJ-"Qz

f)P (r)d'p2@)

)

need to prove the case As in the proof of Lemma 6.2 we see that

(ii) In view of (i), we only

!

< p 1 g < oo'

pr(E(t))

lq@)d'p1@)l

/ o'o

[(;)

: rt'' (lo

(l^,t

: s, pl

/o

/e

no,o-'

(1,*

wr' (E(t)))'/o

+)''')"'

o-r rnhln(Dor)''o

I-, ( Ir*

l tp - (Tzx

p

(t' 1'

1

"

p')

o

"

(1,* rr,lor,("))" cr,pun

d,1t

r@)

to (i), it suffices to prove the ( l. we use again Lemma 6.13 twice and

case, again according


'n'",'1a1,,@y)lq

=

(1.,Q-',,

,rrrr^,r)t,)f)' our@)' Io*

1., (o Ir* tq-

|

(rrx o(t)q @)dtd.ur(d)'

qr/o

(lr*

u-r"rrgrog61,lar)'/o

1cs'lo (r;l-o'o ,,-o'n

@))"

Therefore,usingLemma6.13again,thistimeforthedecreasingfuncthat tion (T2ys11l)("), q replaced by p, o: 1 and p: l' we obtain

l

(r)

: (o Ir* n-' (l-,(rrxne)q@lar,r@) or)"0

(lr* o-' |-"{rr* " r)e (r)duz@)or) (

)p

^,rr2

(iii) In this

< s c pt

f

,r"a)@)at)o a1",@t)"

: (1., (1,* r',r^,v X')at) q or,r,r)"o

rl/c

r',' =

(l

obtain that

Therefore, using Lemma 6.13 for the decreasing function that and with q:7, s : q and p replaced by p/q, we find

(I \./o

",

(1."(1,* t

assertion for 0

(!r* ,'avr,t"))''o : n''o (1"* n""uD+)

/ r@

",

SZT

:

"'Q Ir*

tn-t

Io*

,,r,r"n +)

1't2(E(t))0,)

: r, (lo* !p(r)d,p2@))"'

.

(iu) We use again Lemma 6.18 and similar arguments as above and obtain that

328

Weighted Ineqaalities of

/ r@

I\Jo I

Hady

Integral Opettors on the Cone of Monotone Frnctior.is

Type

(ii)

\l/c

fq@)dp1@)

|

/

holds

s (ocf fr* to-'roroPPllat) =

:

c t q,

q

-''

o

(f;I

r' -,,

if

/

Io*

ro

-'

:: ::E (/- (I' *''''^*)' u6)a*)'/o

Io*

r,r"

a',1 or)''

tn-t r2(E(t\or)

"

:

1,2) mentioned in


_^.

cs ::

/ fr

if

arul only

. c (lr* rrrrrp1,1*'1a*)'/o

::lt

\1/p

e2f)p(x)u({dr)

\l/s

\-rlP \Pu(r)dr) *(r*/ (/or' *r(''s)dv)
/ r@

rr/p

(/, fq(r)u(r)dr) = t (/. fP(r)u(r)dr) holds if and only if \r/q I r, \ -llp ::lE / f, co

The unstont C

if

/ f@

r = (/,

Thenthe inequality

u(o)dn :lB (/, )

/ f@

be a non-negotiue decreas'ing function ond u,u weight functions on (0, oo). (i) Let0 < p S L S q < m. Then the inquality

Corollary 6.1.5. Let f

holds

(I' ,@)da)-"o . *.

(/, fq(r)u(x)da) hokls if and only if

.

For the special cases of M;, T; and pi (i Example 6.12, we obtain the following result:

Let 0

/ f*

(u) We observe that for the decreasing function f (") : X10,"1(c) with any r ) 0, we have equality in each of the cases above' Thus, (i) - (iv) are proved and C: Cr is the sharp constant in all cases'o

(lr* c,ff(r)u@)ax)'/o

Thentheinequality

if

o

o

: r, (lo* fe(r)d,2@))'''

ond only

co

(iii)

", Q

0
/ f*._ \l/c / f@ \l/p (/, (r1f)q(t)u(r)ar) = " (./. fp(x)u(r)tu)

: (n 6-1 11r(E(t))or)'' Ir*

/

Let

:

,@)0,) \J, "@)0,) \J,

Co is

<

oo.

in all cases the best possible.

Remark 6.16. (r) If we choose in Corollary O.lS (i) V,l)@) : It t@at : g(r), i : 1,2, then g is concave decreasing, and we obtain <

oo.

a'n imbedding spaces.

of concave decreasing functions in weighted Lebesgue

330

Weightel Ineqrolities ol Hardy Ilpe

(ii) The result of Corollary

Integnl Operztors on the Cone ol Monotone

6.15

(iv) was derived by a different

holds if and only

method in Remark 6.6.

Example 6.17. If

we choose in Corollary 6.15 the kernels

ki(a,y) re

{

is the Hardy (averaging) operator,

:! .[

(H"!)(x)

and we obtain for functions .f, 0 S (z) Let 0

U- (; l,'

1

::

sup r>0

(ii) Let 0 < p <

(/holds

(; l,'

rroor)o

if and only if

(/-

I

results:

.

"

Jo,lf

,@)a*)"'

(I'

. c(l- (; l,'

,@d,)-''o

(r)lero(a)dr

:

llf

lPo,.

( *),

is bounded if and only if the function tu belongs Muckenhoupt class A, (i.e., u is such that

(1,* raw@dr)

(*t" (r,i))' "o '= :lB (iii) Let0 < p S Q ( @, 1 < g < m. Then (1,* raw(s)ds)'to

f


u@)a,)"'

,t"t*)-"o. *.

operators on U. However, the mapping properties of M were also studied in weighted U-sp:rces, and in fact it was shown by B. MUCKENHOUPT [2] (see also [3]) that M : Uga) + I?(w),p > I (where we now have in mind functions defined on IRN so that f e I](w) mea,ns that

I r(,ildv) ,6va*)'/'

q

(.i,, (r, i))'

Comments and Remarks

(Jfl (min (r, l))q u(r)ac)'/q <m. (Jfl (min (r,;))o u@)ar)t/P

1 and p

'av')"' (I*

6.6.1. The boundedness of the Hardy-Littlewood maximal operator M (see (6.4)) on U, p ) 1, had a considerable significance for the development of analysis, specifically harmonic analysis. one of the reasons is that a large class of convolution operators are dominated by the Hardy-Littlewood maximal operator and hence the boundedness of M on u, p 2 l, implies the boundedness of the class of convolution

rtoou)n u14a')'/o

if and onlv if Cg

[" f(ildv,

Jo

< g < oo. Then the inequality

=" (I* (: holds

6.5.

/ |, the following

(1,'

if

The constant c : cs is in all cases the best possible. Note that for the case (ai) a more general result was stated in Theorem 6.g but without the sharp consta,nt.

ki@,y): it for 03y'-x, ki@'Y) : o for a)t' then

"o

::lt

Rnctiolr g:tl

. *.

roou)o ,@)a,)'/o

c:

s;n

(h l;,r(ddt)''' (h

to the secalled

fo,,-o'1,1a,)'/o'

.*

where Q are cubes in RN with edges parallel to the coordinate axes). operators which are ma-

It follows from this result that convolution jorized by M are also bounded on lp(w).

6.6.2. with the introduction, popularity and use of weighted Lorentz spaces (see (6.1)) in the early fifties and sixties the question a.rose whether the weight characterization of the maximal operator M on

332

Weighted Ineqtalities of

Hodg

Integral Operotors on the Cone of Monotone

Type

U(.)

would carry over to a corresponding result for the weighted Lorentz spaces np(tr). (For u.r : 1 this is trivial!). Since the spacc Ap(u) are defined in terms of the equimeasurable decreasing rearrangement f * of f , it became necessary to relate this quantity with M/ (namely with (Mf).).In fact,

(Mf).(r)

*: I,

f. (t)dt

.

(6.40)

The estimate j in (6.40) was established by F. RJESZ lll as a consequence of his nice sunri,se lemma. Moreover, C. HERZ [1] (under the influence of a paper of E. M. STEIN) proved the inequality J in (6.40). Some further information about these inequalities (and about the corresponding inequalities in terms of the distribution function of M t) can be found in the paper I. ASEKRITOVA, N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [1]. In view of (6.40), it is clear that M Ap(u) -+ Aq(r) is bounded if ' and only if the Hardy averaging operator fI" defined on decreasing functions is bounded ftom IP(u) into lp(u). In 1990, M. ARIITO and B. MUCKENHOUPT [1] characterized the weights u : r.r for which l,t : ltp(u) -+ Ap(u) is bounded (p > 1), or equivalently, for which the operator llo defined on decreasing functions is bounded on I'o(u). Shortly after the result mentioned, E. SAWYER [3] established an explicit duality theorem for weighted .LP spaces on decreasing functions. This duality theorem was used to show that the mapping prop erties ofan operator defined on decreasing functions are equivalent to the mappirrg properties of an operator defined on orbitrory functions but with different weights. Hence he was able to characterize weights u, u for which ? : Ap(u) -+ Aq(u) with 1 1 p,8 1 oo is bounded, in particular, if ? is the Hardy-Littlewood ma:rimal operator M. In this chapter, we have proved Sawyer's duality theorem (see Theorem 6.3), but our proof follows closely that of V. D. STEPANOV [3] which seems to be more elementary. A multidimensional version of the duality principle was recently proved by S. BARZA' H. P. HEINIG and L. E. PERSSON [U.

Rnctiorc

333

6.6.3. The case 0 < q < p, p ) 1, of Theorem 6.8 is also known and is due to G. SINNAMON [1,3], V. D. STEPANOV [B] and others. 6.6.4. The duality theorem extends in a natural way to inaeasing functions so that weight characterizations for operators defined on increasing functions can also be given, see Remark O.1l (iii).

6.6.5. Among spaces related to the Lorentz spaces Ap(r), let

us

mention the spaces

rP(-): -fE

roo,)' ,61a,)"". {r, (l- (; 1,"

-}

)

.f-(u), it is clear that fP(ru) c Ap(u). More over, if ut € Ap, then Ap(ro) C fp(ur), p ) l, so that for such weights and p ) 1, these spaces are equivalent. A d,uality theorem for funcSince

f-JoI

f.Q)at

tions in such spaces has been obtained by M. L. GOL,DMAN [1] and

by M. L. GOL'DMAN, H. P. HEINIG and V. D. STEPANOV [1] with subsequent weight characterizations for the ma>rimal operator and Hilbert transform in these spaces.

6.6.6. Inequality (6.17) (cf. also (6.38)) was probably first discovered by G. G. LORENTZ [1, p. 39]; various other proofs can be found in literature, see e.g. J. BERGH, V. I. BURENKOV and L. E. PERSSON [U, H. P. HEINIG and L. MALIGRANDA [2], V. c. MAZ,JA [lJ and E. M. STEIN and c. WEISS [U.

6.6.7. The proof of Theorem 6.14 is a simplified form of that presented in S. BARZA, L. E. PERSSON and J. SORIA [l]. Let us note that (r) this proof has the advantage that it can be ca"rried over to the multidimensional ca.se where, in comparison to the one.dimensional case, some new problems appear - see the paper just mentioned and the Ph.D. thesis of S. BARZA [1J; (ii) special cases of Theorem 6.14 have been proved (in other ways) by several authors; see M. J. CARRO and J, SORIA [1,2];

334

Weighted Ineqnlit:i.es of Hady Tlpe

H. P. HEINIG and L. MALIGRANDA [2]; Q. LAI [2]' L. MALIGRANDA [1] and E. A. MYASNIKOV' L. E. PERSSON and v. D. STEPANOV [1]; (iii) for the case q > p fairly little is knownl however, for the one dimensional case, V. D. STEPANOV [4] proved a result corresponding to Theorem 6.14 (iv), and this result was generalized in a multidimensional setting in S. BARZA, L. E. PERSSON and v. D. STEPANOV [1].

References

6.6.8. As far as it concerns multidimensional generalizations, we refet to the Ph.D. thesis by S. BARZA [1] where also a fairly complete list of references and some open questions can be found. 6.6.9. Extensions of the results mentioned, e.g. to modular inequalities and weighted Orlicz function and sequence spaces, were given by H. P. HEINIG and A. KUFNER [2]; H. P. HEINIG and L. MALIGRANDA [1] a,nd Q. LAI [1,3]. The following three books carr serve as standard reference sources for integral inequalities (with a,nd without weights). They are mentioned in the text in the abbreviated forms [HLpl, [MpF] and [OK]:

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[MpF] MITRINOVIC, D.S., pEeARJe, J.E. and FINK, A. M., In_ equalities inaoluing lunctions and their integrols and d,eriuotiaes, Kluwer Academic Publishers Group, 1991. MR 93m:26036

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Index

356 Index G general Hardy-type operator 77 geometric mean operator 45 Green function 167

P6lya condition 166 principle

Young 96 intermediate Epa.e 275 interpolation 275ff

duality 303 heuristic 239

K H

kernel, Oina^rov 77

Knopp inequality 42

Hardy inequality conjugate 12 classical cliscrete

l, 2 l, 5l

higher order 23, 165ff.

in diFerential form 2l in modern form 3 limiting case 42

Lebesgue epace, weighted

1l

level function 110 Iimiting case of Ha^rdy inequality 42

weak type inequality 115 weight xiii

s Sobolev inequality 55

indefinite 58 weighted Lebesgue space l1 norm inequality 12, 65 ff. Sobolev space 22 well-determined class 200

Weyl fractional integral operator 79

weighted 22

M

space

multiple 53

matrix, incidence

overdetermined 199

mar
with indefinite weight 58 Hardy-Knopp inequality 57 Hardy operator l1 averaging 45, 279

166

modular inequality 6l moredimensional Hardy inequality 54, 60 moving averaging Hardy operator 125 multiple Hardy inequality 53

12

moving averaging 125

Hardy-Steklov operator conjugate 134

w

space 35

moredimensional 54, 60

conjugate

R

L

fractional order 23, 126,246f1.

standard couple 221 Steklov operator 125 strong type inequality ll5 symmetric boundary value problem l8Z

rearrangement, decreasing 302

Riemann-Liouville operator 79 Robin-type boundary conditions 192

122

N normalizing function 137

moredimensional 155

Hardy-type operator 77 heuristic principle 239 higher order Hardy inequality 165ff.

o Oinarov kernel 77 operator

arithmetic mean 45 geometric mean 45

I

Hardy

incidence matrix 160 indefinite weight 58

ll

averaging 45, 279 conjugate 12

inequality Carleman 42

Hardy-Steklov 122

Fliedrichs 55 Hardy eee Hardy inequality Hardy-Knopp 57 Knopp 42 modular 6l

conjugate 134 moredimensional 155

Poincar6 55 Sobolev 55 Btrong typc i15 weak type 115 weighted notm 12, 65ff.

with indefinite weight 58

Hardy-type 77 Riemann-Liouville ?9 Steklov 125 Weyl fractional integral 79 overdetermined class 199

P Poincar6 inequality 55

weighted Lebesgue ll weighted Sobolev 22

3$7

Y Young inequality 96

Index

A

D

adjoint boundary value problem lg3 amalgam 52 arithmetic mean operator 4i averaging Hardy operator 45,2Tg

decreasing rearrangement 302 differential form of Ha^rdy inequality

discrete Hardy inequality

l, Sl

dietribution function 302

duality 13, 224

B boundary value problem 166

adjoint

183

Robin-type lg2 eymmetric l8Z

E exponent, critical Bb, 36

F

c Carleman inequality 42 Cauchy problem IEO class

overdetermined l9g

well{etermined 2fi) condition, P6lya 166 conjugate Hardy operator 12

critical exponent 35, 36 couple, ctandard 221

fractional integral operator, Weyt Z9 fractional order Hardy inequality 23, 126,245fr. Fliedrichs inequality 55

function distribution 302 Green 167 level 110

maximal 302 normalizing 1JZ weight

xiii

2l

Ind,er

356 Indet G general Hardy-type operator 77 geometric mean operator 45 Green function 167

Young 96 intermediate space 275

kernel, Oinarov 77

Knopp inequality 42

Hardy inequality conjugate

12

cla.ssical 1, 2


fractional order 23, 126' 246fr. higher order 23, 165ff.

limiting

heuristic 239

21

case 42

moredimensional 54, 60

multiple 53 overdetermined 199 with indefinite weight 58

Hardy-Knopp inequality 57 Hardy operator ll averaging 45, 279

conjugate 12 moving averaging 125 Hardy-Steklov operator 122 conjugate 134 moredimensional 155 Hardy-type operator 77

heuristic principle 239 higher order Hardy inequality 165ff.

Lebesgue space, weighted ll level function 110 limiting case of Ha.rdy inequality 42

standard couple 221 Steklov operator 125 stront type inequality ll5 symmetric boundary value problem 187

R

w

rearrangement, decreasing 302 Riemann-Liouville operator 79 Robin-type boundary conditions 192

wea.k

L

l, 5l

in differential form in modern form 3

duality 303

interpolation 275ff.

K H

P6lya condition 166 principle

s Sobolev inequality 55 space 35

type inequality 115

weight

xiii

indefinite 58 weighted Lebesgue space 11 norm inequality 12, 65 ff. Sobolev sprce 22 well-determined class 2fi)

Weyl fractional integral operator 79

weighted 22

M

space

matrix, incidence

166

ma:
N normalizing function 137

o Oinarov kernel 77 operator

arithmetic mean 45

I

geometric mean 45

Hardy

incidence matrix 160 indefinite weight 58

11

averaging 45,279 conjugate 12

inequality Carleman 42

Hardy-Steklov 122

Fliedrichs 55 Hardy see Hardy inequalitY Hardy-Knopp 57 Knopp 42 modular 6l

conjugate 134 moredimensional 155 Ha.rdy-type 77

Riemann-Liouville 79 Steklov 125 Weyl fractional integral 79

Poincard 55 Sobolev 55

overdetermined class 199

strong typc l15 weak type ll5 weighted norm 12, 65ff. with indefinite weight 58

P Poincar6 inequality 55

weighted Lebesgue Ll weighted Sobolev 22

357

Y Young inequality 96