Hardy-type Inequalities (Pitman Research Notes in Mathematics Series)

B Opic and A Kufner

Czechoslovak Academy of Sciences

Hardy-type inequalities

~ JIll ~

JIll JIll JIll JIIIJ111J111 ~

Longman

Scientific &

~ ' 1 .Lechnlca

Copublished in the United States with John Wilev & Sons. Inc.. New York

Longman Scientific & Technical,

Contents

Longman Group UK Limited, Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world. Copublished in the United States with John Wiley & Sons, [nc., 605 Third Avenue, New York, NY 10158

© Longman Group UK Limited 1990

Introduction

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Chapter 1.

The one-dimensional Hardy inequality

1. Formulation of the problem

5 5

2. Historical remarks

14

3. Proofs of Theorems 1.14 and 1.15

21

4. The method of differential equations

35

5. The limit values of the exponents

45

p. q

First published 1990

6. Functions vanishing at the right endpoint. Examples

65

AMS Subject Classification: 26010, 46E35

7. Compactness of the operators

73

H and H L R 8. The Hardy inequality for functions from ACLR(a.b)

ISSN 0269-3674

9. The Hardy inequality for

British Library Cataloguing in Publication Data

Kufner, Alois, 1934­ Hardy-type inequalities 1. Mathematics. differential inequalities I. Title II. Opic, B. 515.3'6

Library of Congress Cataloging-in.Publication Data

Kutner, Alois. Hardy-type inequalities / A. Kufner and B. Opic. p. cm.-- (Pitman research notes in mathematics series, ISSN 0269-3674; 219) ISBN 0-470-21584-4 (Wiley) 1. Inequalities (Mathematics) I. Opic, B. II. Title. III. Series. 1990 QA295.K87 89-14502 512.9'--dc20 CIP

142

11. Some remarks

161

The N-dimensional Hardy inequality

170

13. Some elementary methods

186

14. The approach via differential equations and formulas

204

15. The Hardy inequality and the class

226

Chapter 3.

A

r

235

Imbedding theorems for weighted Sobolev spaces

243

17. Some general necessary and sufficient conditions

243

18. Imbeddings for the case

249

",p",q
19. Power type weights

269

20. Unbounded domains

287

21. The N-dimensional Hardy inequality

304 315

Appendix 22. Level intervals and level functions

315

327

References

DEC 04 lQIL i.'ih

170

12. Introduction

16. Some special results

Printed and bound in Great Britain

by Biddies Ltd, Guildford and King's Lynn

92

129

10. Higher order derivatives

Chapter 2.

ISBN 0-582-05198-3

0 < q < 1

List of symbols

Preface

Inequalities involving integrals of a function and its derivatives

(In the first column, the number indicates the page where the symbol OCcurs)

appear frequently in various branches of mathematics and represent a useful tool e.g. in the theory and practice of differential equations, in the theory

N

the set of natural numbers

of approximation etc. During the last decades, this topic has been further

:R

the set of real numbers

extended, comprising additionally general measures and in particular measures

:R+

the set of positive numbers

generated by weight functions, i.e. functions measurable and almost every­

2

the set of integers

where positive. The authors decided to collect some of the numerous results

(a,b)

an open interval in

obtained in this 'weighted' direction and to present here an - at least

[a. b J

a closed interval in

partial - survey. The one-dimensional case, treated in Chapter 1. is more

(a, bJ

and 3, is as yet solved only partially. Nevertheless. the authors hope that the reader will find here some interesting and/or useful information. In any case, they would

welcome any remarks and comments concerning the content

as well as the exposition. The text is organized in a usual manner. Each of the 22 sections has its own numeration (both for the formulas and for the subsections). The end of a proof is marked by the sign

0 .

We use the occasion for expressing our gratitude to all who supported

:R N

the N-dimensional Euclidean space

II

an open set (domain) in

all

the boundary of

ri

= II

U all

II II

x

=

(x 1 .x 2 ••••• x N)

=

«,xi)

186

x.~/

187

Pi(ll)

187

Ci(ll;xi)

the projection of a cut of II

~(E)

the N-dimensional Lebesgue measure of the set

improved the text by offering valuable comments; Ms. Hana VALACHOVA who read the manuscipt; Dr. Jifi JARNIK who substantially improved the authors' English;

=

}

(xl'····xi-l'xi+l' ... ,x N) II

N

a point in

R

a point in

R N- 1

onto the hyperplane

xi

0

=

E

~-1 (E)

the (N-l)-dimensional Lebesgue measure of the set

XE

the characteristic function of the set

=

XE(x)

an t a

Ms. Ruzena PACHTOVA who carefully typed the text.

}

bn " b

X • X*

A. K.

the closure of

RN

x

to mention Dr. Petr GURKA who not only contributed his own results but also

B. O.

R

186

us and helped us to bring the manuscript to its final form. Mainly. we want

Prague, November 1989

R

half-open intervals in

[a,b)

or less closed. while the more-dimensional case. dealt with in Chapters 2

:R

the monotone convergence a Banach space and its dual space

73

un ---"-.

73

<;

the symbol of the continuous imbedding

73

C;c;

the symbol of the compact imbedding

U

the weak convergence (in

X

E

E

. ",.

Special sets CO ,l

206

CO,K

269

224

C~i~ (x O)

224

C~i~(xo'oo)

N

!1J

288

._

ACi(D)

187

AC i , L(D)

188

AC.1., R(D)

196

ACi,LR(Q)

170

C~(l6)

170

C~UI)

289

~O, 1

170

308

'2J* 1J 0,1 *

210

CM(Q)

218

C:(Q)

w

220

B(x,r)

233

S(x,r)

231

S(0,1)

226

Q(y, r)

{y E RNj ty - xl < r} N the sphere {YGR ; Iy - xl = r} the unit sphere {y ERN; Iy I = l} N the cube {x E R ; Ix.1. - y.1. I < r, i = 1, 2 , ... ,N}

170

supp u

the support of the function

219

oC+ (x O)

219

oC-(x )

the ball

(

special

5 5

ACL(I) = ACL(a,b)

5

ACR(I)

6

ACLR(I) = ACLR(a,b)

= AcR(a,b)

AC(k-l)(a,b) m,n

classes of absolutely continuous functions on

I

=

(a, b) C R

-

46

LP(a,b)

the Lebesgue space

46

Ilull p, (a, b)

its norm

89

Li oc (a,b)

45

LP(a,b;w)

the weighted Lebesgue space

45

Ilull p , (a,b),w

its norm

206

LP(Q)

the Lebesgue space

206

Ilu lip ,n

its norm

240

Li oc (n)

227

LP(Qjw)

the weighted Lebesgue space

227

Ilu j1 p ,l6,w

its norm

228

LP '*(l6jw)

the weak weighted Lebesgue space

228

Ilull;,n,w

its quasinorm

8

~~::o·-._

classes of continuous functions

O

AC (I) = AC (a, b )

_

M+ (1)

=

- __

classes of absolutely continuous functions on

u

Special classes of functions and function spaces

142

__ ",-_ . __

C~(a,b)

C~(D)

J08

""'_ -c-·-" .•,

187

89 special types of domains in R

~

M+ (a, b)

~_

:=,.

---"_";O:_~~

DC R

N

=

W(I) = W(a. b)

8 170

W(ll)

292

WB(I)

292

238

WC(I) A = AR(RN) r Ar('l) A (RN)

240

S

226 226

1

{vO'vI'···.v N}

89

wI·p(a.b;vO'V I )

89

w~'P(a,b;vO,vI)

91

wi,P(a,b;vO'v I )

89

Ilull I.p, (a,b) ,vO,v 1

206

W1,PUG)

206

Ilulll,p.rI

240

76

H* L

76

H* R

213

div g

205

vu = grad u

211 233

IVu I IVu Ip

a collection of weights

205

tlU

weighted Sobolev spaces and the corresponding

Special constants and functions

classes of weight functions

00

=

.J"=e"ww

IT

norm if

1 < P <

p'

if

P

if

p

p' =

the Sobolev space its norm

p' = -.2..­ p - 1

00

289

a*

WI,p(r/;S)

12

~

=

240

W~,p(r/;S)

66

~

= AR(a,b.w,v,q,p)

240

IluiII,p.rI,S

84

A~

=

241

ll!uI 1l 1 ,p.rI.S

241

WI,p(r/;vO'V ) I

241

W~·P(ll;VO,vl)

241

Ilull I,p,r/,vO,v

241

1II u llll,p,n,v

weighted Sobolev spaces and the corresponding norms (seminorms)

I

1

A~(a,b,w,v,q,p)

v't = A (a.b,w,v,q,p)

12

B = BL(a,b,w,v,q,p) L B = BR(a,b.w.v,q,p) R .6 = $] (a,b,w,v,q,p)

109 12

FL(x) = FL(x;a,b,w,v.q,p)

65

FR(x) =

13

k(p,q) = (I + ;,)

F~(x;a,b.w,v.q,p)

I/q Operators 6

H L

6

H R

=

186

D.

186

S = S.(il) i 1

1

00

~(a.b,w,v,q,p)

119

65

=

D.(Q) 1

f) I

(1 +

lip'

00

or

0 < P < 1

Introduction

These lecture notes are devoted to the inequality

(Jn

(O.l)

!u(x)

Iq

w(x) dx

Uq

J

;;; C

[N .~1 1-

rlCl J cl~. (x)

n

I

P vi (x) dx

1

n

and to some of its modifications and consequences. Here the N-dimensional Euclidean space bers (in fact. we will consider

weight functions, i.e.

R

N

~

p

P

•

1 •

]UP

and

q

is a domain in

are positive real num­

q > 0 ) and

w.v 1 ·····vN

are

n

measurable and positive a.e. in

We are concerned with the question what conditions on the data of our i.e.

pPOblem

on the domain

w.v ••.•• v N 1

weight functions

(0.1) for all functions

K~C~(n) with a constant

u

n.

on the parameters

p. q

and on the

guarantee validity of the inequality

from a certain class

K,

•

independent of the function

C > 0

estimates for the best possible constant

Terminological remarks. (i)

C

u . In some cases,

in (0.1) will be given.

The inequality (0.1) will be called here

the (N-dimensional) Hardy inequality, the reason being the following: In 1920. G. H. HARDY proved an inequality (see Section 1) which can be easily rewritten in the form

[J

(0.2)

1

u(t)

t£-P dt Jl!P ;;;

P I

o where

u'

du/dt

[Jlu'(t)/P

t E dtf/P •

o

We shall call (0.2) the (classical) Hardy inequality;

it holds, e.g .• for all functions p > 1

1£-:+11

and

£

~

u

e

C~(o.oo)

provided

P - 1 .

The inequality (0.2) is a special case of the general (one-dimensional) Hardy

inequality 1

b

b

q

wet) dt l/q '" C

[J lu(t) I

(0.3)

J

a ~

where

a

< b ~

00

and

l'J lu' (t) IP

)

vet) dt l/p

which holds (e.g.) for all functions

wet) •

u €

Coo(~)

Friedrichs inequality) or for all functions

a

vet)

is zero: fnu(x) dx = 0

are weight functions.

o

u

(then it is called the whose mean value over

~

(then it is called the Poincare inequality). There­

fore, the inequality (0.1) could be called also the weighted Friedrichs­ So. we obtain (0.2) from (0.3) taking

p

=

q > 1 •

a

=

0

b

=

00.

w( t)

t

£-p

•

vet) = t£

On the other hand. the inequality (0.3) is again a special case of the in­

However. we will show in Chapter 3 that the inequality (0.1) is closely

equality (0.1) for the case N

=

~

1.

poincare inequality, and it appears under this name sometimes in literature. In connection with the inequality (0.4), also the name weighted Sobolev inequality is used for the inequality (0.1).

cqnnected with the properties of the so-called weighted Sobolev spaces, but

(a. b)

It can be said that the conditions of validity of the inequality (0.3) are

nevertheless we will use the name Hardy inequality for the inequality (0.1)

investigated (almost) completely; we will deal with them in detail in Chap­

as well as for the inequality (0.3) and all inequalities appearing here can

ter 1.

be collected under the common name Hardy-type inequalities.

(ii)

In the literature. the following inequality has been intensively

investigated:

[~IU(X) I'

(0.4)

q

'" c[JlV'u(x) IP dxf/P r2

au au au ) and IV'u Ip = x = (x .x •...• x ) , V'u (--~--. -~--, ... , --~-N 1 2 oX oX oX N 1 2 P . This inequality is known, e.g., as the Sobolev inequality and xi

where

=

'xt

I I~u I

i=1

u E c~(n)

holds (e.g.) for <

provided

n

P

N ,

<

and

1 < q

"'~ N - P

is a bounded domain with Lipschitzian boundary ~,

'worse' domains

the admissible values of the parameter

an; for q

may change

(see, e.g., R. A. ADAMS [ 1 ]). The inequality (0.4) is a special case of the inequality (0.1) if we take w(x)

v,(x) :: 1,

=

1

i

=

1,2, ... ,N

a special case of the inequality (0.4) (for

p

q

2 ) is then the in­

equality

(0.5)

flu(x)

n

2 1

dx '" c

2

flV'u(X)

2 1

dx

r2

2

3

'Chapter 1. The one-dimensional Hardy inequality 1. FORMULATION OF THE PROBLEM

J.t.

In the twenties, G. H. HARDY proved the following inequality: Let

1

<

<

P

00

and denote t

.

( 1.1)

f F(t)

f(x) dx

for

E: < P - 1

J f (x) dx

for'

E: > P - 1

~ ~

=

r

t

whepe

f

is

anon~negative

measurable function defined on

(0,00). Then

00

J FP(t)

(1. 2)

~

tE:-P dt

r

P

C J f (t) t

o

E:

dt

o

with a constant

C

>

independent of

0

f

•

For a proof see, e.g., G. H. HARDY [lJ or G. H. HARDY, J. E. LITTLE­ WOOD, G. POLYA [lJ. The exact value of the constant

C was given in 1926

by E. LANDAU [lJ: it is (1.3)

C = [

P IE: -

1.2. Definition.

n

Let

+ 1

I

I

)P. (a,b) ,

~

a < b

~

+

00

,

and denote by

AC(I) the set of all functions absolutely continuous on every compact subinterval [C,dJ

CI

. Further, denote by ACL(I)

and

ACR(I)

the sets of all functions

u E AC(I)

for which 5

,~~=_~,_:.:'_,~:-~~~~~.~~J4:.~~~~~:~~~~'.~~~~~~~~~~~~ ~~~=~~~~~~~~~~~~.;'4:-~

E:

(1.9)

o

u(x)

lim

(1. 4)

x+a+

p - 1

>

u E= AC (0 , '" ) R

Then

and

00

o,

u(x)

lim

(1.5)

J1U(X) IP xE:-p dx

(1. 10)

x+b­

tion

u

Land

R express the fact that the func­

vanishes on the left and right end of the interval

~ith

0

the constant

Proof.

AC (I) AC R(I) . L If it is necessary to point out the concrete form of the interval

J = ~ , then the inequality (1.10) holds trivially. Therefore, let us

assume that the integral

ACR(a,b) ,

AC(a,b) , ACL(a,b)

o If

(a,b) , we will use the notation ACLR(a,b) .

x

=

f f(t)

dt ,

a

b

(1. 6)

(HRf)(x)

=

f f(t)

J

>

f!U'(t)! tE:/p t-E:!p dt ~

o

o

~

xE:-P dx :0: C f

fP(x) xE: dx

E: < P - 1 , and similarly with the help of the operator

[Jluf(t)I P tE: dtJl!P

E-=-l-_

JI/P[

[f o

t

-E:/(p-I)

dt

J(P-l)!P

~

X(P-I-E:)/(P-l)] (p-I)/p

p - 1 - E:

E:

<

P - 1 , i.e.

- E:/(p-l)

> -

1 ). Consequently,

x

JIU'(t)

H for R

I

dt

< '"

for every

x

> 0 .

o Further

P - 1

x

From the inequality (1.2) we obtain the Hardy inequality (0.2) as au

U(x)

since

Let

1.3. Lemma. E:

<

f u'(t)

dt + u(c)

for

c > 0

c

easy corollary:

(1. 8)

we have, by

x

x

(due to (1.8), we have

0

0

for

J1uf(t)! dt

o

'" (H oP (x) L

x

dt .

H ' we can rewrite (1.2) in the form L

'"

x E (0,00)

is finite. Then for

x

~

x

Using the operator

J

H81der's inequality, that

Further, let US introduce the notation

(HLf)(x)

Jlu' (x) IP xE: dx .

J

n

the intersection

from (1.3).

C

Assume that (1.8) is fulfilled and denote

(i)

ACLR(I)

(1. 7)

C J1uf(x) IP xE: dx

I , respect­

ively.) Finally, denote by

I

~

0

respectively. (So, the indices

E:

and

P -

1 < p < 00,

1

and

u

E: ~ P - 1 , and suppose

c-

u 0+

EO

ACeD,"')

Moreover,

u E ACL(O.oo) , and therefore, we obtain for

that

E ACL(O,"')

OP

7 6

-=---

,~~~~~,;;;-,;;;-;;;-

'----------------~-~,

-~-~~--~~--~-~-~--

f

--

1~1.

Finally, x

If

lu(x)!

dtl

u'(r)

x

~

J1u'(t)! dt

(H

Iu I) (x) L

I

o

o

or

W(I)

I

= (a,b)

00

~~~,~~~-~~---~~~~-~~~~~--~~-----

, (HLf)q(x) w(x) dx

HL

~ CL ,

I , i.e.

the set of all functions I

c W(a,b) . Under what conditions

fP(x) vex) dx

riP

the inequality

1/

b

~ cR[f

fP(x) vex) dXJ

a f E ~(a,b) ?

:L8.' Remark. Each of the problems mentioned, 1.5 as well as 1.7, represents in fact a pair of problems: In Problem 1.5 (1.11) on

the set of all measurable functions non-negative a.e. on

I

.

~o

we consider the inequality

two different classes of functions

different

operators.

u, in Problem 1.7 we consider

Nonetheless, using elementary tools, we can

reduce Problem 1.5 (ii) to Problem 1.5 (i) and similarly <

p,q

<

is there a (finite) constant

Let

00

C > 0

b l/q [Jlu(x)!q w(x) dX)

(loll)

~

v, w S W(a,b) . Under what conditions

of the inequality (1.13) can be reduced to the investigation of the in­

x

b lip

c[Jlu'(x)IP vex) dX)

b b

holds

J[J

for every

u E ACL(a,b) , or

(ii)

for every

u E ACR(a,b) ?

1.6. Example and remark. the inequality (1.11) for t~

=-

(i) p

=-

t

iq f (t) dt J w(x) dx

f(s) dSJq

~(y)

dy

and

b =

00,

wet) = t~-P,

vet)

. Consequently, Problem 1.5 is solved in this special case by Lemma

B

~ere

f fP(x) vex) dx

J fP(y)

a

a

(a, B)

=

(-b,-a) ,

Y ~- (0., B) . Obviously

Analogously as in this lemma, Problem 1.5 can be reduced to a

fey)

=

~(y)

(0., B)

for

in Problem 1.5, since the substitution

dy

~(y)

f(-y) ,

f E ~(a, B) ,

an analogue of (1.12) on

1.3. (ii)

By J[J a a

b

a = 0,

s

a x

The inequality (1.10) is a special case of q

y,

yt~lds

a

(i)

the investigation

equality (1.12). Indeed, the substitution

such that the inequality

a

=

p

.

~(a,b)

1

from

a

holds for every

Further, denote by

Let

HR

f E- M+(a,b)

a

measurable, positive and finite almost everywhere (a.e.) on

1.5. Problem.

and

----

~-~-~~~

such that

rr

f/q

b 1/ [f(HRf)q(x) w(x) dXJ q

(1.13)

v, w

Let

•

C ' C L R

rr

W(a,b)

or

1 < p,q <

there (finite) constants the inequality (i)

(li)

denote by

the set of all weight functions on

~(I)

Let

a bolds for every

o

The case (1.9) can be handled analogously.

For

Problem.

.12)

and (1.10) follows from (1.7) for f=\u'l·

1.4. Definition.

--

~-----~~-~------

(1.6). Let us now formulate this second problem.

u'(t) dt.

o

(ii)

~-

problem concerning inequalities involving the operators

x

u(x)

~--~~~-

= w(-y) ,

~, w G W(a,B)

;(y) = v(-y)

for

and (1.13) reduces to

f , v , w . A similar situation occurs x

=- y

transforms

u E ACL(a,b) 9

8

ing

~

u

into

~(y) = u(-y) , and reduces (1.11) to the correspond­

ACR(a,S) ,

(a,S)

inequality on

~ , ;

for

. itt view of (1.14). Since

, ~ .

x

Ju'(t)

u(x)

1.9. Convention. In accordance with the previous remark, we restrict our­ u E ACL(a,b) , and to the investigation of the inequality (1.12) (i.e.

of

u E ACL(a,b)

Moreover,

H ) for f E ~(a,b) . The following lemma states that these L under certain conditions on the weight function two inequalities are v _ equivalent, which means that Problems 1.5 and 1.7 are in some sense

c G (a,b) .

for every

and, therefore, for

c

u(x)

=

I u'(t)

dt .

a

x

1.10. Lemma. Let

1 < p,q <

00

Let

•

v, wE W(a,b)

lu(x)

and assume

v 1-p ' (t) dt <

I : :;

I

(H

lu' (t) I dt

L

Iu' I) (x)

a

x

atRt (1.11) follows from (1.12) for 00

shown that the best constants

C

f

=

lu'l . Simultaneously, we have

in (1.11) and

a

x E (a,b)

with

p'

=

~l . Then the inequality (1.11) holds

Proof.

(i)

C L

C L

in (1.12) satisfy

C :::; C

L

(l~15)

p -

for every u E ACL(a,b) if and only if the inequality (1.12) holds for every f e ~(a,b) . The best constant C in (1.11) coincides with the best constant

we obtain

Further,

equivalent, too.

for every

a+

~

x

the operator

I

dt + u(c)

c

selves in the sequel to the investigation of the inequality (1.11) only for

(1. 14)

u E AC(a,b) • we have

in (1.12).

Assume that the inequality (1.11) holds for

(it)

u E ACL(a,b) . Let

and denote

fEo W(a,b)

"}

b

J =

Assume that the inequality (1.12) holds and denote, for

I

fP(x) vex) dx .

a

u E AC (a, b) , L

b

J

=

Iff" J =

elk

Ilu'(x)\P vex) dx .

a

If

that

J

then the inequality (1.11) holds trivially. Therefore, assume

00

J

is finite. Then we have by Holder's inequality that for x IIU'(t)! dt

=

x E (a,b)

dt

r [f p

obViously belongs to

n

a

1 /p'

dt ]

I

<

f(t) dt

(H f) (x) L

a

:::;

tio

[fx v l-p , (t)

a

x E (a,b) . Then the function

u(x)

,

a

00

x

~",.

vl -p' (t) dt ) lip

XI l' J l/p' < dt :::; Jl/p [ v -p (t) dt

a

'~ ~:i',-':

x

a

10

I f(t)

" '~f

/

is finite. Similarly as in part (i), Holder's inequality yields

x

~,::r>

Ilu'(t)1 vl/p(t) v-l/p(t) dt:::;

[f lu'(t) \P vet)

;;; } / p

.:it~:'. !\tit:\<every

x

:::;

J

x

a

a

then the inequality (1.12) holds trivially. Therefore, assume

00

u

ACL(a,b) . The inequality (1.11) applied to this func­

yields immediately the inequality (1.12). Simultaneously, we have

.shQwn that the best constants

C

in (1.11) and

C L

in (1.12) satisfy

11

fb b

t[lII

C :"; C , L which together with (1.15) completes the proof.

o

wi')

d']

1/

x q [[ v

' r l p )1/ l-p ' (,) d'] 1/ q] - ' (,) d,

r

v

where 1.11. Remark.

(i)

Analogously, it can be shown that under the assumption

(1. 20)

1 q

r

1

P

b

v J x

(1.16)

1

-p ' (t) dt <

co

for every

u E ACR(a,b)

f E ~(a,b)

inequality (1.13) holds for every

s'

=

s s -

(and that

C

= CR ).

Let

1.14. Theorem.

In the proof of Lemma 1.10, the assumption (1.14) was essential. (1.11)

Our aim is to establish necessary and sufficient conditions on

p ,

(1.22)

under which the Hardy inequality (1.11) holds. The corresponding

assertions will be formulated for the inequality (1.11), but

-

E ACR(a,b)

u

~

w(x) dx

q <

] 1/ q

v, w ~

co

B L

W(a,b) . Then the inequality

b

~

cL[Jlu'(x)I

P v(x) dx ]

1/p

a

u E ACL(a,b)

if and only if

= 5 L (a,b,w,v,q,p) <

co

•

C L

in (1.11) the following

estimate is satisfied: (1·13)

(1.12). The corresponding conditions concerning the inequality (1.13) the inequality (1.11) for

1 .

Moreover, for the best possible constant

in view

we will proceed via Problem 1.7. More­

over, according to Convention 1.9 it suffices to deal with the inequality

i.e.

1 + L s s'

Le.

a

holds for every

of Lemma 1.10 and Remark 1.11

,

1 < p

b. q [ J lu(x) I

without the condition (1.14) and (1.16), respectively. (See Remark 3.7.)

q , v , w

1 '

co

if and only if the

Nevertheless, it can be shown that Problems 1.5 and 1.7 are equivalent also

~

s <

<

Recall that for

,

(1.21)

the inequality (1.11) holds for every

(ii)

x E (a, b)

5

L

~

:"; C L

k(q,p)B

L

'

where

will be summarized in

(1. 24)

Section 6.

(1 + .9.,-) p

k(q,p)

1/q

, 1/p'

(1 + E.-) q

First, we introduce some important auxiliary functions and constants. 1.15. Theorem. 1.13. Notation.

For

1 < p,q <

co

,

v, w E W(a,b)

1/

b

(1.17)

FL(x;a,b,w,v,q,p)

FL(x)

[J w(t) x

(1.18) and (for (1.19)

5 (a,b ,w,v,q,p)

B L p

>

L

q

~ = ~(a,b,w,v,q,p)

sup FL(x) a<x
dt]

x E (a,b)

and

1-p'

(t) dt

] 1/p'

,

(1.11)

v, w E W(a,b) . Then the inequality

co

(J llu(x)

q I

w(x)

] 11 q dx

a

holds for every (1.25)

~ =

1/

b

P :"; cL[Jlu'(x) I v(x) dX]

a

P

a

u E ACL(a,h)

if and only if

AL(a,b,w,v,q,p) <

co

•

Moreover, for the best possible constant

C L

in (1.11) the following

estimate is satisfied: (1. 26)

12

1 < q < P <

b

x

q [J v

Let

denote

q

1/q (~) r

1/' q A

L

:"; C :"; q1/q(p,)1/q'~ L

. 13

1.16. Remark. Theorems 1.14, 1.15 will be proved in Section 3. Let us note that they can be extended, in a modified form, also to the 'limit' values of

q , namely

and

p

1

'" ; we will deal with these special results

or

°.

=

P. R. BEESACK [lJ investigated the equation (2.3) under certain additional

in Section 5. Moreover, an analogue of the assertion of Theorem 1.15 holds

restrictive conditions on the solution (growth conditions for

also for the case

x-+a+) and on the weight functions (differentiability of

O
is not a variational one, although

the inequality (1.11), i.e.

y

if

v); his approach

the equation (2.3) is the Euler-Lagrange

equation for the functional

(see Section 9). However, let us first give some historical remarks concern­ ing

d d p-1 -1 -dx [vex) (~) ] + w(x) yP (x) dx

(2.3)

the inequality (1.12).

b J(y)

f [(y'(x»)P

vex) - yp(x) w(x)] dx .

a

2. HISTORICAL REMARKS

It should be mentioned that P. R. BEE SACK dealt not only with the case p > 1 , but also with

~

Isolated partial results have been appearing in literature from the

p <

°

and even with

°< P < 1 ;

in this last case,

the inequality sign in (2.2) must be reversed.

very beginning. Let us mention here the result of G. A. BLISS [lJ from 1930, who extended the classical Hardy inequality (1.2) to the case for the special value for

(a,b)

(0,"') ,

=

E =

vex)

° . He showed that the inequality -l-q/p' =1

,

w(x)

x

=

x-l-q/p'

dx

] l/q

(1.12) holds

u E ACL(O,"') , too:

;': C

L

[llu' (x) I

P dX]'

Ip

A systematic investigation of

the (generalized) Hardy inequality starting in the late fifties and early six­ connect~d

with the name of P. R. BEESACK. He dealt with the case

y

and on the weights

of K;. TALENTI [lJ, [2J, who considered a little more special

v, w

we~ght

func­

contains the assertion that the solvability of the equation (2.3) is not only sufficient but in a certain sense also necessary for the validity of the inequality (2.2). We will deal with this equivalence in Section 4 for
the more general case

p = q

and showed that the inequality (1.11), i.e. in this case the inequality

fIU(X)

IP w(x) dx ;':

fIU'(X)

2.3. Differential equations; the case

p < q.

The approach mentioned in

Subsection 2.2 was extended to the case

b

b

a

Some of the restrictions on the solution

Moreover, the paper of G. TOMASELLI [lJ implicitly (see his Lemma 2)

2.2. The approach via differential equations.

(2.2)

or D. T. SHUM [lJ.

tions.

° ties was

the Hardy inequality (2.2) as a special case; see, e.g., P. R. BEESACK [2J

were removed by G. TOMASELLI [lJ in 1969. He followed up the earlier papers

'"

00

[flu(x) Iq

BEESACK's approach was extended to a class of inequalities containing 00

, and consequently, the

corresponding inequality (1.11) holds for

(2.1)

1 < P < q <

IP vex) dx

1 < P ;': q <

00

a

by P. GURKA [lJ. He investigated the inequality (1.11), i.e. holds if there exists a positive solution y'

on

(a,b»

y

of the differential equation

(with a positive derivative

b

(2.4)

[Jlu(x)l q a

14

l/q w(x) dx ]

b

;': C [f I u' (x) I p v (x) dx J

1/ P

a

15

-----:---.,---------::-,::-----::

on

-

ACL(a,b)

on

in connection with the differential equation

A~

(2.5)

dx

(a, b)

.

c:---~-----~O_::==_""':'~~--~____:_~_::__::::""~=~___=_~c_=:-~~=="'==.=~~=~~~__.;;,-,,-~-""'=-'"''''''-':;;;;;,:::_-=::-~~'''----__:::===_"''-=~'''''''_':,-,:c:....---:,-~''"'~~~~-~~~~-.,'"'';;,-''''''''':'_~''''''''--=,;.''''-''''' -=~~~==-____=:_~=-~,,;;,:....~----~-"",~'=.-~-=--

-------=---------:-:--

[vq/p(x)

with

y'

~

(-~.Z)

q/p'

dx

] +

AC(a,b) ,

holds for every

w(x) yq/p' (x) = 0

y

and

y'

is closely connected with the constant

C

~5. Comments.

positive. The number

A

>

0

E

u

(i)

.

Obviously, we suppose that all terms appearing in the

-

I

(2.10)

ties, a number of authors dealt with the inequality (2.4)

v

(t) dt <

and

co

(ii)

(almost at the

u

E

ACL(a,b)

Formulas (2.6) and (2.8) are in a certain sense

vex)

coincides with

= q

= A ~(x) w

w

from the formula (2.8), we obtain that

from (2.6). Therefore, we can omit the tildes. A

Assume

v

E W(a,b)

l'

is given and define

[xII' v -p (t)

v -p (x)

w(x)

dt

)-P

Then there exists a constant

I

~

u

E ACL(a,b)

a

(2.8)

x E (a,b) .

such that the inequality

C f 1u '(X) IP vex) dx 1

a

is given and define

~) with the definition of the function

stead of

[f wet)

b flu(x) IP ;(x) dx

in­

from (1.17), we

v, w

according to (2.6) im­

mediately yields co

,

so that our problem is solved by Theorem 1.14. (iv)

Nevertheless, the approach just described

PORTNOV

ppecedes the approach

[IJ in 1964, by F. A. SYSOEVA [IJ in 1965 and in the same year by

dt JP

,

x E

(a,b) .

(v)

1

>

0

such that the inequality

b

c\J1U'(x)

ClJ who published this result later.

In the formulas (2.6) and (2.8), the weight function

pressed in terms of

~

FL(x)

w

give the original (direct) proof of the assertions from 2.4(i) and 2.4(ii):

V. N. SEDOV ClJ and A. KUFNER

b

Then there exists a constant C

16

will be given in Subsection

via Theorem 1.14. Formulas of the type (2.6) have been derived by V. R.

~ E W(a,b)

~(x) = ;1-p(x)

a

co

If we compare the integrals appearing in (2.10) (with

BL(a,b,w,v,p,p) <

x

(2.9)

(iii)

b

flu(x) P w(x) dx

(ii) Assume

q <

it can be easily shown that the choice of C > 0 1

b

holds for every

~

full

sef that the conditions (2.10) are quite natural. Therefore, we will not ,

a

(2.7)

1 < p

w

2.6.

was investigated. The result can be formulated in the following form:

(2.6)

mutually invepse:

A , denoting

description of the formulas for

(i)

xE(a,b).

x

and computing the function

First, the case p

for

00

in (2.4),

othep one so that (2.4)

find formulas expressing the holds, say, for every

v, w

dt <

Multiplying, for instance, the right-hand side in (2.8) by a suitable con­ stant

one of the weight functions

f ;(t)

In the six­

same time but independently) in a way which can be described as follows: Given

in view

b

I-p'

a

2.4. The approach via formulas connecting the weight functions.

-

that

x GURKA's results will be dealt with and slightly improved in Section 4.

....~:=~-~7=:"'.~-~.'-~""""~-""-"'~:"':"'~-:"':'=''''''=_':'''':::~"=:;=--''''''''--:::-:,;===...;:=,--"",=

foregoing formulas are meaningful. In particular, this means of (2.6) and (2.8)

in (2.4).

ACL(a,b)

~--_.:-~~-~~==c"~,.,,...~~=---=

v

and vice versa. A. KUFNER,

following formulas in which both functions

w

is ex­

H. TRIEBEL [IJ gave the

v , ware expressed

simuZtane­

ously in terms of an auxiliary function; namely, P

I

vex) dx

a

17

_______________

==================================== w(t)

eA(t) A' (t) [eA(t) - eA(a)r p

v( t)

e (l-p) A(t)

A

2.7. Comments.

[A' (t)] I-p

, (a,b)

such that

lim A(t) and A(a) is lim A(t) = 00 • Here A(a) t+a+ t+ban admissible value. If we choose v , w in the form (2.11), the inequality

(2.7) holds for every

u (:

2.6. Formulas; the case

p

with the constant

ACL(a,b) ~

Let

q.

~

1 < p

q <

00

,

C

=

and either

f

(ii)

v , w

chosen

~

The formulas (2.11) mentioned in Subsection 2.5 (v) were ex­

tended by P. GURKA [lJ also to the case wet)

1

<

p

~

q <

00

he obtained that

:

eA(t) A' (t) [eA(t) _ e A(a)r 1- q / P '

=

while the function (t) dt <

instead of (2.17). His proof is

see (1. 18) .

(2.18)

W(a,b) ,

I-p'

BL(a,b,w,v,q,p)

[p / (p - 1) ] p

x

(2.12)

pl/q(p,)l/ P '

according to Subsection 2.6,

A'(t) > 0 and

v

The foregoing result is due to P. GURKA [lJ, who in

based on Theorem 1.14; it can be easily shown that for

is a continuously differentiable function on

v €

(i)

fact obtained the constant

(2.11)

where

.....

=_=_=_=_=~~-=-C_====_"""=~~=----=--==_L_~~,===-=''''''~-_=-'''''----''-"_~",_==,=-"=",===--=-...,..._ ~--=-""""---====""",,,,-=,,===",,,=-=-="=_=._

v

is the same as in (2.11).

00

(iii)

a

Formulas (2.13) and (2.15) cover also the result of G. A. BLISS

[lJ mentioned in Subsection 2.1.

and x

(2.13)

CL v 1- p ' (x) p'

w(x)

(f

v

1-'

p (t) dt

J -l-q/p'

equality in the form (2.7), i.e. for

a

for

W(a,b)

rai weights

f w(t)

dt

<

FL(x)

00

v, w ), expressed in terms of the boundedness of the function

from (1.17), appeared probably for the first time in G. TOMASELLI [lJ

(1969). Apart from the condition

x

and

B 1

(2.15)

vex)

q •

Necessary and sufficient conditions for its validity (for the case of gene­

,

b

(2.14)

=

p

x E (a,b) , or

wE

b

(J wet)

P') -p 1-p (qw (x)

dt

L

q JP-l+ P /

<

00

he derived also another necessary and sufficient condition of the form x

x

for

x €

B L

(a,b) . Then the inequality b

(2.16)

(flu(x) I

q

b

l/q w(x) dx ]

~

k(q,p) (Jlu'(x)/P vex) dx ]

a

holds for every (2.17)

k(q,p)

Let us consider the Hardy in-

(i)

2.8. The approach via Theorem 1.14.

a

u E ACL(a,b)

(1 +

~)

P

lip

sup a<x
[f wet) a

x

t

(f

1 p 1 p v - ' (y) dyJP dtJ.[J v - ' (t) dtJ-1 < a

a

He started from the differential equation of the type (2.3) and substantial­ ly used the auxiliary constant t

x

with

l/q

, lip'

(1 + P-)

(2.19)

K

1

P - 1

inf f

sup a<x
1 f (x)

J wet) a

[f(t) +

J v l-p' (y) a

q

where the infimum is taken over all positive measurable functions 18

J

dy P dt

f

on 19

~~c~P1_~:!!I!!!!!~~c!!:,:il!i!n:~~¥:~..~~~~-".-

;;:-_..

(a,b) . We will use an analogous constant for the case

p

:;i

q

..

'

_ .. "

(see Section

who considered the case v , w

(0,00)

(a,b)

w(x)

=

"

..

-

..._:_ -,_C:;,_

.-.-,.

-

[~([x,OO») f/q

sup x>O

._ ..,

,~~~~~.!.~~~U-}~

.... .;:"~,_

_,,_~;:.;~~.___

-- -_.. --

n

[~~*r-p' dt f/pl

<

-;._ ,

-_._--"..__

00

~~

__ .

.--_ __ ._----_. ..

•

o

This result can be found also in V. G. MAZ'JA [1].

and more special pairs of weights

connected by the formula x- P v(x),

'.- .,---_

B L

4 ).

G. TOMASELLI extended the ideas of G. TALENT I [1], [2] (1966, 1967),

_ __ ._-- __

2.9. The case

x E (0,00) •

q

This case has been treated in literature only in

P

<

the way indicated by Theorem 1.15, which was proved by V. G. MAZ'JA [1] in

1979 (again for the case of Problem 1.7 (i». In 1972, B. MUCKENHOUPT [1] published a direct proof of Theorem

(ii)

p = q . Moreover, he considered (for

1.14 for the case

was also investigated by V. G. MAZ'JA

the more general inequality

the inequality (2.22) for

The case of Borel measures, i.e.

(a,b) = (0,00) )

q

<

p,

[1], and the corresponding necessary

and sufficient condition for the validity of the inequality (2.22) reads

d~(x) ~ c

f(HLf)P(x)

(2.20)

o ~

where

f fP(x)

dv(x)

AL- :~~ H[, ([x.~) lJ' {q [I [:~TP'

0

and only if

(iii)

<

00

r = pq/(p - q) .

where sup [ x>O

B L

v*

where

t"' [:~*r-p' dr'

are Borel measures, and showed that this inequality holds if

, v

(2.21)

dt

~(Cx,OO»)]

II

p[f[~~*r-p' dt ] I/P' x

<

00

o

2.10. The case 1 the case

denotes the absolutely continuous part of

v.

1

p,q

<

<

P

<

00

<

0

00,

<

q

1.

<

Up to now we have dealt only with

Recently, G. SINNAMON [1] investigated Problem 1.7 see Section 9.

alfo for the case mentioned above

For the case 1 < P

~

q <

00 ,

3. PROOFS OF THEOREMS 1.14 AND 1.15

the first proof of Theorem 1.14 was published in 1978 by J. S. BRADLEY [1]. He considered the interval

(0,00)

Let us start with an auxiliary assertion.

and, similarly as other authors mentioned

above, investigated in fact Problem 1.7 (i). One year later, the same result was published independently by V. G. MAZ'JA [1] and V. M. KOKILASHVILI [1].

3.1.

The latter author considered also the corresponding analogue of the in­

integpal inequality b

equality (2.20), i.e.

(2.22)

(f o

(H f) q (x) L

d~(x) f/q ~ C

a

where the necessary and sufficient condition reads as follows:

d

[J [J

(3.1)

lip fP(x) dv(x) ]

,

o

a

c

K(x,y) dy

r

J

dx

J l/r

d

~

b

~

1

(for

r

=

1I r

f [f Kr(x,y) dx c

J

dy

a

holds for every non-negative measurable function r

The Minkowski

A modification of the Minkowski integral inequality.

K

on

(a,b) x (c,d)

and

1 , we have the equality sign since (3.1) is in fact a

consequence of the Fubini theorem)

-

see, e.g., G. H. HARDY, J. E. LITTLE­

WOOD, G. P6LYA [1] (Theorem 202) or N. DUNFORD, J. T. SCHWARTZ [1] (Chap. VI,

20

21

Section 11). We will use (3.1) in the following special form b

x

[f ~ (x) [f ~ (y)

(3.2)

a

dy

r

b

1/ dX]

~

r

a

I

b

~(y)

[f

a

x

f

1/ ¢

(x) dX]

r dy

I

J [Ja

a

1

I

v -p (y) dy

)-1/S

1

I

v -p (t) dt

a

y

x

¢, ~ E ~(a,b)

where

t

x

1

I

h- P (t) v -p (t) dt

s

=-l

s

[I

v

l-p '

(y) dy

)(S-l)/S

__s

h(s-l)p' (x)

s ­

1

a

In what follows, we will use the notation from Section 1, in particu­

and consequently

lar the constants and functions introduced in Subsection 1.13. 1 < P ~ q < 00

Let

3.2. Lemma.

and

v, w €

[

W(a,b) . Assume that the num­

ber

BL = BL(a,b,w,v,q,p) from (1.18) is finite. Then the inequality (1.11) holds for every u E ACL(a,b) and the best possible constant c in (1.11) L ~

C L

with

L

The assumption

V

f(t) dt)

B L

< 00

implies that the integral

w(x) dx

JP / q

~

a

b

(3.5)

II

b

x

a

a

/

x

[I ( I

-p (y) dy

a

:;;

J

~ (

t E (a,b) . Consequently, the function

S

s - 1

)

pip I

I

b

fP(t) v(t) hP(t) [f h(s-l)q(x) w(x) dx ]

f l-p' Jl!(PIS) (y) dy [J v

s

From the definition of the number

B L

see (l.18) and (l.17)

o Let

b

is a fixed number from h(t)

<

f E ~(a,b)

h(s-l)q(x)

satisfies

L

[I

we have -(S-l)/S

w(y) dy ]

x

f

f(t) dt

I

1 p f(t) v / (t) h(t) h- 1 (t) v- 1 / p (t) dt

b

b

x

~

h(s-l)q(x) w(x) dx

[f fP(t)

b

~ B~S-l)q/s f [f

t

t

a

-1+1/S w(y) dy

v(t) hP(t) dt l!P [xI h- P I (t) v 1 -p I (t) dt Jl/ J a

Pl

L

w(x) dx

J

x

b

s B(s-l)q/s

x

a

dt.

and hence

a

~

[hs(x)] (s...,l)q/s :;; B(s-l)q/s

t E (a,b) .

Then Holder's inequality yields

x

I

for every

< 00

(1,00)

P/q

t

a

a

where

r =

a

t

h(t)

q

p/q f (t) d t ) q w(x) dx

b

is finite for every

(3.4)

q

q/p ), we obtain

t

I a

a

xI

Estimating the right-hand side with help of the inequality (3.2) (for

from (1.24).

k(q,p)

Proof·

k(q,p)B

(

P ~ (s ~ 1) p/p' [I [I fP(t) v(t) hP(t) dt ]q p h(s-l)q(x) w(x) dx J /

satisfies the following estimate: (3.3)

~J

[I

,

w(y) dyJ

1/s .

t

This estimate together with (1.17) and (1.18) implies

Further, we have 22

23

b

[J

b

h (s-l)q (x) w(x) dx ]P/q ~ sp/q Bi S- 1 )p/S

[J w(y)

t

::;;

t t

<

= s

[BL(f 1

p/q B(S-l)P/S

L

)-1/P'J P/

ity

B L

(3.7)

= sP / q BE h-P(t)

a

Proof.

(3.6)

and

q

, 1/q

wet) dt J

[ J ( f(t) dt) a a

(3.8)

~

s

1

l/q

UfP(t)

BL

vet) hP(t) h-P(t) dt ]

l/ P

(3.9)

a

~

C . L

Ja v

1-p'

(t) dt

<

l/p

[J

b 1/ [f(HLf)q(x) w(x) dX) q

b 1/ cL[J fP(x) vex) dX] P

~

a

holds for every

fP(t) vet) dt ]

a

sl/ q

(

a

s - 1

s

was arbitrary,

s

>

b

~

1 , and since

k(q,p) ,

[J w(x)

(f

dX)

[J (f f(t) a

0.10)

1/

dt)

w(x) dx

] 1/q

a

~

[J w(x)

dX)

q

f [f

f(t) dtr w(x) dx

f(t) dtf

b

i;

[f

f (x)

dX)

k(q,p) BLU fP(t) vet) dt ] a

.

f E ~(a,b)

for every

f E ~(a,b)

a

f(x) =

{

:

for

x

for

xc [i;,b)

part of Theorem

i;

1.14. Let us point out that we approached Problem 1.5 via Problem 1.7; in

f

Lemma 3.2 we proved that if

a

24

BL

<

00

,

then Problem 1.7 (i) has a solution.

E

(a,E;)

(x)

This function obviously belongs to 'if'

~ cL(f

Let us take

1-p I

and the assertions of Lemma 3.2 follow from Lemma 1.10.

In Lemma 3.2 we in fact proved the

l/ P fP(x) vex) dx )

l/p

Thus, we have shown that the inequality (1.12) holds for every

3.3. Remark.

~

a

a

i;

b q

x

This estimate together with (3.9) implies

we finally conclude from (3.6) x

be fixed. Then

a

b

b

(a,b)

i;

i;

inf g(s) = g(l + 3,) s>l P

~

b

i;

_S_)1/p

i; E:

Let

f(HLf)q(X) w(x) QX I

Here

f E ~(a,b)

b

where =

x E (a,b) .

for every

00

a

b

g(s)

cL for every u E ACL(a,b) .

Then it follows from Lemma 1.10 that the inequality (1.12), i. e. b

~ ( _S_) 1/p I

g(s) B L

W(a,b) • Assume that the inequal­

v, w €

x

xJ

S -

00

Assume in addition that

(i)

and from (3.5) we have b

1 < p,q <

(1.11) holds with the (finite) constant

Then

S

v -p I (y) dy

Let

3.4. Lemma.

P/(qS) dy ]

~(a,b)

, we have

i;

f(x) dx =

f

1 p' v (x) dx

a

and moreover 25

o <

[J

fP(x) vex) dX)

1

[J

p

I

v -p (x) dx

)

lip

<

ro

•

The last integral is finite due to (3.8) and positive since

Further, for a.e.

v

~

W(a,b) .

b

-

~

x E (a,b)

(3.17)

lim v

V 1-p

I

(x) dx

JlI

n 1-

1

I

Let

n

I

p (x)

=

pl

and the monotone convergence theorem together with (3.16) yields that

C . L

$

be a general function from

v

vex) + ~(1 + x 2/ (p/-l»)

Obviously

v E W(a,b) , too, and for

=

b

a

I

$

fv a

f

we have

1-

n

I

P (t) d t

~

P '-1

)J p' -1

J p/-l

I

1

<

ro

If we use Lemma 3.4 for

$

v (x)

for

n

x E (a,b)

and

only if the number 3.6. Comments.

Let

US

3.2

-

n E :N ,

(3. 15)

[flu(x)

I

w(x) dXJl/

a

for every

1/

b

q $

[f Iu

C L

I

$

-

C

L

a

p

$

o

q, we have in fact proved the 'only

B L

from (1.18) is finite.

mention that the proof given here is

(x) Ip vn (x) dx J

-

especially as concerns Lemma

a modification of the former proofs given by B. MUCKENHOUPT p

q ) and J. S. BRADLEY [l J, V. G. MAZ' JA

=

k(q,p)

[l J

[IJ

. An important

from (1.24). Instead of the in­

equality (3.3), the authors establish the following estimates (3.18)

q

)l/ pl

Lemmas 3.2 and 3.4 provide a complete proof of Theorem 1.14.

---r

we obtain b

I

-p (x) dx

lated also in the following form: Problem 1.5 (i) has a solution if and

role is played by the constant vex)

1

1.14 is proved completely. Let us mention that this theorem could be formu­

(the case

Using the inequality (1.11) and the estimate (3.14)

V

(a,b) , which immediately implies the estimate (3.7).

n

dt + f n dt + --2--- dt t I t -1

4n P -

[f

p

if' part of Theorem 1.14, and consequently, in view of Remark 3.3, Theorem

dt 1 2 I (p I -1 ) [v(t)+-(l+t a n 1

sE

1..:5. Remark.

J

=

w(x) dx J

~

for every

b

- 1P, -1 $

n E:N define

x E (a, b) .

x E (a,b)

n

v I-p (t) d t n

W(a,b) . For

[f

=

~

1I

b

vn(x)

xI

I

a

(3.12)

(3.13)

we have

n+l 1- , v p (x)

FL(~;a,b.w,v,q,p)

(ii)

cf. (3.11).

v -p (x) $ v -p (x)

n-+ ro

~

[f

dXJ l/q

w(x)

1

o$

Consequently, from (3.10) we have

[f

(a,b)

a

a

(3.11)

SE

holds for every

~

II

b

p

a

C L

$

pl/ q (p/)l/ p l B

L

(J. S. BRADLEY [1], V. S. KOKILASHVILI [l

J and - by another method ­

P. GURKA [IJ) or

u E ACL(a,b) . (3.19)

Since (3.13) is nothing else than the condition (3.8) for the function v n ' we obtain from part (i) of our proof that in view of (3.15)

C L

$

ql/ q (q/)l/ pl B

L

(V. G. MAZ'JA [1]). All constants mentioned above are closely connected via the function

b

(3.16)

[f w(x) ~

26

dX)

1I

S

q

[J a

V

l-p' (x) dx )l/

n

pl $

-

C

L

(3.20)

g(s)

sl/q( _S_)l

/pl

s - 1 27

-----.---------­

introduced in the proof of Lemma 3.2. For the constant from (3.18) we have

'?1~"'; - ----~--

---,---=---~~-~~:;;;;;:---=:--==--=----=--=:;;;;=.----::-~-:::::=---=::;;:;:-::=--==.=~ ~~~---:--=:----.--=~-==:;;;;:-~---==---=:

Lemma 1.10, we have (3.15) and due to the proof of Lemma 3.4, the estimate B ~ C which implies that B < ro • L L L Now, we are ready to give in two lemmas the proof of Theorem 1.15. The

(3.16), too. Consequently

pUq(p') 1/p'

g (p)

,

while

first concerns the 'if' part. q 1/ q(q') 1/' p = g(q) ;

both constants are greater than the constant

k(q,p)

from (3.3) defined as

3,)

= inf g(s)

s>l

p

provided 1

<

P

<

q

<

00

•

q

<

P <

<

and

ro

v, w E W(a,b) . Assume that the num­

~

L

satisfies the following estimate:

Consequently, the constant

k(q,p)

leads to the

C ; this estimate is due to B. OPIC and is L published here for the first time.

up to now best estimate of

(3.21)

Proof. For

1

= ~(a,b,w,v,q,p) from (1.19) is finite. Then the inequality (1.11) holds for every u E ACL(a,b) and the best possible constant C in (1.11)

ber g(l +

Let

3.8. Lemma.

p = q , all three estimates

(3.3), (3.18) and (3.19)

-

coin­

C L in (1.12). This follows from the fact that the right-hand side in any of

= t€

~ <

The assumption

ro

implies that the integral

t

cide and, moreover, give the best possible estimate for the constant these estimates is (for p = q and for the special weights vet) €-p wet) = t ) equal to the best possible value C from (1.3).

C ~ q1/ q (p,)1/ q 'A L L

1

f v -p

' (y) dy

a

is finite for every to show

t €

(a,b) • In accordance with Lemma 1.10, it suffices f E

that the inequality (1.12) holds for every

~(a,b)

the estimate (3.21) holds for the best possible constant For such

1.10 holds without the assumption (1.14). Indeed, according to Remarks 3.5 and 3.3 we have to show that

the following implication holds:

Problem 1.7 (i) is solvable (with a finite constant

B

L

Let us prove it. Suppose

v E W(a,b)

<

b q

[J (HLOq(x) w(x) dX)

n

28

fP(x) vn(x) dx )

f E ~(a,b)

(with the same constant

fulfil the condition (1.14)

-

C ). The functions L see (3.13). Therefore, according to

a

y

[f

fey) dy

r

w(x) dx

a 1

f(t) dtr-

fey) d Y ] w(x) dx ,

a

b

(3.22)

1/p

a

a

holds for every

~cL[J

I [I

and the Fubini theorem yields

n

b

x

I [f a

is general (not necessarily satis­

from (3.12), we obtain in view of (3.14) that the inequality

x

a

ro

and Problem 1.7 (i) is solvable with C < ro • This means L that the inequality (1.12) [i.e. (3.9) ] holds. Using the functions v

1/q

w(x) dx =

a

fying (1.14»

b

b

J(HLf)q(x)

C L

in (1.12).

f , we have

b

\~

v

C L

The proof of Lemma 3.4 implies that the assertion of Lemma

3.7. Remark.

and that

I

(H f) q (x) w(x) dx L

a

q

f [I a

a

b

q-1 f(t) dt }

fey)

[I

w(x) dX] dy .

y

Rewriting the right-hand side in an appropriate form and using Holder's in­ equaiity for the product of three functions (with exponents and

p/(p - q) ,

p

p/(q - 1) ) we obtain

29

inequality (1.11) holds with the (finite) constant u E ACL(a,b) . Then

b

(3.23)

q (p _ 1) (l-q) / p

f(HLOq(x) w(x) dx a b

b

Y

f[fW(x) a

dX]

a

y

[HLf(Y~:

• (p - l)(q-l)/p

Y

-1

J v l - p' (t)

dt]q

v(l- p ')(q-l)/P(y) dy S

where r

1

1

q

p

b

Ai[f

l)O-q)/p

Proof.

_

(3.25 )

for every

] (q-1) /p

)

I

p, q

are arbitrary numbers from

(1,00»

and consequently x

wet) dt < 00 ,

I

v

l-p I

(t)dt
a

c

x

(a,b) . Let us choose two sequences an + a,

numbers such that

a

b

n

b

t

and for

{a}, n

n EN,

(b} n

x E (a,b)

of real let us

introduce auxiliary functions

where the new weight function

w is given by

y ;(y) = (p -

1)

U

V

1

I

-p (t) dt

]

-p

v

1-

(3.26) I

P (y) ,

y E (a, b)

f

n

(x)

.

b

a

However, for this function

wand for

v

[

we have

[f(HLOP(Y)~(Y)dY]

q = p ),

b

1/

P S k(p,p)

[J fP (x) v

lip (x) dx

[I(HLf)q(X) w(x) dX]

k(p,p)

b

ql

AL[I fP(x) vex) dx

1

<

J

o q

<

p

<

00

]r/(pql)

1

I

v -p (x) X(an,bn)(x)

(a ,b ) , and hence n

I f~(x)

n

vex) dx > 0 .

v, w t. W(a,b).

Assume that the

An

=

[I

Uwet)

rJ v 1-p

dt J r/q ,

an x

I

(t) dt

r/

q I

v 1-p I (x) dx f/r

an

we obtain in view of (3.25)

0.29)

and

a.e. in

bn b

lip

a

a

which implies (3.21).

n

pl/P(pl)l/pl (3.28)

q q S ql/ (pl)l/

I

a

J

we arrive at the inequality l/

1

v -p (t) dt

If we define

Using (3.23), the last estimate and the fact that

b

[xI

b

(3.27)

a

a

]r/(pq)

an

f (x) > 0

Obviously,

and consequently, according to Theorem 1.14 (with b

I

wet) dt

x

BL(a,b,~,v,p,p) S 1

30

<

x

[f (HLOP(y) w(y) dy

3.9. Lemma. Let

00

for every

~ = CL

b

•

a b

l/' q

«

B S C L L

lip

fP(y) v(y) dy ]

r

Due to Lemma 3.4 (where

we have

a

S q(p -

q

v l - p' (t) dtr-l vO-pl)(p-q)/p(y) f(y) v l / p (y) •

[f

l/q (~)

(3.24)

CL

[ b r A S J wet) dt n an

]'/q bJn [fx v 1-p an

I

/ I (t) dt \r q v 1-p I (x) dx j

an

31

b

=~

I

b

bn

[J wet)

[J v 1-p

dt J r / q

r I

(t) dt )

[J w(x)

~

/p' < "" •

r/(pq)

YJ

[t

r

(3.30)

J

fP(x) vex) dx = n

a

w(x) dx

)r/(pq)

n

E ~(a,b) . ~~e can deal with (1.12) instead of (1.11) in view of

1/q

[f (H

(3.31)

r [' [ J J w(x) dx

[q

=

a

1

(J

f (f a

fn(t) dt)q-

)r/

q

[Y.

Jr/(qpl)

l/q (~) q r

1/ fn(y) d Y) w(x) dX]

~ q1/q(~) 1/

q

l

r

1

I

v -p (x) dx

)r/

qI

v 1-p (y) dy I

]l/ q

an

an y

a

y

) q (x) w(x) dx ) L n f

bn b

1/ b (f(HLfn)q(X) w(x) dX) q

x

(x) dx

a

Lemma 1.10 and the second condition in (3.25).J Since

b

l-p '

an

b

f

v

and consequently,

an

Now, we will derive lower estimates of the left-hand side in (1.12) for

[JY

Y

r fP(x) vex) dx = A n n

f

~[f r

=

1 I ]r/(pql) 1 I -p (x) dx v -p (t) dt

an an b

Further, we can easily verify that bn b

V

J

Y

an

an

dx )

1/' q Ar/q n

q, This inequality together with the formula (3.30) used in the inequality

a

(1.12) yields the Fubini theorem yields

1/

b

[f (HLfn)q(x) w(x) dX)

q

r

1/' q Ar / q ;;; C Ar / p n L n

This implies q1/

q

b

y

[f [f f n (t) a

ql/ q

q-l dt)

b

f n (y)

a

bn

[f

y

~

[f w(x)

1/ q

(3.32)

dX) d Y ]

y

[J fn(t)

since

b dt )q-l fn(y)

(f

q

0 < A < n

00

l/q I A;;; C n L

in view of (3.30), (3.29) and (3.27). Letting n

(3.32), we obtain (3.24).

}.10. Remark.

-+

in

00

o

Lemmas 3.8 and 3.9 give a complete proof of Theorem 1.15.

The procedure used here is a small modification of the original proof of

(a ,b ) , we obtain from (3.26) that n

1/q(~) q r

w(x) dx J dy ]l/

y

an an Further, for

(~)

q

a

=

l/q

n

V. G. MAZ'JA [1].

y

f fn(t) an

dt y

32

assertion, which will be useful in the sequel.

b

f [J w(x) an

In the conclusion of this Section we give a proof of an auxiliary

=

t

r/(pq) dx )

t

[J

v

l-p'

(x)

dx

)r/(pql)

v 1- p I (t) dt ~

3.11. Lemma.

Let

1

< q < p <

00

and

v, w E W(a,b) . Assume that

an

33

,==,,-==--_.-

'-_-'!!O'~~~~-~'~-"_.:=::"'~T!i::.,,~~~~~;a:$~~~"F"#§i-_.;;;;~~::::__

b

.

~_....__:':.._:

..__ :~~~=-'--~:;:-:-

~ ~ '.::::"i!.~:""_'::,_,':;:':""':::i~

__ .:..... l£.I¥;:k":'::;'~~;

b

f w(x)

dx

<

and

00

a

v 1-p' (x) dx

f

b <

00

•

(3.37)

n

1

1

q

p

b

(3.38)

{a} , {b n } be two sequences of real numbers such that n and fop

t b

and

n e:N

An

= [f [f

w(t) dt

r/

q

an x

+a

dt

x

q

b

x

1-p'

(t) dt

r/(pq')

(x) X(a

[f w(t)

~ A~r/p

•

An-rip

f

wefconclude that

a

gn

E

ACL(a,b) ,

Since

f(g~)p(X)

dt

r/(pq)

[f [f }-p ,(t) dt r/(pq') , ) j }-p (x) dx an an

dt

r/(pq)

n

n

[fbnv 1- p ,(t)

dt

r/(p'q)

<

00

,

an

e ACL(a,b) . , we have (3.34) from (3.37); further, since

(3.35) follows from (3.38) and the lemma is proved.

b

(3.34)

g' = f

gn

~

an

an

f n (y) dy .

fn(y) dy , and since

bn x

b , [f = q~ A~r/p w(t)

-"

x

gn(x)

n

an

b )(x) n' n

f

A- r / p f fn(y) dy n

f (y) dy

b

and (3.33)

An

a bn

a

an 1-p'

A~r/p f

a

v

gn(x)

b

f fn(y) dy

[1

l/p'

x

l ' (t) dt r / ' v l -p' (x) dx f/r ,

r / (pq)

~ q1/q(~)

a

,

an

[f w(t) • v

n

l/q

[f(HLfn)q(x) w(x) dx )

Moreover, in view of (3.33) and (3.36),

[f v -p

b

fn(x) =

a

define

x E (a,b)

bn b

Then

1 .

According to (3.31), we have

r

b

v(x) dx

a

Denote

Let

f~(x)

r J

a

gn

=

HLf n '

0

v(x) dx

a

and

4. THE METHOD OF DIFFERENTIAL EQUATIONS b

0.35)

[f

g~(x)

1/

w(x) dX)

q >- q l/q (~) r

l/q' A

n

Let us just formulate the main result.

a

for evepy noof·

We have used the functions

of Lemma 3.9 0.36 )

4.1. Theorem.

n E:N

f

f

n

and the numbers

A

n

(4.1)

in the proof

n

=

n

n

(x)

,

(4.2)

we obtain from (3.30) that 34

<

p ~ q

<

00,

v, w e W(a,b) . Moreover, assume that

v E AC(a,b)

x

r

A- / p f

1

and

see the formulas (3.26) and (3.28). Therefore, denoting (x)

Let

f v 1- p ' (t) dt

<

00

faY'

x E (a,b) .

a

,i

Ii'

35

together with (4.5) implies

Then the Hardy inequality (1.11), i.e. b

1/

[Jlu(x)lqW(X) dX]

(4.3)

x

b l/p q;;; cL[Jlu'(x) IP v(x) dx ]

(HLf)q(x) w(x) x

holds for every u E ACL(a,b) with a (finite) cons~ant c L if and only if there is a number A > 0 such that the differential equation A

[v q / p (x) (~) dx dx

cL

has a solution

[J f(t) [y'(t)r1/p' [y' (t)] 1/p' dt a x q/p ;;; w(x) [f 'l'(t) dt]

o

] + w(x) yq/p' (x)

y(x)

>

y' (x)

0,

0

>

for

x G: (a,b) .

;;; w(x) yq / p ' (x)

Consequently, denoting

via the inequality (1.12) is correct due to Lemma 1.10.

solution

b 1/ [I(HLf)q(x) w(x) dX] q;;;

(4.6)

c[I

fP(x) v(x) dX]

(4.10)

p

f

E

~(a,b)

with the constant

x

(I

a

1/r

r 'l'(t) dt)

dX]

a

b

1/

[J(HLf)q(x) w(x) dX]

b

f E ~(a,b)

1/

b

[f ~(x)

r ;;; J 'l'(t) a

dX]

r dt .

t

~(x)

r dX]1/

;;; \p/q v(t) [y'(t)]p/p' ,

t

satisfies

the inequality (4.10) together with (4.9) implies

fP(x) v(x) dx <

~

•

b ] 1/q [bI '¥(t) v(t) [y'(t)]p/p ' dt ] 1/p ;;; \l/q [ J (HLOq(x) w(x) dx

a

y

b

[J ~(x)

Since according to (4.8)

b

Let

[J 'l'(t) dtf/P

we have

a

a

Clearly, we can suppose that

I

x a

r = q/p

b

1/

[f

Proof.

~(x)

'l'(t) dt]q/P =

a

C = \l/q .

(4.7)

;;;

and the Minkowski inequality (3.2) yields

b

a

holds for every

q/p' dt]

\ > 0 such that the differential equation (4.4) has a

satisfying (4.5). Then the inequality

y

[1

b 1/ [f(HLOq(X) w(x) dX] r;;;

Let the assumptions of Theorem 4.1 be satisfied and assume that

there is a number

w(x) ;;;

a

f E ~(a,b) ; the approach

deal with the inequality (1.12) for functions

r

a

The assertion of Theorem 4.1 is a consequence of several lemmas, which

4.2. Lemma.

x

[f y'(t)

a

satisfying the conditions

y

y' E AC{a,b) ,

(4.5)

q/p'

dt)q w(x)

a

a

a

(4.4)

[J f(t)

be the solution of (4.4) satisfying (4.5). For

- \ cL dx

q p [v / (x)

(~)

(4.8)

iP(x)

(4.9)

'l'(t) = fP(t) [y'(t)r p / p ' .

Then (4.4) yields that

dx

x, t E (a,b) denote

q/p' ]

iP(x) = w(x) yq / p ' (x)

a

a

= A1/q

[Jb fP (t) v(t)

dt

)l/ P

a

This is the inequality (4.6) with the constant and Holder's inequality

C

from (4.7).

D 37

36 ):.··

:,.

I.

.;;.;.~ :;_:':,:_'.¥.:-:,,=""'_""~:~~.::'"O~'_

,

f~X)

K = L inf sup

(4.11)

Theo~em

Let the assumptions of

4.3. Lemma.

q

a<x
f

4.1 be satisfied. Let us denote

J wet)

[f(t) +

f

the~e

numbe~

exists a

A > 0

such that the

a~e

diffe~ential

(4.14)

f(x) >

Further, on equa­

~ Aq

K

K

<

00

<

00

(a,b)

let

[f(t) +

J v 1- p ' (s)

f

Let Z

on

diffe~ential

then the

,

(4.15)

equation (4.4) has a solution

,

nE'N,

x

x zn+l (x) = ~q,

A > K

y

f

wet) z~ / p '+1 (t) dt +

n'z, = L

W Z

Aq

zn(x) > 0

Obviously,

q/p'+l

Z

+ v

is a positive solution of the equation

v 1-p ' (t) dt .

a

and in view of (4.14)

x E (a,b)

for

x

1-p'

(4.16)

f wet) Z6 /p '+1 (t)

dt <

00

•

a

(a,b) . Since Moreover,

x

~

f

be the solution of (4.4), (4.5) and put

= (~') v 1 -p '

z(x)

z (x) n

a

It can be easily verified that (4.13)

ds]q/p'+l dt .

v 1-p' (t) dt ,

a (i)

there exists a positive

define a sequence of functions

US

).

fo~ eve~y

satisfying (4.5)

Proof.

..............

x

(and, consequent ly, If

. ,,_:-:::::

a

zO(x) = f(x) +

(ii)

"~"~~:..__ "!!...~-~_.. ~. ~ ._~~.:<'!:

t

Jwet)

K ;;; A

(4.12)

..

by the formulas

satisfying (4.5), then

y

-"~":-~:~.',.. _ ~~:::;:~_::::~~:.:::. _ _

such that x

which

f EO W(a,b)

__ ..

A > K • According to (4.11)

a

tion (4.4) has a solution

y

function

Let us fix

(a,b)

If

(i)

(ii) dt ,

a

a

where the infimum is taken over the set of all positive on

J v 1- p ' (s) ds]q/p'+l

m

In view of (4.11), we immediately obtain (4.12).

t

x

':_....

f

x

z'(t) dt, ZO(x) - Zl (x)

a

f(x) -

wet) zci/ P '+l(t) dt > 0

LAq f a

we have according to (4.13)

according to (4.14), and consequently

x z(x) >~ ~ Aq

x

J wet)

zq/p'+l(t) dt +

f

x

v 1- p ' (t) dt

Zn(x) - Zn+l (x)

a

a

p---'­ Aq

f

wet) [zq/Pl'+l(t) - zq/p'+l(t)] dt > 0 . n-

and,consequently, denoting Thus we have shown that the sequence

x

f(x)

=

z(x) -

f

f(x) > 0

A

~> qp'

on

(a,b)

non-negative function z(x) and

~~;

,l

x 1

r

fci) J wet) [f(t) + a

f v 1-p' (s) t

ds]

q/p'+l

dt .

~j 1/':

-be;'

": J

,~ip

a ,

38

{z (x)}

which together with the positivity of

v 1 - p ' (t) dt

a

we have

n

a

t'

'\

:.

on

n

zn(x)

is decreasing on

(a,b)

yields the existence of a

(a,b) ,

z(x) = lim zn(x) n+ oo

Taking into account (4.16) and applying the monotone convergence theorem, we obtain from (4.15) that 39

~:~~~~_to..ri;~q;:"~~~~~?~':;;:~:P"~-~"'~~;:t:~,

x

~

z(x)

x

J wet)

Aq

f v 1- p

zq/p'+l(t) dt +

This formula implies that the function

z

(t) dt .

is positive, belongs to

AC(a,b)

If

b

o

The proof is now complete since it can be shown that the function

f w(y)

<

-1 1-p' (t) dt ) [f z (t) v

exp

dy < 00

Consequently, the function

c

with

c

(a,b)

is the solution of (4.4) satisfying

f(t)

=

A E (0,00)

(a,b)

K fpom (4.11) is finite if and only if SBr

such that the ppoblem (4.4), (4.5) is solvable. Con­

,

b

> Br

C ~ L

(4.17)

1 q K /

f(t) > 0

,

b

f wet)

K ~ E..:- sup q a<x
[f w(y)

p dyr

k(q,p)

=

(1 +

.9,-) p

! (4.20)

The first inequality in (4.17) will be proved by contradiction. 1 q To this end, let us assume that K / < C ' and choose A such that L O

=

~ q

. , q/p'+l (SBr )

(i)

K <

A6 '

the problem (4.4), (4.5) is solvable for

Lemma 4.3. Formula (4.7) in Lemma 4.2 implies that

A

A6

q]

dt

[f w(y)

-p'/q d Y]

sup

a<x
t

f )-p' (y)

dy

a

hex)

"i' ~ [J

'(Y) d Y] p'

Iq

~ v,~p' (y) d Y]

~

x q/p'+l

p' ) , (sB L

<E..:p, - q (s - 1) B L

l/q < A0 < C . K L

(4.19)

q/p'+l

,/

t

b

x

(4.18)

Since

[SB(

a __ SBr'

, l/p' (1 + E..:-) q

yields

see (4.11)

K

x

C be the L

1.Jhepe

Ppoof.

t

[f1' v -p (y) dy ) ,

t E(a,b)

for

K be

~ k(q,p)B L l/q

we

a

TheI definition of the number

B be the numbep fpom (1.18) and let L best possible constant in (4.3) Then

s

which implies

K. This will follow from the next assertion (and Lemma 3.4).

defined by (4.11), let

dy

t

show that the validity of the Hardy inequality implies the finiteness of

Let the assumptions of Theopem 4.1 be satisfied. Let

r

,/q

, ~ [f w(y)

sequently, using in addition Lemma 4.2, Theorem 4.1 will be proved if we

4.5. Lemma.

s E (1,00) • Moreover, for such

for every

have

According to Lemma 4.3, we have shown that under the assump­

tions of Theorem 4.1, the numbep

the number

f

,

1 p v (y) dy

a

t

is continuous on

thepe is a

t

SBr' [f w(y) d Y)

o

4.4. Remark.

_p' /q

b

a fixed number from

(4.5) .

t E (a,b) .

for

t

x =

~!:!~:"'~'Fz~r¥!

B = 00 , then the second inequality in (4.17) holds trivi­ L ally. Therefore, assume B < 00 • Then, due to the definition of BL and the L fact that w E W(a,b) , we have (ii)

and satisfies the differential equation (4.13).

y(x)

I"t""",':,

which contradicts (4.19).

,

a

a

:.....\"~_,.;~~:p%~.,¥.1~T~~i{~~!!;;- ;.;"" :ill~~~'1"'r'\'t-"'.c,"-:Ii:'·'

sup hex)

a<x
due to where

CL -s: 1\,1/q = A0 ' 40

41

x

(J w(y)

J wet)

hex) =

a

t

[[J 1 -

dy

( dt

J w(y)

t ) by the formulas

P,/q dy J

(4.25)

r

p

w(y) dy

' /q

[J

w(y) dy

r'

for

g(s)

)1/

_s

/q] .

Kl/

~ g(s)

J

z = q,-I(x)

1 p v - ' (y) dy ,

s

lip'

q,

lim

-1

b

B L

(4.26)

inf g(s) ,

l<s
lim

L

q,

-1

0

~

and consequently

Combining the inequalities (4.17) and (3.7), we have

C L

~

1 q K /

~

L

(ii)

z IE (O,L)

ties considered.

w

by

1 < p ~ q < 00,

v, w

c W(a,b)

a

This enables us to introduce new variables

(4.29) z

and

s

(instead of

x

and

in~quality

=

I

[f f(s) dS)q

~(z)

dz

fP(z) dz .

°

(4.6) can be transformed into the inequality

z

1/ q

L

[f (I ° ° °

in fact the inequality from Problem 1.7 f (s) ds)

the interval

x E (a,b) .

fP(x) vex) dx

L

which is

I

L

Therefore, the

and

x

=

z

° °

a

(4.28)

for every

f(t) dtJq w(x) dx

a

J

purpose, let us make some slight transformations of the data and inequali­

I

L

b

will show that the less natural assumption (4.1) can be removed. For this

Let us assume that

x

and

but now under the restrictive assumptions (4.1), (4.2). In what follows we

(4.24)

f ,

~ (z) = w(q, (z) ) v P , -1 ( q, (z») ,

I [I

(see Remark 4.4).

1-'

v P (t) dt < 00

new functions

s E (O,L)

a

From the inequality (4.23) we again have the estimate (1.23),

(iii)

a

p f(s) = f(q,(s»)v '-I(q,(s») ,

b

Theorem 4.1 is proved completely: the validity of the

C < 00 ) implies B < 00 due to L L the first inequality in (4.23), while the last inequality in (4.23) then K < 00

(y) dy

we easily obtain k(q,p)B

Hardy inequality (with the best constant

implies

,

I v 1- p

(x)

and introduce for. f E ~(a,b)

(4.27)

B L

I

-p (y) dy .

If we denote

the second inequality in (4.17) follows immediately.

(4.23)

J

V

a

x-+b­

4.6. Remarks. (i)

s

°.

(x)

x-+a+

Since k(q,p)

1

-I(t)

a

s E (1,00) , we have from (4.20), (4.21) q

t

x

~}Obviously,

q

s - 1

q, (s)

t

where

If we denote (4.22)

= q,(z)

x

a

(

x

x

(4.21)

~'

) (-p , / q) (q / p , +1)

b

(O,L)

~(z)

q

~(z) dz

)

~

C

[f

fP(z) dz

l/p

J

,

see (1.12)

on

and with the new weight functions

and ~(z)

=

1 .

Obviously, these new weight functions satisfy all assumptions of Theorem 4.1, and,therefore, the following theorem holds:

42

43

Let

4.7. Theorem.

1 < p

~

q <

v, wE W(a,b)

00,

(4.24) be satisfied. Let the number

and let the condition

and the weight function

L

~

defined by (4.26) and (4.27), respectively. Then the Hardy inequality b

(f

~

!u(x) Iq w(x) dXJ l/q

b C [flu'(x)I P vex) dx )

a

BL ~< CL ~< K- 1/ q = < k( q,p )B­ L

(4.30)

be

Using the substitutions (4.25), we can easily show that BL = B , L

(4.31)

l/P

where

a

if and only if there is a number

u E ACL(a,b)

A

>

0

such

and consequently also the inequality (4.6)

+

dz

w(z)

yq I P ' (z) = 0

y

y(z) > 0,

y' (z) > 0

for

z E (O,L) .

Note that in Theorem 4.7

p, q

the condition (4.1) was not

mentioned. This is a consequence of the fact that we reduced our problem to an analogous problem for weight functions

~, ~

from (4.29) where the

conditions analogous to (4.1) and (4.2) are fulfilled automatically. (ii)

For the same reason, a formula analogous to (4.23) holds for the

5.1. Convention.

Up to now, we have dealt with

equal

on

to

00.

sup O
[f

z

) 1I q

[f ds riP'

w(s) ds

o

z

L

sup z lip' [f w(s) ds )l/q O
p ,q

is

Therefore, let us introduce the following 'arithmetic'

convention: 000

00 L _

p, q E (1,00) . Now, we

will consider also the case when at least one of the exponents

numbers

B L

(and, of course,

1 < p ~ q < 00 ).

5. THE LIMIT VALUES OF THE EXPONENTS (i)

v, w E W(a,b)

satisfying the conditions

y' E. AC(O,L),

4.8. Remark.

holds. Besides,

Due to (4.30) and (4.31), we have shown that the estimate (4.23)

holds for every pair of weight functions for

has a solution

-

is the best possible constant for which the inequality (4.28) holds. (iii)

A -d [d (-~.Y) q/P']

dz

C

L

that the differential equation

is the best possible constant for which the Hardy inequality

C L

(4.3)

holds for every

C = C L L

K = K and

and, for

x >

_ (_ 00)

o, x

x

o,

x +

ooX

00

z

K

L q

inf f

sup O
1

fez)

f ~(s)

[f(s) + s]q/p'+l ds

5.2. Some function spaces.

'L = '~Pf {I0

'

[!

f(s) dS)q

~(z)

r

J fP(z) o

dz

=

f E W(O,L)

w E W(a,b) , let us denote by

which are the weighted Lebesgue space defined as the set of all measurable functions u

q

on

(a,b)

for which the norm

dz

where the supremum is taken over the set of all L

and

~

LP(a,b;w)

(5.1) (O,L) , and

p

o

where the infimum is taken over the set of all positive on

For

II

b

\ [[1"(') I

P

f E W(O,L)

such that

(5.2)

1 . This formula reads

Ilull

p,(a,b),w

= ess sup a<x
w(,)

Iu ()I x

dxl

P

for

1

for

p

~

p <

00

,

is finite. 44

45

For

w(x)

~ 1

we obtain the classical (non-weighted) Lebesgue space

ql/q (~)

l/Q'

LP(a,b) • whose norm will be denoted simply by p = 1

reduce, for

(5.3)

or

~ ~ C ;;; q L

q

11

q(p')

11'

q A L

00 , to the simple identities

l/u11p,(a,b) .

B =C L L

u·wl/ p belongs to LP(a,b;w) if and only ~f the product belongs to the Lebesgue space LP(a,b) and Ilull p,(a,b),w-- Ilu·wl/PII'p,(a,b)' Since

u

(cf. Lemma 5.4) and

q1/q ~

=C

it follows immediately that LP(a,b;w) is a Banach space with respect to the norm (5.2). (Note that we have used, for p = 00 , Convention 5.1.) Now

(d. Lemma 5.6). [Note that for

the Hardy inequality (1.11) as well as the inequality (1.12) can be re­

from (1.24) equals

written in the following form:

(p'q/r)l/ q ' = 1,

II u I q, (a, b ),w ;;; C il u ' II p ,

L or

1

=

q

the number

ro

k(p,q)

1 according to Convention 5.1, and similarly (p,)l/ q ' = 1 for q = 1 or p = ro .J

Now, we shall show that our considerations are not purely formal. How­

(a, b ), v

ever, we start with an auxiliary lemma.

or

(5.4)

II uw

l!qll

1 (D II q, (a, b ) ;;; C liu'v " p, (a,b)

w(x)

(5.7)

wE W(a,b) . Then

Let

5.3. Lemma.

and

~

Proof.

or

(5.5)

IIHLf'W

l/q II

q,(a,b)­

~ cLl/fvl/pl/p,(a,b) ,

B L

ess sup wet) a
Sex)

from (1.18) can be expressed as follows:

A

sup Ilwl/qll .I/v B = a<x
= -----E.-. p - 1

l/P

n

II

p' ,(a,x) !'lI11,

under Convention 5.1 the numbers

B L

~

p,

qE

;;; q < p

~

such that

nO E R

a < 6

o<

[l,ooJ

and the assertions of Theorems 1.14 and 1.15 can be extended to the cases

and

E I; sex) +

.

ln

< w(x)} , of

IAI

A

0

such that for

An

Let

00 , respectively. Moreover, the estimates

(1.23) and (1.26), i.e.

X

IAn I ;;; 6 - a <

o

00

n

eR

A = U An and there n=l Further, we can choose numbers

0

A nO

n

[a,6J

J x(t) dt,

we have

x

A nO

g

is continuous,

o

I

>

0

and define

ER .

B ;;; C ;;; k(q,p)B L '

L L The function

IAn

•

be the characteristic function of the set g(x)

.

is positive,

> 0

I An I

and consequently

from (1.19) as

well as the inequalities (5.4), (5.5) are meaningful even for

exists

a, 6 E I ,

p, q e (1,00) • Nonetheless,

from (5.6) and

= {x

Then the Lebesgue measure

.

These relations have been derived for

1 ;;; p ;;; q ;;; 00

(a,b) ,

I

A = {x E I; Sex) < w(x)} ,

Similarly, the number

p'

x E (a,b) .

Suppose that (5.7) is not true, and denote

respectively.

(5.6)

for a.e.

ess sup wet) a
I HL f II q , (a, b ),w ;;; CL II fII p, (a, b) , v

where

=

p

g(a)

0,

g(l3)

IAn

o

I .

Therefore, 47

46

.

.'>

",I',"'·,·,,· :"

l'

such that

z E (a,S)

there exists a number

:21 1-Anal. putting

g(z)

x E C

IC I (a,z)n

B =

~

(z,S)(1An

C

a

Since

g

21 IA- no I

a

k

2- )

(l -

=

IBI .

B

E (l,ooJ . Let CL

be the best pos­

(5.11 )

C < 00 L

L

C L

<,;;

B

B

<,;;

C L

L

k E :N •

,

for

-'/

L

B and C are either L L simultaneously finite (and equal) or simultaneously equal to 00.] [If (5.10) and (5.11) hold, then the numbers

(i)

Sex) + -- < w(x) nO

P

B L

»

< 00

(5.10)

' we have 1

00,

Proof· Obviously, it suffices to prove the following two implications:

k IBkl = g(x ) - g(x _ ) = 2- 1BI > 0 k 1 k

C. ~O

=

q

B = C . L L

(5.9)

, such that

Then obviously

B k

Let

or

q E [1,ooJ

there exists an increasing sequence

[a,zJ

nB

p = 1 ,

be defined by (5.6) and let sible constant in (5.5). Then

and define B = (x _ ,x ) k 1 k k

Since

Let

v, w E W(a,b) .

> 0 .

k = 1,2, ...

k

k

o

=

x E (a,z) , g(x )

X

Ici

is nondecreasing on

of points

Set

> 0

5.4. Lemma.

=

, since

o

we have

IBI

w EO W(a,b)

However, this contradicts the assumption

x E Bk

The case

p = 1,

1

<,;;

q < 00

In this case

which together with the fact that the function

S

is nondecreasing on

I

(5.12)

b

B = B (a,b,w,v,q,l)= sup L L a<x
[[f wet) x

yields 1

S(x _ ) + nO < w(x) , k 1

x E Bk '

dt J l/q

ess sup v -1 (t) ] a
B < 00 • Using the inequality (3.2) for L

f E ~(a,b) , we obtain

Assume that

k E:N • '!'

r = q ,



.

w,

Consequently,

b

1

Sex k-1 ) + -s: ess sup w(x) n O- yEB k

<,;;

S (x ) k

(5.13)

and,therefore, (5.8) For

S ( xl ) x E C

and

[f

(I

a

a

f (t) dt]

q

w(x) dx

] 1/q <,;;

I

b

f(t)

[I

a

q w(x) dXJ1/

dt .

t

Lemma 5.3 yields

+

--n;k -

1 <

1

~

S

(

X

\

k)

for

k c:N .

b

I

we have

k EO:N

S(x ) + -k nO

<,;;

[f w(x)

q dt

t

J f(x)

vex)

b

w(x)

for all

<,;;

I a

[J

wet) dt

r/

q v- 1 (x) dx

<,;;

x

a

k E:N , we have

dXJ

b

k

S(x ) + -- < w(x) . 1 nO

Since this inequality holds for every

1/

b

f(t)

a

1

Sex) + -- < w(x) , nO

which together with (5.8) implies

48

b

x

1/

b

f(x) vex)

[J wet) x

dt]

1

q Cess sup v- (t)]

dx

<,;;

a
... ~'-"-.-.-

-----'-----=;~~

.,-_..... -

...

BL

•• _ ~ _•• _ _ ,,, ..

f

b

~

'-r_-'",n-~·~"'~·--'''-·-·-''--·-'-~~

f f(x)

vex) dx

n (x)

x~ (x)

=

.....

b

~

b

(5.18)

00

b

q [J f(x) dX] a

~

Sex)

Then

Sex) > 0

ess sup v a
for every

X E (a,b)

for some

o

~

implies that for

o (i-I)

Then

o

<

(5.17) then

< S(O

n

SO

f fn(x)

vex) dx

a

r

1

p

=

1 ). Denote

Letting

00

] 1/ q ~

n

-+

S(x ) = 0 O v E W(a,b) . This

1

n J-

1

we obtain

00

1/

w(x) dX]

S(~) ~

q

C ' L

which in view of the definition of

B (a,b,w,v,q,l)

(5.19)

such that

Let

v

B

L

see (5.12)

yields

C

~

L

(i-2)

L

be a general function from

W(a,b)

and for

n E

~

define

1

v (x) = vex) + - , n

x E (a,b)

Obviously, v

and if we denote

> 0 . Therefore, there exists a subset

CL [S(~) -

~

(a,b) , since the assumption

n . for every

w(x) dx

b

1

-

I

[I

n E~

IMn I '

~

Since M eM n

n

such that

w, v

n

n

ess sup v (t) < O
n

(i.e.

~

x E (a,b)

E W(a,b) , too. and -1

(5.20)

I = {x E (a, 0; v (x)I > S (0 } - , n

o < 1Mn I Define

x €

Mn

[

(t)

E (a,b) , there exists

S(~) - 1 < M

1Mn I

-1

I dx

a

b

CL [J f(x) vex) dX] a

leads to a contradiction with

00

n

CL

we immediately conclude from (5.18) that

Assume in addition that

Sex) <

(5.16)

~

J fn(x) dx

b

(cf. the formula (3.10) for =

~

~

•

~

1/

[J w(x) dX]

(5.15 )

fn(x) dxJ

Since, moreover,

C < Then the inequality (5.5) holds for every L and analogously as in the proof of Lemma 3.4 we obtain

(a,b)

[f

q

Mn

B . L

Now, assume that

~ E

b

CL J vex) dx ~ CL IMn I [ S (0 - }n

=

a

The implication (5.10) is proved.

with

....."''''·'----'-·~ ....~ ; ' i ; ; Q , ~ " _

a

However, in view of (5.5) the last inequality immediately implies that

(5.14)

dXJ

~

BL J f(x) vex) dx .

a

~

1/

[f w(x)

b 1/ [J(HLf)q(X) w(x) dX) q

f ~ M+(a,b)

_ _ .. __ • _ _ "._.... _ . , _ . _ ......

Then (5.14) and (5.17) yield

and from (5.13) we have

~

~_w

x E: (a,b) .

,

a

C L

tW----~·-~·---~·-·-·~-;,;r~;>;;~

,, _ _ .•,_ _ ~ _ ,

00

for every

x E (a,b) .

vex)

on

(a,b) , the inequality (5.5) holds for the pair

with

v

instead of

n

v) with the same constant

CL . More-

over, (5.20) is nothing else than the additional assumption (5.16) for <

00

•

and consequently ~;

I~ II,

vn ' we can use all arguments from (i-I) arriving at the in-

equality 51

~

B (a,b,w,v ,q,l) n L

x

CL

sup a<x
as an analogue of (5.19). Letting

n -~

00

we again obtain (5.19) for an

,

P = 1,

q =

00

b

•

In this case

(5.21)

sup ess sup v a<x
-1

(t)

ess sup v a
-1

~HLf w

l/ q l

~HLf Iloo, (a,b)

I

f (t) dt

a

f f(x) dx i

f(x) dx

I

Lemma 5.3 yields

I a

(x) dx

b

~

BL

I

I

; ; cL[f

f(x) dx

CL ~ BL

v

and the implica­

f(x)

sc

and let

(a,b)

f

E

~(a,b)

be such

l/p

fP(x) vex) dx

J

.

l-p'

(x)

for

x

for

x E [s,b) ,

=

I;

(a,O

we obtain

•

s

s

r f(x)

• v 1-p ' (x) dx

s I fP(x)

a

a

a

In this case

52

;;; B L

Putting

a

(5.26) 00

L

a

a

the case (i). 1 < P <

Let

•

C

f(x) vex) dx ,

The proof of the implication (5.11) is completely analogous to that of

(5.23)

00

0

q

) l!p' l' v -p (x) dx

s

tion (5.10) is proved.

The case

I

Then the inequality (5.24) has the form

s

(5.25)

f(x) vex) [ess sup v -1 (t)] dx a
<

C

L suppfC(a,O

~

which in view of (5.22) leads to the inequality

(iii)

[b

a

Assume that that

f(x) vex) v

l/p

a

b

~

)

~:,,'

a

a

vex) dx

and in view of (5.24) we have

-1

~

b 1/ BL[I fP(x) vex) dXJ p

(

b

b

I

•

1 p 1 p f(x) v / (x) v- / (x) dx

a

a

00

I

f(x) vex) dx .

a

B < L

I

dx

b [ fP(x)

;;;

a

I

f(x) dx ;;; CL

Then Holder's inequality yields

•

a

b

b

b

Assume that

00

b

f f(x)

the inequality (5.5) has the form

I

B < L

b

ess sup a<x
) l!p'

a

a

q,(a,b)

(t) dt

b 1/ C [] fP(x) vex) dX) p L

~

dx

Assume that

(t),

x

(5.22)

f f(x)

f C ~(a,b)

and since for

l-p'

a

a

il~oo,(x,b)·~v-l~oo,(a,x)

sup a<x
[I

v

and the inequality (5.5) has the form

(5.24)

B = B (a,b,w,v,oo,l) L L

(t) dt

b

P )l/ '

a

arbitrary v E W(a,b) . The implication (5.11) is proved. (ii) The case

v 1- p '

[I

(iii-l)

B = BL(a,b,w,v,oo,p) L

'i l'.,

dx

r

s vex) dx

I

v 1 -p ' (x) dx .

a

Assume in addition that 53

I I

i:'

:,

.'

•

",),.

x

from the case

v I-p' (t) dt <

J a

x €

for every

00

',<J~:L

(a,b) .

[r

V

i

l -p' (t) dt

r

For generaZ

(iii-2)

by using the functions

v

E

v (x) n

(a,b)

CL caused by the fact that the weight functions w

W(a,b)

~

B L

BL ~ CL (5.29)

BL(a,b,w,v,oo,oo)

f

f(x) dx

Assume that

B < L

b

f

f(x) dx

~

a

(b - a)

[Jr

sup a<x
d tJ

fP(x) ~(x) dx

lip

J

a

=

wq(x) ,

~(x)

=

vP(x) .

f E LP(a,b;w)

~ ) f·w

E

LP(a,b) .

Nevertheless, this approach has the following consequences: The number

C ess sup f(x) L a<x
00

•

Then for

f

.

B L

E ~(a,b)

(b - a) ess sup f(x) a<x
C

L ~

<

BL

,

=

- B (a,b,w,v,q,p) L

sup a<x
~w~

q,(x,b)

=

-l/q 'II--l/P~ sup ~w ~q,(x,b) v p',(a,x) a<x
ijv-l~

p',(a,x) ,

and this expression contains the weight functions

B ess sup f(x) L a<x
w, v

even if

p = q

= "'.

B again fully characterizes the L validity of the inequality (5.29). This can be shown by the methods used in

Using (5.28) for

C ess sup L a<x
f(x)

=1

the foregoing subsections. See also J. S. BRADLEY [lJ, V. G. MAZ'JA [lJ.

, we obtain

Now we shall consider the limit cases corresponding to the Hardy in­

CL '

equality for

p > q . Let us recall that in this case an important role was

played by the number

C . L

A = AL(a,b,w,v,q,p) L

The investigation of the Hardy inequality is reasonable as

far as the best possible constant

54

cL[J

of the weighted Lebesgue space:

a

(5.30) 5.5. Remark.

~

Let us note that the finiteness of

B L

~

(H f) q (x) w(x) dx L J

This approach looks very formal; in fact we slightly modify the definition b - a

in view of (5.28).

Assume that

B L

[f

b

associated with the inequality (5.29) has the form ~

a

i. e.

we have

on unbounded intervals. This apparent discre­

q

l/q

;(x)

b

~

p = q

with

q

and the inequality (5.5) has the form

C L

=

b

from (3.12) and proceeding analogously as

x

and

do not appear in

v

following inequality:

, we obtain the inequality

In this case

B L

p

a p

and

pancy can be avoided if instead of the inequality (1.12) we consider the

C L

in the case (i). The case

in this case. This is

is unbounded, and consequently also

inequality for

which immediately yields the inequality

(5.28)

is infinite if the interval

see (5.27)

Consequently, it seems not to be reasonable to investigate the Hardy

/p' S C ' L

a

(5.27)

q

formulas (5.5) and (5.6): due to Convention 5.1, for w 1 / q = wO = 1 , v 1 / p = v- 1 / p = v O = 1 .

Using (5.26) in (5.25) we obtain

(iv)

=

p

. "l,;

C L

is finite. The number

ilL = b - a

;1

55

i

b b "l/q

l [ [l

where (5.31)

l/q' r 1 p v - ' (t) dt] ] v 1-p'(x) dx

[I

w(t) dt]

p

a

= -1 - ­ 1

r

=

)

which is the inequality

q

IIHLf'wlll,(a,b)

q E [1,(0)

00,

q

1,p~0,(0)

As

CL

Let

C L

, using again Convention 5.1.

q = 1 ,

or

P E 0,(0)

p =

$

A L

A be defined by (5.30) and let L possible constant in (5.5). Then q

l/qA

C L

Now assume that

be the best

C < L

=)

00

q

l/qA

The case

In this case

r

p'

=

~

.j, a, b n n Denoting

b

a

t

,

dt)P

A

P vl -p' (x) dx Jl/ ' .

(with

00

•

b

b

J

$

[J a

56

1 P fP(y) v(y) dYJi /

b

[J w(x) y

b

[f [' w(x) J

a

)l/(P-1)

1 ' v -p (x) X(an,bn)(x)

n

(in fact,

f

is the function from 0.26) for

n

,

b

[f

an

are two sequences of real numbers such that

n

w(t) dt

q = 1 ).

l/p'

PI' v -P (x) dx

J

J

x

analogously as in the proof of Lemma 3.9, cf. (3.27), (3.29), that

0.30)

bn

b

f(y) J [J f(y) dyJ w(x) dx = a a a b

dt

{a}, {b}

[f

f E ~(a,b)

x

j' w(t)

x

Using the Fubini theorem and Holder's inequality

p , p' ), we obtain for

b

n

we obtain ~ <

[

bn

[ J [J w(t)

A (a,b,w,v,l,p) L

Assume that

(x)

and b

(5.35)

n

n EN, where

a

p E 0,(0) .

q'

f

x

for

q

-s CL

Therefore, it is meaningful to define

L

(cf. (5.10), (5.11». (i)

E (a,b)

i;

s C

L -

l/p' dxJ

b

(5.36) (5.34)

Analogously as in the proof of Lemma 3.4

•

a

C s l/qA L - q L

)

00

[f v 1- p ' (x)

dx

i;

for every

~ < '"

1 .

=

i;

Jw(x)

implications: (5.33)

C < L

b

Analogously as in Lemma 5.4, it suffices to prove the following two

Pl'oof·

q

we obtain

C . L

L

q1/

q E [1,(0) . Let

00

v, w E W(a,b) . Let

(5.32)

.

is the best possible constant in (5.5), we have

which is (5.33) since 5.6. Lemma.

~llfvl/Pllp,(a,b)

$

p

Now we extend the formula (5.30) also to the limit values or

b 1/ AL[J fP(y) v(y) d Y] P

l/r

x

y

dx

r'

(5.37) dXJ dy

o

$

vl -P' (y) dy

<

f

f~(x)

AP

v(x) dx

n

, <

00

•

an

f/

In view of (5.36), the Fubini theorem yields P

'

b

b

x

J(HLfn)(X) w(x) dx

f [f fn(t) dt] w(x) dx

a

a

a

57

[J

J fn(t) a

and a comparison with (5.39) yields

bn b

b

b

w(x) dX] dt

J

p'

[J

v

w(x) dX]

1-p I

(t) dt

p'

C ,;; q1/qA

A n

an t

t

Assume that

This inequality together with (5.37), used in the inequality (5.5), yields

f

L n

and consequently

A

n

Letting (ii)

n

--+

00

,

f

In this case,

1

=

q < P

~

p'

00

AL(a,b,w,v,q,oo)

b

[f

[f

(x - a) q-1

b

x

[f

dt)

1 dtr-

~

:N •

dtr-

C ess sup fn(x) L a<x
~

l/q

b

X(an,b ) (y) n

[f w(x)

an

Consequently, letting

wet) dt) dX]

q1/

dX) d Y]

~

y

CL

=

n

q

[)

[J dt a

--+

r-

the monotone convergence theorem yields

00

1/q

1

[f wet) dt ) dy ]

q

1/ q A ~ C L L

y

l/q w(x) dx )

~

<

00

•

C ess sup f(x) . L a<x
and (5.34) is proved.

Using (3.22) and (5.38), we obtain for

o

5.7. The equivalence of the inequalities (5.4) and (5.5) in the limit cases.

that

In Sections 1 and 3 we have shown that Problem 1.5 is equivalent to Problem

1/

(HLf)q(x) W(X») q

1.7 provided

1 < p,q <

(see Lemma 1.10, Remark 3.7). In this section,

00

we have extended Problem 1.7 (i) to the limit values of

p, q , rewriting

the inequality (1.12) in the form (5.5):

a

1

= q /q

b

[J

[J

a

a

f (t) d t J

~ q1/ q ess sup f(x) a<x
b

I q-1

Y

f (y )

[f

l/q w(x) dX) d Y]

~

y b

b

[f [f w(x) a

ql/q A ess sup f(x) L a<x
be the numbers from part (i) of the

l

[f

a

dX]

a

Assume that

[J

n €

b )(x)

y

[J

l/q

a

x

a

b

n

define

an' n

x

[f (HLf) q (x)

~(a,b)

, b

n

l/q

b

e

X(

a

b 1/ [J(HLfn)q(x) w(x) dX) q

The inequality (5.5) has the form

f

=

Let

,we obtain

1 ).

b

b

a

(5.39)

(x)

q1/ q

[f [f wet)

=

n

and

a (5.38)

•

a

= q

r

l/q

•

b

=

q

we obtain (5.34) (again

The case

~

00

L

Using again the formula (3.22) and the inequality (5.39) for the function

C L

~

n

C <

x c (a,b)

proof, and for

I '/ AP ~ C AP P

n

-1- •

L -

y

~

CL II fv1/P I p, (a, b)

Analogously we can extend Problem 1.5, i.e. y

dX)

I HLf.w1/q I q, (a, b)

(5.5)

[f d t)

q 1 -

the Hardy inequality (1.11)

l/q dY]

lIuw1/q~q,(a,b)

(5.40)

S

CL~ulv1/Pllp,(a,b)

a for

u

e

ACL(a,b)

(see (5.4». 59

__ ._--_."""""""".-,..-........,.."....

.=~.

~-"""""""~,-~:

..

..

__

~~:~..:""~~~,~ ;~ ~~----,,,-"'C~

define

In this section we have dealt up to now with the inequality (5.5), i.e.

with the (extended) Problem 1.7. Therefore we should investigate

v n (x) = vex) + 1, n

whether its equivalence with Problem 1.5 occurs also in the limit cases.

Since

vn(x) ~ vex)

(5.44)

I uw

for every

u E ACL(a,b) . Let us show that then the inequality

(5.45)

IIHLf.wl/qll q , (a,b)

An important role in the equivalence proofs was played by the condi­ tion x

(5.41)

J

V

1

I

-p (t) dt <

cf. (1.14). This condition can be rewritten in the form

(5.42)

I

, ( a,x ) <

which is meaningful for

p

and (5.40) holds, we obtain

1

I q I q:J ( a:J b)':;; CL Ii u 'vn Ill, (a, b)

00

a

Iv -l/PII P

n ~ R

x E (a,b),

holds for every

~(a,b)

cLllfvnl11, (a,b) . Denote

b

00

J

~

f €

~

[l,ooJ . In particular, for

p

it assumes

= J f(x) v n (x) dx .

n

a

the form ess sup v a
(5.43)

-1

(t) <

If 00

•

Jn

that

=

00

Jn <

than the inequality (5.45) holds trivially.

,

00

•

Since the proof of the equivalence in the limit cases uses almost literally

x

the arguments of the proofs of Lemma 1.10 and Lemma 3.4, we will give it

J f(t) dt

here only for one case:

a

p

= 1,

q E

Therefore assume

Similarly as in the point (i) above, we obtain

~

-1

In'ess sup v (t) n a
~

n·J

n

<

00

and consequently, the function

[1,00) •

x

(i)

Assume

that the inequality (5.5) holds. Then, according to Lemma

5.4, the number

B from (5.12) is finite, and consequently L (5.43) is fulfilled. For u E ACL(a,b) denote

= J1ul(x)

I

that

00

J

J

then the inequality (5.40) holds trivially. Therefore, assume

BL(n) = CL(n)

x

x

J1uI(t)ldt

J lu' (t)

a

a

60

~

CL(n)

the

C . L

BL(n) ~ B = B (a,b,w,v,q,l) for n ~ 00 , we have that B ~ L L L CL < 00 In view of Lemma 5.4, the finiteness of B implies that the L inequality (5.5) holds, too.

Since

I

vet) v -1 (t) dt ::; J'ess sup v a
and the inequality (5.40) follows from (5.5) with

(ii)

Denoting by BL(n) the number B (a,b,w,v ,q,l) and by L n best possible constant in (5.45), we have from Lemma 5.4 that

is finite. Then

arguments as in the part

ACL(a,b) . For this function we obtain the inequality (5.45)

from (5.44).

vex) dx .

a

If

= J f(t) dt a

belongs to

b

J

u(x)

the condition

f = lull

-1

(t) <

~

00

by the same

The other limit cases can be investigated in an analogous manner.

(i) of the proof of Lemma 1.10.

Assume that the inequality (5.40) holds for

u E ACL(a,b)

and

5.8. Summary.

In the foregoing subsections, we have shown in fact that the 61

.....

_--"""-~

.........""'....

.,..,,~-",=",,~ '~~~~~~=- ,~-

•. _.:Y-··

-~ ~~

assertions of Theorems 1.14 and 1.15 can be extended to the limit values of

J

provided we consider the Hardy inequality (1.11) in the form

q (5.30). [(5.40)J and the numbers BL and AL in the forms (5.6) and (5.4) For the convenience of the reader, we will list the forms of the Hardy.

p

and

inequality as well as the formulas for the best possible constants the

~ ~.. ~~. :-~~~.=;"'.. ~ --_._~_.,.~:-:;;;,;;~~

necessary

and

CL

(i)

q ~ <1,00)

p = 1, b

[flu(x)l

q

1/q ;;; C L

w(x) dx ]

J Iu I

(x)

I

vex) dx

1,

p

b

a

x

IU wet)

sup a<x
q

L

,

[J (J wet)

q t:: [1,00) ,

1 p' v (x) dXJ

dt)P

l/p'

< 00 .

P

a

B L

,'_

I

a

C

b

(ii)

A = L

b

1/ b q [flu(x) I w(x) dXJ q;;; CL ess sup lu' (x) a<x
b

a

C L

~

sufficient conditions of the validity of the inequality (vi)

~

a

CL

C < 00 ) for the individual cases. L

(i. e.

u"."":=,,..~,, __ ~ _C'~'-" C••"";;;;;;;;;;_

b l/p lu(x) I w(x) dx ;;; cL(Jlul(x) IP vex) dXJ ;

a

and

:':-"C-:~.,:-~--~·

L

=

q1/q~

b

= q1/

q

x

[f(X - a)q-1 [f wet) dt] dX] a

dt] 1/q ess sup v-1 (t)] < 00 .

a
x

l/q <

00

a

Theorems 1.14 and 1.15 together with the results listed in Subsection 5.8 enable us to formulate the following two more general theorem:

= 00 b

ess sup !u(x)1 ;;; CLUlu/(X)! vex) dX]

a<x
a ess sup v a<x
C = B L L

-1

(x) < 00 .

(5.46 )

,

ess sup lu(x) a<x
q

a

b

rJl

C = B L

L

1/ b P CLU lu' (x) I vex) dX] p

I ;;

v

1-

P I (t) dt

FL(x;a,b,w,v,q,p)

v, w E W(a,b) . Define

=

Ilw 1/ q I'Iq,(x,b)' I v -1/ p I p',(a,x)

B

L

= BL (a,b,w,v,q,p)

~uw1/q~ q, ( a, b);;; CL!ulv1/P~p, ( a, b) ~

PI

< 00 .

a

sup FL(x) . a<x
Then the Hardy inequality (5.48)

]l/

FL(x)

1;;; p ;;; q ;;; 00,

and (5.47)

p E (l,oo)

(iii)

Let

5.9. Theorem.

holds for every (5.49)

u E ACL(a,b) [and, consequently, the inequality

< [I 1/ P I II HLf ·l/q'l w Iq,(a,b) - CL fVp,(a,b)

(iv) P = q = 00

ess sup lu(x) a<x
B L

I ;;

C

L

b - a < 00

ess sup ju' (x) a<x
I

holds for every (5.50)

B L

f E W(a,b)

J if and only if

= BL(a,b,w,v,q,p) < 00 .

Moreover, the best possible constant

C L

in (5.48) [in

(5.49)J satis­

fies the estimate (v)

62

q

pE(l,oo)

63

°6

L _-

B L

(S. Sl)

$

°

-

C L

-o.em.£

-_.

__ Uu

@

!1M

Ja;UJM~_~~lMi~~~:i~!t1kL~~jt~"ki~.~~1t~ffiLj:.~;;£:Al§;;j@Ai$:.&~:;Ax~tC;.,*_~t~l,:e@fg~1?gL=~-:-.~tt'fii~<j.t~%~t.iftS~, _ _ii~

k(q,p)B L

$

(5.57)

(L) q

v(x)

b

1-p

w1 - p (x)

where l/q k(q,p)

(S.S2)

Let

1

(1 + L)

q

q < P

$

$

(we use Convention 5.1! ), we can show that the inequality (5.48) holds

[ l [! wee) del l[

b

1/

b

x

Q

{

with the constant

u E ACL(a,b)

C L

=

k(q,p)

from (5.52).

v, wE W(a,b) . Define

00,

~ = ~(a,b,w,v,q,p) =

(S.S3)

)

x

for every S.10. Theorem.

rf

, l/p'

(1 + !l,) p

=

P-1+P/q

w(t) dt

1 P v - '

(el der

/Q ']'

1 P

v -

6. FUNCTIONS VANISHING AT THE RIGHT ENDPOINT.

ll/r

' (x)

dXj

~

EXAMPLES

Up to now, we have investigated the Hardy inequality (1.11) for

functions

u E ACL(a,b) ,i.e.

the Problem 1.5 (i), which is the same as

to consider the inequality (1.12) from Problem 1.7. In Remark 1.8 we have

where

u E ACR(a,b) , i.e.

shown that the Hardy inequality (1.11) for functions 1 q

r

1 p

the inequality (1.13) from Problem 1.7, can be reduced to the former case. Therefore, let us now shortly summarize the results concerning Problem

Then the inequality (S.48) holds for every the inequality (5.49) holds for every ~ = AL(a,b,w,v,q,p) <

(5.54)

00

u E ACL(a,b)

[and, consequently,

f E W(a, b) J if and only if

Theorem 1.15, extended to the limit values of

in (5.48) [in (5.49)J satis­ Let

6.2. Theorem.

(~)l/q

A L

$

C L

in the sense of

section 5.2. C L

fies the estimate q1/ q

p, q

Section 5. Note that we use Convention 5.1 and the spaces introduced in Sub­

•

Moreover, the best possible constant

(5.55)

1.5 (ii). The following two theorems are analogues of Theorem 1.14 and

$

q q1/ q (pl)1/ '

~

.

(6.1)

1

$

p

$

q

$

v, w E W(a,b) . Define

00,

FR(x) = FR(x;a,b,w,v,q,p)

Ilw

=

l/ q

I

Iq,(a,x)' v l

-l/ p

rIp',(x,b)

and 5.11. Formulas connecting

v

and

w.

In Subsection 2.6 we introduced

formulas connecting the weight function w. v from (1.11) for

(6.2)

1< P$ q<

cf. (2.13), (2.15). The main tool for the proof was Theorem 1.14. Theorem 5.9, the extended version of Theorem 1.14, allows at least one of the formulas just mentioned to the case

Assuming

wE W(a,b)

and

1

$

.p

uS

$

B R

~uw1/q~ q, ( a, b)

(6.3) 00

:

holds for' every

b

(5.56)

J w(t)

dt <

00

,

x

and putting for

E (a,b)

sup FR(x) . a<x
$

CR~u'v1/P~ p, ( a, b)

u E ACR(a,b)

[and, consequently, the ineqaul1:ty

~HRf'W1/q~q,(a,b) S CR~fv1/P~p,(a,b)

(6.4)

with x

BR(a,b,w,v,q,p)

Then the Hardy inequality

to extend q <

=

00

H

R

from (1.6) holds for every B = BR(a,b,w,v,q,p) < R

00

f

E

~(a,b) ]

if and only if

•

65 64

C in (6.3) [and in (6.4)J satisfies R

Moreover, the best possible constant the estimate

b

(6.11) ~

B R

(6.5)

and

~

C R

k(q,p)B

(1

=

+

v

[J

I-p'

(t) dt

]-I- Q/PI

x

x E (a,b)

I IIp'

l/q

k(q,p)

(6.6)

I-p' (x) V

R

for

where

~I P

w(x)

3,) p

(1

+

L) q

(ii)

1

~ p ~ q <

w E W(a,b)

CD,

,

x

Let

6.3. Theorem. (6.

n

1

~

q

<

p

~

(6.10*)

v, w E W(a,b) • Define

CD

{I [[I

w«)

<

CD

and b

dt

dt

a

= AR(a,b,w,v,q,p)

~

J w(t)

f/q [J

V

1

I

-p (t)

ql dt ]l/ ]r

1/ r 1

I

v -p (x) dx

x

1

(6.11*)

(L

v(x)

x

I-p

I-p (x) w

q )

[J w(t)

dt] p-l+p/q

a

for

where

x E (a,b) .

Then the inequality (6.3) holds for every

1:

(6.8)

1. _ l

=

r

q

constant

p

Then the Hardy inequality (6.3) holds for every

u E ACR(a,b)

sequently, the inequality (6.4) holds for every

f E ~(a,b)

C = k(q,p) R

u E ACR(a,b)

with the

from (6.6).

As concerns the approach via the differential equations, the following [and con­ analogue of Theorem 4.1 holds: J if and only

if ~

<

CD

(6.12) C in (6.3) [and in (6.4)J satisfies R

l/ q '

~ ~ C ~ q

II

R

Similarly, we can extend to

u

~

q(pl)

II

q ~ . I

ACR(a,b)

mulas connecting the weight functions

wand

the approach via the for­ v

(see Subsections 2.6 and

5.11) :

(6.13)

1 < p

~

q <

CD,

v E W(a,b)

00

v, w E W(a,b) • Moreover, assume that

J vI-pI (t)

,

V

1

dt

<

CD

for

x E (a, b)

•

Then the Hardy inequality (6.3) holds for every (finite) constant

u E ACR(a,b)

C if and only if there is a number R

A> 0

with a such that

the differential equation

d i d q/p' I I

] - w(x) yq p dx dx

A -- [v q p(x)(- ~)

has a solution (6.15)

b

66

<

v E AC(a, b)

Let one of the following two assertions be fulfilled:

6.4. Theorem.

x

q

~

x

(6.14)

J

p

b

q1/q (P~g)

(6.10)

<

and

the estimate

(i)

1

•

Moreover, the best possible constant

(6.9)

Let

6.5. Theorem.

= AR(a,b,w,v,q,p)

y

o

satisfying the conditions

y' E AC(a,b) ,

y(x) > 0,

y'(X)

<

0

for

x E (a,b) .

I

-p (t) dt <

CD

6.6. Remarks.

(i)

The restrictive condition (6.12) can be removed in the 67

same way as in Section 4 (ii)

cf. Remark 4.6 (ii) and Theorem 4.7.

only for

If we introduce the number

KR

L q

inf a<x
f

b

w(t)

[ f(t)

+

x

B

R

1

J

I

v -p (s) ds

]q I p' +l

dt ,

then the inequality (6.16) holds

t

(i)

K from (4.11), we can again derive the estimates

which is an analogue of

p ~ q < 00 ;

1 :;

b

(6.18)

fop

u €

a=sg-~-1 p p' ,

J3
~ C ~ K~/q ~ k(q,p)B R . R (ii)

fo~

if and only if

ACL(O,ro)

u E ACR(O,oo)

if and only if

For illustration, we will now give several examples in which we will consider particular weight functions

w, v , conditions of the validity of

(6.19)

B > P - 1

a=sg-3-_ 1 p p' .

the Hardy inequality with these weights and, sometimes, also estimates for the best possible constant. Although it would be useful to know the exact

The constant

vaLues of the constants

B , B ' A ,A (which would make it possible L R L R to obtain good estimates for the best possible constants in the Hardy in­ equality -

see, e.g., (6.5) and (6.9», in most cases we will give only

C

C

where

k(q,p)

can be chosen in the form

k(q,p)B

is defined in (5.52) and

B

is

B L

or

B R

from (6.17).

necessary and sufficient conditions for their finiteness.

6.7. Example. (6.16)

p, q E [1,00)

Let us consider

[flu(x)l

q

l/q

s x

] l/p

dx

(6.20)

=

(0,00)

w(x)

,

~

x

a.

vex)

= xB

q , then the numbers

1_

B L

(E) l/q

+ lip')

for

J3 < P _ 1 ,

BR

(E) 1/q (p _ l)l/p' (8 + 1 _ p)-O/q + lip')

for

B

8)-O/q

q

q

>

,if

and

A are R

we can assert that the inequaLity (6.16)

does not hoLd (with a finite constant AC~(O,ro)

A L

C) on the class

ACL(O,oo)

E.

consider the

~

C

[J I

u' (x) p x J3 dx ] l/p

I

o

1 ~ p ~ q

<

00,

if and only if one of the following two con­

B

:>q
<

P - 1,

B
o~

0 ~ 8 g

3- _ 1 , p'

0>8.9.

3- _ 1 p'

P

(ii) for u € ACR(O,b) ditions is fulfilled: p > q , then the numbers

a, B €

b

]l/ q

P _ 1

(recall Convention 5.1).

both infinite, and consequently

dx

and

p

(6.22)

(6.17)

If we consider the case

a

(i) for u E ACL(O,b) ditions is fulfilled:

(6.21)

(p _ l)l/p' (p _

[Jlu(x)!q x

0 < b < 00

This inequality holds

a.,J3EE..

B and B from L R and (6.2), respectively, are given by the formulas

(or (5.47»

p

p, q E [1,00) ,

o

o (a,b)

For

b

~ C [flu'(X)\P

xa. dx )

If we consider the case (1.18)

and the inequality

00

o Then we have

6.8. Example. inequality

(6.23)

I :> p

~

q < 00

if and only if one of the following two con­

and

J3

or

B~

>

P _

a~s.9.-~-l

p

p -

o > ­

p'

1

I ~ q < p < 00 • Therefore, it is natural to consider (6.16)

68

69

,-

-a

- ",-"

n

W·-

-,,"'"_. ""-'W-

~--<

".",.. _;;'~ _...

-T-=·:i[iit~""'"ii-~'fi'-i"'jjjitqiiJ_Iiuiij~-:~-.fAij(-,~r:"'fii1fjj--i-;ff~ijrjjijfii~~M4i+-

=-;~-'-"'.,.".

·'~-"'''"i,r ~~=;~'~:':"-iriW

-1

""Tty"

1

(6.24)

~

q < p <

00

and

B

or

B~

.9....- - 1 p'

a > B .9­

> P -

p

p -

a > -

1

in comparison with Example 6.7, the set of admissible values of

a,

B

is

[1,00),

~

p, q

0

< a <

00

and

a, BEE. consider the

/q

x

a

~

q dX)I/

B c[J1ul(x)I P x dx

11/P

u €

~

(6.27)

(ii)

~

p

~

p

~

q <

B<

00

the corresponding numbers

~

(6.33) q

<

<

00

00

and

B< P - 1 ,

u~B!l_tL-l

or

B ;;; p - 1

u < - 1

and

B < P - 1 ,

a
or

B ;;; p - 1 •

u < ­

u E ACR(a,oo)

P

a=B!l-.9....--1 p p' .

P - 1

~

~

and

are infinite. The constant C = k(q,p)B

p > q

C in B = B L

with

from (6.17).

(i)

p = q E (1,00)

and

U

= B - p , the inequality

It can be easily shown that the inequality

(flu(x)l

q

1/ q x (In x)u dx )

~ C

[J lu'(x)IPx p-

1

(In x)

can be transformed into the inequality (6.30) for the function = u(l/x)

(ii)

are the same as the conditions for the vali­ AC (O,I) L

(6.28)

1

~

p

q < 00 ,

B>p-l,

(6.29)

1

~

q < P < 00 ,

B > p - 1 ,

u
function

(AC (O,I».

R

Here it is important that we consider the interval

(1,00)

the interval

(0,1)

or

If we investigate the corresponding Hardy inequality

(0,00)

for instance on the interval

u~B.9.-~-I, p p

~(x)

. Consequently, the conditions for the validity of (6.33) on the

class AC (I,oo) (AC (I,oo» R L

dity of (6.30) on the class

if and only if one of the following two con­

B) l/p dx

1

1

p'

ditions ffi fulfilled: ~

B = B R

6.11. Remark.

q < P

for

1

if and only if one of the following two condi­

ACL(a,oo)

tions is fulfilled:

1

i f and only i f

(6.30) was investigated by J. KADLEC, A. KUFNER [IJ, in the early sixties.

This inequality holds

(6.26)

E AC R(O,I)

For the special case

a

for

- .9....- - 1 u= B !l p p'

B>p-l,

(6.30) can be chosen as in this example, i.e.

a

(i)

(6.32)

or (Jlu(x)

u

q <

It can be easily shown that similarly as in Example 6.7, in the case

inequality

(6.25)

~

b :

substantially richer.] For

P

~

for

(ii)

[Let us emphasize the influence of the finiteness of the right endpoint

6.9. Example.

1

(6.31)

w c W(O,oo)

and ask whether there is a weight

[or a weight function

v

E

W(O,oo)]

such that the

inequality

(6.34) 6.10. Example.

(6.30)

p, q E [1,00) ,

For

1 [J1U(x) /q

~

lIn xl

o

u

dxl

q

~

1 P c[J1ul(x) I 0

70

for

u E AC (O,I) L

w(x) dx

1/ )

if and only if

q

~ C

l(J Iu

I

(x) Ip x P-

1 lIn x IB dx11/ P

o

[or the inequality P - 1 lIn

x

xl

B ] l/p dx

00

(6.35)

q

[f lu(x) I ~1 lIn xl

holds for every

u

1 q

dx J /

~ C[flul(x)I P vex) dXr/p]

o

°

This inequality holds (i)

[q

o

a,BEE. consider the inequality

1/

[flu(x)

u E ACL(O,oo)

or for every

u E ACR(O,oo)

provided 71

1

<

~

P

q

00 , then the answer is negative: it can be easily shown that

<

inequality on the class

the corresponding numbers

BL , BR are infinite for any choice of w or v , respectively. On the other hand, the answer is affirmative if we con­

w(x) =

We can choose a

=

~ - 1 = 83.p p'

6.12. Example.

ACL(O,oo)~

lx

lIn x/a

f

R

x

dt,

J

e

at2

J

dt,

e- 8t2 /(P-l) dt ,

1

e- 8t2 /(p-l) dt

x

which appear in the definition of the numbers

p 1 vex) = x lIn xl 8

8

with

> P -

1 ,

bers are infinite if

a

~

0

or

8

~

BL , BR ' AL ' AR . These num­ 0 . Consequently, the condition

see Section 8, Example 8.6 (v).

0.<0,8>0

(6.40)

For

p, q €

[1,00)

0., 8 ER consider the inequality

and

is necessary for the validity of (6.39) on the classes mentioned. The con­ dition

(6.36)

e

at2

x

ACR(O,oo) . or

leads to the calculation

AC (_00,00)

or

L

of the integrals

sider the inequality (6.34) or (6.35) on the narrower class of functions ACLR(O,OO)

AC (-00,00)

[ flu (x) Iq

e

ax

dX] 1/ q ~ C[ flu' (x) Ip

e

8x

dx

0. = 8

mentioned above seems to contradict our necessary condition

(6.40). But, in fact, TREVES investigated the inequality (6.39) on the more

Jl/ P

C~(_oo,oo)

special class

defined in Subsection 7.11. We will resume the

study of this inequality in Section 8. This inequality holds (i)

u E ACL(-oo,oo)

for

if and only if

7. COMPACTNESS OF THE OPERATORS (6.37)

1 ;:; P

~

00

q <

8

<

0

a

HL

~ND

HR

= 8 3. p

7.1. Notation and some auxiliary results. (ii)

u E ACR(_oo,oo)

for

(6.38)

;:; P

The constant

C

~

00 ,

q <

if and only if

8

>

0

a =

8 .9. p

For two Banach spaces

X, Y

we denote by

eX, y]

(7.1)

•

in (6.36) can be chosen in the form

C

= k(q,p)B , where

or

K[X,Y]

the set of all linear mappings from

X into

Y which are continuous or

compact, respectively.

(P.) 1/ q

B L

(p _ 1)1/P'(_8)-(l/q+ 1/p')

for

8 < 0 ,

(p _ l)1/P' 8-(l/q+ 1/p')

for

8

q

B

B R

(P.) 1/ q

If

q

For

p, q

[f

X

C, ~

Y

denote that the identity mapping ~

(1,00)

and

a, 8 E R consider the inequality

I

I

p = q = 2,

a = 8

>

0

in F. TREVES [l]

for

u EX, belongs to

K[X,Y] , respectively. We will say that the imbedding

and

I

is

X is continuously (compactly) im­

X.

The symbol (7.3)

for

I , Iu = u

continuous (compact) or that the space

1 q 2 q x lu(x) eo. 2 dx J / ~ C [f- lu'(x) P e 8 x dx J1/P ;

this inequality appears

and

0 •

bedded into (6.39)

Y , then the symbols

X~ Y

(7.2) >

[X,Y] 6.13. Example.

xC

U

--">.

U

n

will denote the weak convergence of

(see L. HoRMANDER [lJ, p. 182). The investigation of the validity of this

u

n

to

u

(in

x ).

The symbol 72

73

_

...

x

(7.4)

__

_.

fF

Ill<

___ ~_

_

=_rttllm

_~

.. '_

'".__

1£.-JSIt

.. _.,,~_

_ ._ _ __, .... .,-"'~,-'-__

=,!!!/ilIU

___._.""_,,,,,,_•• __. .____

T ~ [X,yJ

and

X*, y*

are the dual spaces to

X, Y ,

.

~.-'"

_

u_,,·,".~~·

T, acting from

Y*

into

(v)

(R. A. ADAMS [lJ, Theorem 2.21)

S~ LP(a,b)

in N. DUNFORD, J. T. SCHWARTZ [lJ.

T E [X,yJ ,then

If IITII

Let n

.•.

-'''_.,_~~_.~~""""""'."

,..-'--~"'.'.~,~"-~-"""".'

...

,""-'--."'~.~"---_---'"'~

is precompact in

and

if and only if T* E K[y*,X*J.

T E K[X,yJ

{u} n

ex,

u

n

f

(We define

00

•

A bounded set

hER

G

with Ih

=

I

[c,d J < <5

£ > 0

~ (a, b)

we have

=

0

for

e

x

(a,b) .)

(N. DUNFORD, J. T. SCHWARTZ [lJ, Theorem IV.B.7)

The sequence

{u} n

c=

~

P

<

00

u E LP(a,b)

if and only

for every measurable subset

M ~ (a,b) .

converges weakly to

LP(a,b)

Let 1

•

if the following two conditions are fulfilled:

introduced in Subsection 5.2. For

1

<

p

<

00

and

LP(a,b;v)

(0.)

v E W(a,b) , the mapping

(8)

uv 1 / p

I

un(t) dt

is obviously an isometric isomorphism of

LP(a,b;v) into LP(a,b) , and

1 p LP ' (a,b) into LP ' (a,b;v - ').

This fact together with Riesz' representation theorem leads to the following

or on

assertion:

ACR(a,b) ) is equivalent to the inequality

I

g(K) u(x) dx

for every

u E LP(a,b;v) .

q

1/

w(x) dX]

q

~

Tf

1/ P

a

and H are defined by the formulas

L

R (1.6». The inequality (7.9) means in fact that

=

HLf

b

C(flf(X)I P v(x) dX]

a

with

b

(fITf(X)l

(7.9)

u(t) dt

Let us note, that the Hardy inequality (1.11) (on

b

(iv) Let 1 < P < 0 0 , G E [LP(a,b;v)]* . Then there exists an element , l' P g E L (a,b;v -p) such that

00

M

7.2. Introduction. ACL(a,b)

f

---+

M

simultaneously an isometric isomorphism of

<

sup Ilunl1p, (a,b) n

defined by

G(u)

<

lu(x) IP dx ;'; £P .

u(x)

(vi)

---"- u . Then

We w~ll work mainly with the weighted Lebesgue spaces

=

and a closed interval

0

S and every

u E

1;'; P

a

(7.8)

Tu .

---+

>

<5

Let

if and only if for every

LP(a,b)

and

T* E [y*,X*J

Then

T EK[X,yJ,

~(u)

..

(a,b)\G

T E [X,yJ

Let

~,,--''..,.'''-''''.,..,,:,

Ilu(x + h) - u(x) IP dx ;'; £P

(7.7)

IIT* II •

=

...-'-'''"'.'''...

b

X* .

We will use the following assertions whose proofs can be found, e.g.,

(7.5)

•• '_~ ',_,," '''",' '~':'~--"C' ",~"."'''''.''''''''''''''-'-~'.''---'''''''''''''-'''--""",~'''''''''''~~~'"'-"''~'~~''''~'''',.''''"---"~"~="''''.'''''''''~·~'O'''-''''='·'''C'=-''''''-'~_''''''·''''.='.''''''·:·'''''''''=,,...'",."......~~,."-.""""-=""'.""_~''''~-''''2_'''''"=--=-:,

...........; " ~ t t ~ ~ ; ~ £ : ~ ~ ~ ~ ~ ~ , ~ ~ ~ l f ~ ~ ; ~ > ; W ~ ~ ~ < t ' Q _ ~ ~ : ~ ~ ' ! ~ I l ~ ~ ~ ~ : ~ ~ . m

such that for every

will denote the adjoint operator to

~

....o••__u.

there exists a number

T*

Tu

..~·.~

Yare isometrically isomorphic.

X and

then

(iii)

_ .

Further, the following two assertions will be used:

Finally, if

(ii)

__._

Y

2:

will denote that the spaces

(i)

~_.

asw;u::~!~~J&~J>Wo!.J'll!

or

Tf

=

HRf

(H

a (7.10)

T E [LP(a,b;v), Lq(a,b;w)]

Moreover, According to our foregoing results, the (necessary and sufficient) condition I GI

=

I g I P' , (a , b) , v 1-p ,

Consequently, (7.6)

[LP(a,b;v)]

for (7.10), i.e.

for the continuity of the operator

HL or

HR , reads as

follows:

*

, l' ;: LP (a,b;v -P ) . 75

74

--

B

L

(7.11)

~

<

00

<

00

or or

B

R

A

R

<

<

00

00

if

~

if

~

~

p

q

q < p

~ ~

--

-----

-- --------

f~(x)

00

=

1f ~

I-p' (x)

00

Therefore, it is natural to ask what are the additional conditions guaran­ teeing the compactness of the operator

--

RL or

= [J v

The corresponding results will be formulated in Theorems 7.3, 7.4 and

Denoting

(7.15)

We will need some information about the operator

T*

adjoint to

T

from (7.10). According to the assertions (i) and (iv) from Subsection 7.1 (cf. (7.6», it is T*

(7.12)

E

I 1 I I 1 I [L q (a,b;w -q ), LP (a,b;v -p

I-p I

Z

J f(t)

ZI; ~ 0

in

LP(a,b;v)

M

be a measurable subset of

)J

J zl;(x)

dx =

M

[J

I

[Izell (b) ,v = 1 . Therefore, let <, p, a, and denote MI; = M (a,1;) . Obviously,

n

(a,b)

fl;(x) dx)/llfl;llp,(a,b),v

M

[J

(RR£) (x) ,

I-p I

V

[J

(x) dxJ /

dt

The last integral tends to

(ilL £) (x) .

1

I

v -p (x) dx

Jlip

I;

[J

~

0

1 pl

V

1-p I (x) dx J /

.

a

a

MI; f(t)

-

I; dt

x (il;f)(x) =

I; ~ a+ .

for

dition (S), since (0:) is fulfilled

x

(7.13)

1I p

we have

f I; Illf II I I; p,(a,b),v

I;

b

=

)

(t) dt

Indeed, according to the assertion 7.1 (vi), it suffices to verify the con­

Moreover, it can be easily shown that

(R~f)(x)

I; ;:; x < b

a

sidered a little more general operator. The proofs given here are slight modifications of his proofs.

for

I;

If I , I; p,(a,b),v

[2J who in fact con­

a < x < I; ,

Obviously,

RR'

7.6. These results are due to V. D. STEPANOV [lJ,

for

I; -~ a+ , since

for

B < L

00

•

Thus, the con­

dition (S) is verified and (7.15) holds.

a

The assertions 7.1 (iii) and (7.15) imply that

Let

7.3. Theorem.

1 < p ~ q <

00,

v, w E W(a,b) . Let

FL(x)

and

BL

(7.16)

be defined by the formulas (1.17) and (1.18). Then the operator

b [

B = BL(a,b,w,v,q,p) < L

(ii)

lim

FL(x)

B

L

<

00

(I)

I [J

b

x~b-

Let

H be compact. Then ilL is continuous and consequently L

(see Subsection 7.2). Now, we will show that lim FL(x) = 0 . Let

us choose a fixed

76

I HL Z I; II

,

= lim FL(x) = 0

x~a+

Proof·

00

I; ~ a+ .

x

is compact if and only if (i)

for

Since

ilL : LP(a,b;v) ~ Lq(a,b;w)

(7.14)

IIHLzel1 ( a, b) ,w ~ 0 <, q,

~ I;

J

~, (a, b) , w

I;

fl;(t) dt

r

a

Jfl;(t)

dt

]q

a f

w(x) dx

~

" l;" p ,(a,b),v

w(x) dx

a

II f I; I ~ , (a, b) , v

=

b. J w(x) dx I;

I;

[J v - p' (x) 1

Jq -

q/p

dx

a

x~a+

~ E

(a,b)

and define

Fq (I;) , L 77

the equality

= 0 follows from (7.16).

FL(~)

lim

~

~+a+

lim FL(x) = 0 x+b-

We will show that

b

f [J

as well. Due to the assertion

q,

R

l'

P,

(a,b;w -q ), L

l'

(a,b;v -p

:(x)

for

a

<

x

~

~

for

~

<

x

<

b ,

~

~

E (a,b)

and

(II)

B

we have b

II g I

~ q' , (a, b) ,w 1-q

Denoting subset

[J

,

J

~

dXJ~[J w(x)

The last integral tends to

(7.17)

II;

0

for

II , = 1 ~ q' ,(a,b),w 1- q

z~ ---"- 0

in

~

Let the conditions (i), (ii) of the theorem be fulfilled. Let

S

=

dX)

~

q

q

o<

(7.20)

b-

due to the condition

If

uES,

p ,(a,b),v

lim ~+b-

F (~) L

o

for

~

-+

1-p'

0

-+

~

for

a -+

r J g~(t)

II HL

78

Z

~

f a

H f'wl!q L

<

E >

4 k(q,p l/q

with u

fEB. According to

E S the conditions (7.7),

p) are fulfilled. 0 . The condition (ii) of our theorem c, d €

E

FL(x)

3 1/ q

)

(a,b) ,

for every

c

<

d,

such that

x E (a,c) U (d,b) .

,then

lu(x) I

q

dx

~ b

x

(flf(t)

I

dtf w(x) dx +

a

x

f d

[Jlf(t) I dtJq w(x) dx

1

1

+ 1

2

•

a

b- . We estimate the integral (a,c)

11 =

dt

p'

x

h<-II q',(a,b),w 1- q ,

1

'

v -p (x) dx ~ ]

(7.22)

1

1

using the inequality (1.12) for the interval

and (7.20): c

b

Lq(a,b;w) , and

q

J

~ J

follows from the estimate

,p' , II p' , (a, b) , v l-p

its image in

(a,c)u (d,b)

b- .

b

*-

/

instead of

u = HLf.w

(7.21)

The assertions 7.1 (iii) and (7.17) imply that

Finally,

1

For this purpose, choose

B < 00 L and in view of the assertion 7.1 (vi),

, l' Lq (a,b;w -q )

IIH~ ;~II ,

(H B)w L

~

~ -+

HLB

the assertion 7.1 (v) we have to show that for

c

(7.18)

LP(a,b;v) ,

the set of all functions of the form

(7.8) (with

b 1/ [J w(x) dxJ q

~

Mfl(Cb)

Moreover,

a

F( (~)

w(x) dXr' -p' /q'

implies the existence of numbers

1/'

b

w(x)

v 1-p ' (x) dx J [f

be the unit ball in

(7.19)

M

J

b

lrJ

w(x) dx

[J g~(x) dxJlllg~11 q , ,(a,b),w l-q'

dx =

M

[

(x) dx

denote by

1/q'

;~ = g~/IIg~II , 1-q' , we have for an arbitrary measurable q ,(a,b),w M C (a,b)

f ;~(x)

l-p'

~ q' , (a, b) ,w l-q'

)J

(see (7.12), (7.13», is also compact. Choosing again

={

v

J

h liP'

= H E [L

g~(x)

dt

~~

7.1 (ii), the operator H~

p' g~(t)

f a

x

[flf(t) I dtr w(x) dx

~

a

J

~ [k (q , p) B (a , c; w , V , q , p) • I f /I q ~ L p,(a,c),v

<,

79

~~,-.

__

~.~ ~- -:?'fiiIH -:~_"#~~-~~~~~~~:;~~~~~~':;:;'.i~:,';'L~!t"~;'~;':.:'T~~~~~,:"",~).~~ ..-------··--·-----·---·-··--r'T:"-·:-·-·---·--·-··-·---··---·--··-_.-_._ ~,~!:_'_~':'"":-_~:~:-"';~~;~~~~~~:;~~~~~~~~~~~;~~~"T;~~~ -.-.. -..-.-.- -.--_.,...---.-.-.-.---.. - ..----- ---- --.-.-- - ---- - -----..

.-.---------.--------.------~-.----.---.-.---

~

[k(q,p)

sup

a<x~C

~ ~q

FL(x)]q

~

3.4

(b) ~ 1 since u ~ S and consequently fEB). p. a, ,v Using H~lder's in~quality, the inequality (1.IZ) for the interval (d,b)

a

If

I

b =

Z b

~

d

f

[flf(t)1 dtr w(x) dx +

1~

x [Ilf(t) I dt]q w(x) dx

(7.25)

Ilfll~,(a,b),v +

~

dy

~

E:

J

J

c

~

JI u (x) I q dx

1

1

q

3

2

I

=

Iq

lu(x + h) - u(x)

dx ~

(7.26 )

b

~

q 1 2 - {

I

b lu(x + h) I

q

I

dx +

d'

I

due to (7.Z3), since

[u(x)

E (a,c)

c'

d' E (d,b) . Then

c'

b

d'

Ilu(x + h) - u(x) [q dx

I

I

I

a

a

a

1

=

Ilu(x + h) - u(x) I a

d'

for

x

q

~

c' J

+

dx

¢

+

Finally, let

Iq

dX} ~ 2

q

E:

12 ~

q

"3

d'

b

b

b

=

11 '

1 '

!u(x + h)

q

I

dx

~

I

Z '

Ihl < H/4

J

[u(x)

Iq

dx

~

l

Z

d'

d'

u(x)

~

<­

and (7.23).

(recall that we define

I u (x) I q dx

a

d'

~

p ) now follows from (7.Z1), (7. ZZ)

instead of

Further, choose some points

I

Analogously we have

q sup F (x)] q ~ _E:_ q L 3·Z d<x
lu(y) I

b

zq-1 {Fq(d) + kq(q,p) [sup F (x)] q} L d<x
The estimate (7.8) (with

I

which together with (7.22) implies

a

d

~ [z k(q,p)

(7.24)

dx

a

d

[f

w(x) dXJ

I u (x) I q dx

a

+ [kCq,p) BLCd,b,w,v,q,p) Ilfllp,ca'b»)q},

~

I

c

q

a+h

I

d, q/p'

1 p v - (t) dt]

b

[f

I

b

d

a

d

{

lu(x + h)

c'

~

d

a

zq-1

c'+h q

a

x

(7.Z3)

~

I

a

zq-1 {

min (c-c', c'-a, d'-d, b-d') ,

=

c'

[flf(t)l dt + Ilf(t) I dt]q w(x) dx

d

~ H , where

then

d

I

H

[flf(t)1 dtJq w(x) dx

d

Ih I

Z x

I

a

h E: R is such that

(with the corresponding estimate for the best constant) and (7.Z0), we have for the integral

C f lu(x) 'I q ) dx

ju(x + h) I q dx +

I

~f~

(recall that

I

c'

q 1 2 -

c

and denote

=

c' - H/4 ,

d = d' + H/4 . Then

d'

J

+ 1

J

J

+ J3 2

3

J

=

lu(x + h) - u(x) I

q

dx

c'

c'

d'

(a,b) ). We have

J

(7.Z7)

1 q 1 q

I(HLf)(X + h) w / (x + h) - (HLf) (x) w / (x) jq dx

~

c'

d'

~

q 1 2 - {

J

I (HLf)(x + h)

[w 1/ q(x

+

h) -

w1/ q(x)] Iq

dx +

c'

80

81

d'

d'

JI

+

[(HLf)(x + h) - (HLf)(x)] w1 / q (x) IqdX}

s

J ~f~q

s

q 1 2 - (J 31 + J 32 ) .

c'

x+h I-p'

I Jv

p,(a,b),v

d'

Holder's inequality yields q q J ( J If(t) I dtf Iw1/ (x + h) - w1/q(x) I dx

=

c'

d'+h

J ( Jlf(t) I dtf c'+h

Iw1/q(y) - w1/q(y - h) I

q

J I II ~ , f

( ~J v 1- p ,(t) dt )q/P' IwI /q(y) - w1/ q (y - h) Iqdy

(a, b) ,v

c'+h

[J

S

v

I-p' (t) dt

/ ' d q 1 q 1/ _Iw q(y) - w / (y - h) I dy .

J

J

provided

c

a

For v

Since number

I-p'

1 ­ S L (a,d)

H O

0

>

and

w1/ q

such that for

E Lq(c

hER,

a

C

l'

v -p (t) dt

]q/P']

d'

q

(t) dt

P

I/

'

S

c;q/ [3'2

q

[J w(x)

dX)]

c'

E

q

32 - 3'2 q

Ih I < min (H/4, HI) Ihl < 0

=

min (H/4, H ' HI) , (7.27), (7.29) and (7.31) yield O

3"

J3 S

Now, the estimate (7.7) (with

d

('J

we have

c;q

(7.32)

[hi < H we have O

q 1 L3'2q f1w1/q(y) - w / q (y - h) I dy S c;q / 1

1 p v - '

HI > 0

s-­

- H/4, d + H/4) , there exists a

d

q p / ' .

which together with (7.30) implies (7.31)

q p ]

r

x

a

d

dt

I-p' - ­ v E L 1 (c,d) , there exists a number

Ihl < HI

IJ

sup c'<x
a

d '+h

s

w(x) dx S

X

x+h

dy S

(7.28) S

hER,

I J v 1-' P (t)

sup c'<x
and

w EL (c',d')

such that for

y

dX]

1

Since

a

x+h

[Jc' w(x)

S

d' x+h 31

q/p'

x

c'

J

(t) dtl

,

instead of

q

in view of (7.25), (7.26) and (7.32).

p ) follows from (7.24)

o

Let us formulate the 'right endpoint' analogue of Theorem 7.3.

which together with (7.28) implies q

(7.29) provided For

J

7.4. Theorem.

~ -(­

31 - 3'2 q

H : LP(a,b;v) R

we have

d' x+h J 32

=J c'

(7.30)

JIJ c'

82

x

v, w E W(a,b) . Let

00,

FR(x)

and

BR

IJ

f(t)

I

dt -

J f(t) a

a

dtlq w(x) dx S

(i)

w(x) dx S

Lq(a,b;w)

(ii)

B = BR(a,b,w,v,q,p) < R lim

FR(x) = lim

x~a+

q

If(t) I dt/

-+

is compact if and onZy if

x ,

d' x+h S

1 < p S q <

be defined by the formuZas (6.1) and (6.2). Then the operator

Ih I < min (H/4, H ) O

Ihl < H/4

Let

For the case

00

,

FR(x) = 0

x~b-

P

>

q , the situation is quite different:

83

-"- --

1

Let

7.5. Theorem.

•. -_._.

P

< q <

<

00

v,

,

W E

----

-

W(a,b) . Let

A L

-- -

b

be the number

H : LP(a,b;v) L

is compact if and only if A L

AL(a,b,w,v,q,p)

=

<

AL(a,b,w,v,q,p)

00

A comparison with Theorem 1.15 shows that

H is continuous if and L only if it is compact. Therefore we can reformulate Theorem 7.5 as well as its counterpart for the 'right endpoint' with A from the formula (6.7) R as follows: 7.6. Theorem.

Let

1

q

<

p

<

<

v, wE fl-'(a,b) . Then the following

00,

[Jb

1

<

q

<

00

•

<

P

<

00

1 q

-1 =

,

~

r

w(x) dx

fir

.

-

1 p

-

,

v, w E W(a,b) . Let

A L

Then

(ii)

, 1/r (~) FL(E;) ~ ~L(E;) ~ AL

(iii)

(3.)

(iv)

lim

r

r/Pr

x

A~ = A~(a,b,w,v,q,p)

1/r

w(t) dt

-","",.-=-_.~',.'--~-~.--=---,...------:-~

<

00

,

FL(E;) ~ ~~(E;) ~ A~ ,

~L (0

= 0

E;+a+

(i)

H. E [LP(a,b;v), Lq(a,b;w)] ,

(ii)

H. E X[LP(a,b;v), Lq(a,b;w)] ,

(iii)

A. (a,b,w,v,q,p)

lim ~1'(0 = 0 L E;+b­

(v)

1

i

riP'

(i)

conditions are equivalent:

where

Let

7.7. Lemma.

vl -p' (t) dt

- - ---'---

a

E;

Lq(a,b;w)

-+

x

{ J [[J

defined by the formula (1.19). Then the operator

- -'-

1

<

Proof·

The condition

(i)

A

00

L

<

00

is equivalent to

1

L

or

i

=

H E [LP(a,b;v), Lq(a,b;w)]

R .

L

According to Subsection 7.2, this implies

The proof of Theorem 7.5 is based on a lemma in which the following

notation is used with (7.33)

1 =

r

H~

1 _1 q

p

However, this is equivalent (due to Theorem 6.3) to

,', _ 1-p' 1-q' , , Ai* -_ AL(a,b,w,v,q,p) - AR(a,b,v ,w ,p ,q )

b [[x l ' { f f v -p (t)

dt

) 1/ p ,

a

a

and for

E; E (a, b)

(7.34 )

\(0 E;

[b

f w(t)

dt

)1/ P] r w(x) dX}

AR(a,b,v 1 -p ' ,w 1-q ' ,p',q') 1/r

~

~~(O

00

and the assertion (i) follows from (7.33).

x

(ii)

E; E (a,b) . Then

Let

~L(E;;a,b,w,v,q,p)

x

b

x

d t ) 1/ q

[f v 1-p' (t)

dt f I q ,

r

:i';

vl -p' (x) dx }1/

r

b I E; X 1 p [f w(t) dt f q [ f [ f v - I (t) dt E;

,

, 1/r

a

(E..:-) r

(7.35)

<

~ :i'; ~L (0 :i';

{ f [ [f w( t ) a

, l' , l' H E [L q (a,b;w -q ), LP (a,b;v -p )] R

=

~~(E;;a,b,w,v,q,p)

b [

JE; w(t)

a

a

dt

)1 I q

[E;J

'

r

Iq'

v 1- p (t) dt

I v 1-p I (x) dx r r

) 1/p ,

(~') 1/r

F (0 L

,

a

which is the estimate (ii). 84

85 ",,:,

j${i'.:

-------,-;;:-­ - - -

~ ~ ~ = ~ '~"".-~-=~~---"~,-" "--------­

(iii)

The proof is an analogue of that from part (ii).

(iv) and (v) are consequences of the absolute continuity of the integral (see (7.34) and (7.35); moreover,

~L(b)

A < L

=

00

¢~(a)

and

There is a close connection between the numbers ~

(1.19) and

~

from

from (7.33), namely, -- t.~) q

~(a,b,w,v,q,p)

0

>

£

(7.37)

i'

~

¢L(c)

¢~(d)

$

The assertions (iv), c, d E (a,b)

,

(v) of Lemma

c < d , such that

i

where

, l/r

(7.36)

For this purpose, choose

7.7 imply the existence of numbers

~
7.8. Remark.

7.3, we have to show that for U E S = (H B)w l/q cf. (7.19) the L

conditions (7.7) and (7.8) (with q instead of p) are fulfilled.

K

* AL(a,b,w,v,q,p)

4'3

=

-

k(s,t)

1/

q(~)

1/r

q

=

s

max (k(q,p), k(p',q'»)

lis ·(t') lis'

,

s, t E (l,oo)

for

This can be easily shown if we suppose that

w, v

I-pi

II

from (7.21) the estimate

Indeed, integration by parts then yields

I [f

b

~

=

b

II $ [ k(q,p) AL(a,c,w,v,q,p) ] q $ [k(q,p) ¢L(c) ] q

x

b

a

wet) dt J r/q

[f

x

l p v - ' (t) dt

r/q

J

' v 1-p ' (x) dx

a

b

~ Ia [fx wet)

dt

r Iq - 1

J

r/q If

1

rip,

=

w(x)

(~, +

1)­

1 [xI

v , w n

v

n

n

J

w ,v n

ij~L~

, we introduce auxiliary

1 (1 n

I

Z

86

ij~~~ ~ k(p',q') A~(d,b,w,v,q,p) S k(p',q') ¢Z(d)

S [Z k(p',q') (~)

cf., e.g.,

H L according to Theorem 1.15.

Let

=

0.13)

(i)

If

AL(a,b,w,v,q,p) <

00

•

1

2

~ Zq-l {F~(d) + ~~Lijq} S 1/r


is compact, then it is continuous

Similarly as in the proof of Theorem

from (7.Z3) we have the estimate zq-l {Fi(d) +

Lk (p' ,q ')


~ (d) ] q} ~

q

s

E

3'Z

and consequently,

q

The rest of the proof is the same as in Theorem 7.3.

7.10. Examples.

(ii)

in

--*

q

1

PEL (a,b)

7.9. Proof of Theorem 7.5. 00

L

Using this estimate, the first inequality in the assertion (iii) of Lemma

+ x 2/ (p'-1»)

instead of w , v . The formula w , v n n (7.36) then follows by the monotone convergence theorem.

A < L

-

7.7 and (7.37), for the integral

the identity (7.36) holds for

and

C

the upper estimate of

~

by the formulas

min (w(x), n/x , n)

n

k,

the analogues of the operators H and H and H = H L L R L only, then for their norms we ' but now acting on the interval (d,b)

If we denote by

r/q' + 1 = rip' .

E W(a,b)

1-'

of the function

C ~ k(q,p) A ] . L L

Z

wn(x) Obviously

[note that with help

q

H R

have the estimate

vex) +

(x)

q

3'4

l' r I q , +l v -p (t) dt dx

a

v , ware general functions from W(a,b)

functions

£

~---~

(1.Z6) can be rewritten into the form

(f) (~/ since

and (7.37) we have for the integral

Using the above introduced notation

1 L (a,b) .

€

o

The examples from Section 6, in which the Hardy inequalitj

was considered, give at the same time necessary and sufficient conditions for particular pairs of weights H L

and

H R

v , w , which guarantee the continuity of

as operators acting from

LP(a,b;v)

into

Lq(a,b;w)

. Using

87

--' .­

~:-~:':::"-':""~~~:~ _." .. __ "';:"~:::~.~~ o.c~::~~~;_~~,,::_:_:_

~_~::_~__.-:_"':'- ;_~"::,,,::~:c~~~:.~.:

the foregoing results, we can give conditions under which these operators

7.11. Weighted Sobolev Spaces.

are compact. For simplicity, we will deal only with the operator

us define the

(i) (cf. Example 6.7) w(x) = xu,

vex) = x

B,

1 < P

Let u, S

$

q < 00,

=

(a,b)

(0,00)

H . L

being described by (6.18). The continuous

can never be compact since

u Lq(O,oo;x )

~

FL(x)

= const

and thus conditions (ii)

of Theorem 7.3 cannot be satisfied.

H L

a = 0,

b < 00 ,

Lq(O,b;x )

= (11 u" p

p,(a,b),vo

(iii) (cL Example 6.9) Let p, q E. (1,00) a S w(x) = x , vex) = x . Then the operator

a >

°

(7.41)

va' VI

E

1 Lloc(a,b)

Consequently, under the assumptions (7.40), (7.41), the space

u Lq(a,m;x )

w~'P(a,b;vO,vI) defined as the closure of the set

a< S .
assertions, see A. KUFNER, B. OPIC p -

1,

a < -

1

fore, when

a < min (- 1 , .9.( S - p p

+ 1) - 1)

If

J.

I

I

I

similar to that of case (i) above and the operator

H L

E

is satisfied on

~u~

cannot be compact. for every

88

w, VI

W(a,b)

II u II q, (a, b ) ,w

(iv) Let p, q EO,"") . In the case of (a,b) = (0,1) and

-1 a w(x) = x In x , vex) = x p-l In x B or (a, b): (_00,00) and Ux Sx w(x) = e , vex) = e (cf. Examples 6.10 and 6.12), the situation is

I

with respect to the norm (7.39)

,"', I ,:0"

'\,

~'

'.; ,:',.

o '

VI

we will always assume

satisfy the conditions (7.40), (7.41).

are such that the Hardy inequality

$

ACL(a,b)

q,(a,b),w

v

(For the proofs of these

[2J, Lemma 2.3 and Lemma 4.4.) There­

dealing with weighted Sobolev spaces,

that the weight functions

B.
S EO R,

C~(a,b)

is also a Banach space (normed again by (7.39».

or ~

a , b . Then the inclusion

holds if and only if b

is compact if and only if

S

+ I u ' I! pp,(a,b),v ) 1 I p 1

C~(a,b) C w1 ,P(a,b;v O'V 1 )

(compare with the continuity conditions (6.21), (6.22».

B
and

(a,b) ,

in the neighbourhood of the endpoints

--+

E W(a,b) , let

,

loc

VI

a>S.
H ; LP(a,oo;x S ) L

vo' VI

­ (a,b) , which means that VI-lip E Lp' (c,d ) for every interval [ c,d ] C l,p m then W (a,b;v 'v ) is a Banach space. Further, denote by CO(a,b) the O 1 set of all infinitely differentiable functions u on (a,b) which vanish

is compact if and only if S
-lip E LP

(7.40)

a

--+

l,p,(a,b),vO,v 1

If, in addition,

(ii) (cL Example 6.8) Let p, q EO ,00) u S w(x) = x , vex) = x . Then the operator

; LP(O,b;x S )

I! u I

(7.39) B L

and

00

weighted Sobolev space

as the set of all functions u E AC(a,b) such that u E LP(a,b;v ) O u' E L P (a,b;v ) . On this space, we define the norm by the formula 1

operator H ; LP(O,w;x B) L

~ p <

For

w1 ,P(a,b;v O'V 1 )

(7.38)

,

'~.

C I u ' I p, (a, b) , V 1

or

S C~u~

ACR(a,b)

then obviously

l,p,(a,b)'V ,vl O

V E W(a,b) . This means that o

89

.

In the sequel, we will establish conditions on the weights which guarantee

is defined by

not only the continuity, but also the compactness of the imbedding (7.42).

Du

Let

7.12. Lemma.

--+

HL , HR

If

, p

(i)

(ii)

1:::; p, q ;;;

o

FrO, and

H is continuous 3 then

As a consequence of Lemma 7.12 and Theorems 5.9, 5.10, 6.2, 6.3, 7.3, 7.4 and 7.6 we immediately obtain

If

7.13. Theorem.

00

and

H

is compact 3 then

Let

w~ ,p (a, b i v0' vI) ~ ~ Lq (a, b jw) • Obviously it suffices to give the proof for the case

follows from the continuity (compactness) of the operator numbers

B (a,b,w,v ,q,p) 1 L

or

A (a,b,w,v ,q,p) 1 L

HL

If

H = HL . It that the

E

1

L (a,x)

for every

x

E

i

satisfy (7.40), (7.41).

00,

-I

p

00

B (a,b,w,v ,q,p) i 1

,

<

00

,

then

are finite, and conse­ (ii)

(a,b) .

This implies, together with the density of the set u

v o ' vI

=R

1;;; p :::; q :::;

If

1 < P :::; q <

lim x~a+

'w~'P(a,bivO,v1) ,that

YO' vI' wE W(a,b) ,

w~'P(a,bivO,v1) ~~ Lq(a,bjw)

(7.44 )

quently

1-p I vI

Let

= L or

i

(i)

Proof.

u'

w~'P(a,bjVo,V1) ~ Lq(a,bjw) . 1 < p,q <

LP (a,bjv 1)

Now, the assertions of our lemma follow from the formula (7.43).

Lq(a,bi w)

be one of the operators

=

--+

and is obviously continuous.

YO' vI' w E W(a,b) . Let

LP (a,bi v ) 1

H

W~'P(a,bjvo'V1)

D :

w~'P(a,bivO'V1) C> Lq(a,biw)

(7.42)

C~(a,b) in

from the latter space fulfils

00,

B (a,b,w,v ,q,p) 1 i

<

and

00

F (x i a,b,w,v ,q,p) = lim F (x j a,b,w,v ,q,p) = 0 1 i 1 i x~b-

then

w~'P(a,bivO,V1) ~~ Lq(a,bjw) .

(7.45)

x

u(x)

=

J u ' (t)

dt

for a.e.

x E (a,b)

(iii)

If

1;;; q < p <

00,

A (a,b,w,v ,q,p) < 1 i

00

,

then (7.44) holds.

(iv)

If

1 < q < P <

00,

A (a,b,w,v ,q,p) < 1 i

00

,

then (7.45) holds.

a

(cf. the proof of Lemma 1.10). Thus, the identity operator I : w~'P(a,bjVo,v1)

--+

Lq(a,biw) 7.14. Remarks. (i)

can be expressed as the composition (7.43)

I =H a D L

where the operator

Let us define the weighted Sobolev space

W1,p( a,bi v ,v 1 )

L O as the closure of the set of all and

u '

E LP(a,biv1) . If we take

u E ACL(a,b) i

such that

u E LP(a,bivO)

= L in the assumptions of Theorem

7.13, then the assertions (7.44) and (7.45) hold with

wi,P(a,bi v O'V 1) 91

90

.. _

__,____

_

._~=

=~ ~__~__~~~~-e=c-"""-

-

~_-=--~~~=-~==--:~-;~

-

_ _=----

"i-c:.-'0"':~-_'-"'2220i.'-

,J...

w~'P(a,b;vO,vl) . The similar consideration concerning the

instead of

-~=-

.;::':;~~_

;"~':;Z-~"'~:?=--=-'-:'--:-'-=._ ~~~-~

•

-":--:=:-c--,::"~

c~'".-~

~ -~'i

.."~..,,

S>p-l,

(8.3)

~--=~-"

= .

:;''''"~

;'-:::~-F--- '~"""~=

• ,.

a

,:..~-=

~~_~~_~=

"",_"",~c,==,"""---------'-="~~-=""",,,,", __ ~ -;:c-

--

=

\11

-

~c-""",,,",,=::-~:·.-=;~--~"'~- -;;'7"~_~","~-

-.

S .9. - .<:L - 1 P p'

right endpoint is left to the reader.

in the case of the inequality (8.1), or (ii)

The results mentioned in Theorem 7.13, part (ii), generalize a > 0

certain results from the paper B. OPIC, A. KUFNER [3J. On the other hand,

(8.4)

the results contained in Theorem 7.13 can be derived also under some modi­

in the case of the inequality (8.2) [cf. Example 8.16, formula (8.97)J.

see Theorem 8.23.

ficated assumptions (iii)

The invalidity of these inequalities on the classes

All conditions guaranteeing the continuity or compactness of

the imbedding

AcR(a,b)

w~'P(a,b;vO,vl) C Lq(a,b;w)

or

is a consequence of the fact that in our particular cases B L

(B.5)

ACL(a,b)

=

and

00

B R

This indicates that the obvious condition for the validity of the general

and appearing in Theorem 7.13 (and also in Theorem 8.23) are only suffi­ Hardy inequality

cient. One of the reasons is that the weight function

V

o

does not play

(8.6)

II uw

any role in these conditions. In Section 11, we will mention some recent results concerning necessary and sufficient conditions for the validity of

function

v

w~'P(a,b;vO,vl) ,

from Examples 7.10, and

V

oE

vI

W(a,b)

plays the role of the is arbitrary except for

~

C I u ' v 1/ p II p, (a, b )

ACL(a,b)(~

ACLR(a,b)

Examples to Theorem 7.13 can be easily constructed using

Examples 7.10. Note that in

q , (a , b )

on

(7.44) and (7.45) (see Subsection 11.3). (iv)

1 Iq II

ACR(a,b) ,

namely the condition min (B (a, b, w, v, q, p), B (a, b, w, V , q, p)) R L

(8.7)

<

00

,

is only sufficient.

the conditions (7.40), (7.41).

Therefore it is meaningful to look for necessary and sufficient con­

ditions for the validity of the Hardy inequality (8.6) on the class 8. THE HARDY INEQUALITY FOR FUNCTIONS FROM

on the class of functions vanishing at both endpoints

ACLR(a,b) ,i.e.

ACLR(a,b)

of the interval

B.l. Two examples.

1 [f lu(x) jq i lIn

(8.1)

For the case

If we consider the Hardy inequalities

00

q a Jl/ xl dx ~

1 ~ p ~ q <

00

,

such a condition was derived by P. GURKA

[2J. He has shown that the corresponding necessary and sufficient conditior C

[f

Iu' (x) Ip x P- 1 lIn xl S dx J 1/p

o

o

(a,b)

reads

(B.8)

or

B(a,b,w,v,q,p) <

B

00

where

[ f !u(x) I

q

(B.2)

e

ax 2

Jl/q ~ dx

C

[ flu'(x) IP

eapx2/q dXJ1/P

(B.9)

B

=

q 'I -l/PI/Ip',(a,c)' 'IV I - l/P II p',(d,b) }] sup [llw1/ llq, (c,d) min { Iv

the supremum being taken over all pairs for 1

~

p

~

q <

00

,af~

ER , then we have shown in Subsections 6.11, 6.13

that they hold neither on the class

ACL(a,b)

nor on the class

ACR(a,b).

provided

such that

a < c < d < b .

Simultaneously, P. GURKA obtained the following estimate for the best pos­

sible constant

On the other hand, we have mentioned in these examples that both in­ equalitiesholdonthedass ACLR(a,b)=ACL(a,b) (I ACR(a,b)

c, d

(B.10)

C

in (8.6):

l B ~ C ~ 4B . 2

93 92

b

In the following theorem, his result is extended to the case 1

~

p

$

q

00

$

Let

8.2. Theorem.

finite constant

1 ~ p $ q $

C

>

v, w

~

W(a,b) . Then there exists a

:N

(8.12)

(8.13)

:N 2

2- 1 / p B ~ C ~ h(q) B

for

I v~-p' (t)

{nE:N' '

=

I v~-p' (t)

c

b

dt

~ f v~-p' (t)

dt} ,

dt} .

d

C

~

:N 1 . Further, let

znE[d,b)

b

c

f

(8.17)

~

v -p' (t) d t

f

and for

4B

v

i-p' (t) dt,

n

n E :N

1

zn

n E:N

1

and

x E (a,b)

define

x

q

(8.18) 8.3. Remark.

2- 1 / p

n

B

~ C

dt

a

where 2(2 q - l)l/q

$

(8.19)

B

which follows from the fact that there is a number So € (1,2), So = sO(q) , . s q l/q such that the funct~on g(s) = s-:-T (s - 1) decreases on (l,sO) and increases on

I gn(t)

u (x)

q E (1,00) , the estimate (8.11) can be replaced by a

For

little rougher estimate (8.15)

$

a

a

~

dt

be such that

s s =-1 (s q - l)l/q

2-l/ P B

b

d

f v~-p' (t)

{n E:N ;

, :N 2 by

of generality we can suppose that it is the set

and (8.14)

~1

At least one of these sets contains infinitely many elements; without loss

inf g(s) , 1<s<2

g(s)

00

a

with

h(q)

=

1

C in (8.6) satisfies the esti­

mate

q E [1,00)

<

c

B from (8.9) is

if and only if the number

Moreover, the best possible constant

for

P ' (t) dt

(cf. (3.13)). Let us define two sets

finite.

(8.1l)

fa v n-

such that the Hardy inequality (8.6) holds for

0

u E ACLR(a,b)

every function

00,

1

o<

(8.16)

and the estimate (8.10) is improved.

(sO,2J; thus

h(q) = g(sO) < g(2) = 2(2 q - l)l/q

gn

[X(a,c) - X(Zn,b)]

v 1- p ' n

Obviously (8.20)

un E AC

LR

(a, b)

and c

8.4. Proof of Theorem 8.2. fix numbers

c, d E (a,b) , v (x) n

Obviously

94

(i)

v

=

v(x) +

E W(a,b) n

l

n

and

Let

C

<

00

and

1 < p

c < d , and define for (1 + x 2 /(p'-1))

~

q < 00 . Let us

(8.21)

u (x) n

n E~ functions

1 P ' (t)

fa v n-

dt

for every

x E [c,d] ,

which implies '

x E (a, b) •

d

(8.22)

[

I w(t) dt

c

c

Uv~-p' a

(t)

dtf ]l/

q

d

[flun(t) I

q

w(t) dtf/q

c

95

In view of (8.20), the Hardy inequality (8.6) holds for of the estimate v(x) ::;; v (x)

for a.e.

n

un ' and in view

which is in fact the first estimate in (8.11).

x E (a,b)

(ii)

we have b

(8.23)

dxr/q : ; C [Jlgn(x) IP vn(x)

a

dX]

p. (8.26)

(J Ign (x) Ip vn(x) dx ]

n E N

and

ro

1 < P <

=

q

,

ro

•

For the function

r

1-'

J v n P (t) dt

n

from

1I q I ' I lunw q,(c,d)

Ilunt, (c,d)

which is an analogue of (8.22). If we rewrite (8.23) in a form meaningfull also for

=

q =

00

viz.

,

b

b

(f IX(a,c) (x)

u

we obtain in view of (8.21)

1

a

(8.24)

ro

a

The formulas (8.17) and (8.19) imply l/ P

C <

c

a

b

Let

(8.18) with

1I

b

(Jlun(x) I q w(x)

1/p C ,

B ::;; 2

(8.25)

x(zn,b) (x) IP

v~-p' (x)

Ilunw dX] lip

1/q'I

I q, (a,b) ::;; C (Jlgn(x)

IP

1/P vn(x) dx

a

=

a

J

,

we obtain in view of (8.26) and (8.24) the estimate

c b (J v 1- P ' (x) dx + vl - P' (x) dx ] lip n n a zn

f

c

=

2 II P

[J vnl- P' (x) dx riP

c

,

f v~-P

a

c

1-p' J1/P 1/p C (t) dt ::;; 2 (f vn (t) dt a

a

which together with (8.23) and (8.22) yields as in part (i)

and we again d

c

(J w(t) dt]l/q

[f

c

v~-p' (t)

c dt] ::;; 2 1/p C

a

(f

v~-p' (x)

(iii)

dX]l/P .

(iii-I)

a

d l/q c , lip' (J w( t ) d t ] (J v -P (t ) d t ] ::;; 21/P C

~

and letting

-+

ro

n EN 1 ' we arrive at the estimate

(J w(t) dt] c

II

c

q

(f v l-P' (t) a

Thus, in view of (8.9) we have ~6

dt J

=

P

ro

<

ro

Assume in addition that

-1

(t)

<

ro

Me

1 = (a,c) , 1

d ,denote

(8.27)

1 ::;; q ::;;

1

5. = ess sup v 1 tEl.

-1

1

2

(a,b) . Let us fix numbers

= (d,b) l/P

and define

, i =

1, 2

v E W(a,b) , we have

5.

(t) = II v -

c, dE (a,b) ,

I , I P , i

1

'I ­ l/P II I w1/qll q,(c,d)'IV P' ,(a,c) = d

<

for every measurable subset c

a n

C

ess sup v tEM

In view of (8.16), we obtain the estimate

c

Let

arrive at the inequality (8.25).

(recall Convention 5.1). 8ince

1/P

loss of generality we can assume that

' ::;; 2 1/p C .

1

>

0 . Without

51 ::;; 8 2 and, in view of our addi­

tional assumption,we have (8.28)

o

<

8

1

There exists

~

8

2

<

n E N

ro

such that 97

a

(8.29)

<

1

Ii

51 -

(8.34)

ql i'I W1/ I/ q, ( c, d)

If we denote

(8.30) for

M,

ln

1,2

i

1

a .

>

(iii-2) Therefore, there exists a subset

Min

C

Min

Let

v

u

n

<

1M In I = 1M 2n I

<

v n (x) = vex) + , n

00

Obviously

by the formula

Then

J [XM a

v

n

ess sup v (t) tE M n

(t) - XM (t)] dt . 2n

In

$

00

M C (a,b) . Thus, the function

(8.6) holds with the same constant c

J XM

u (x) n

a

of

In

(t) dt

IM ln I '

IIw 1/q l 1M I , q, (c,d) In'

(iv)

J lx M

a

J vex)

J

dx +

MIn

In

(8.35) M2n (x) I vex) dx

(x) - X

1 -1 1 -1 vex) dx :;; 1M 1(5 - -) :;; + 1M2n I (S 2 --) In 1 n n

(8.36)

,

-+

00

Let

,

C <

we finally obtain the inequality

~w1/q~ q, ( c, d)

un

2C(S 1 - 1)-1 n

w, v

v

B

n ~

instead of 2C

which is in

p = 1 .

= q = Let us fix numbers c, d E (a,b) , u E ACLR(a,b) , which is constant on (c,d) , we

P

00,

00

(8.6) that for

~

•

x E (c,d)

lu(x)

I ~ ~uw1/q~ q, ( a, b) ~

C

~u'vl/p~p,(a,b)

Iu' (x) I .

min (c - a, b - d) = min

and the choose a sequence

estimates (8.32), (8.33) yield

IIw 1/q l q,(c,d) ~

fulfils

Denote

M2n

The validity of the Hardy inequality (8.6) for the function

00

n

Cess sup a<x
:;; 21M In I (S 1 --) n

-+

n

obtain from

1 -1

n

~

v (x)

c < d . Taking a function

b 1 IPI'j Ilu~v I p , (a,b)

n

instead n vex) . Consequently, we can proceed literally

as in part (iii-I) arriving at the estimate (8.34) with

Further, using (8.28), (8.29), (8.30) and (8.31), we obtain

Letting

since

C also for the pair

fact the first estimate in (8.11) for

Il unw1/qj'Iq , (c,d) =

(8.33)

w, v

v . Letting

and consequently (8.32)

v

the additional assumption from part (iii-I). Moreover, the Hardy inequality

E [c,dJ we have

x

x E (a,b)

n <

for every measurable subset

un E ACLR(a,b) . For

nE"N

as well, and

E W(a,b) -1

x u (x) = n

and for

W(a,b)

define

1

a

Define

- 2C .

be a general function from

such that

(8.31)

<

n

1

'

1M.ln I

then

' 1/q" IIW ~q,(c,d)Sl

5. - -1 :;; v -1 (x) }

={xEI.·

{II v -I/PII p, , (a , c)' ~v - l/P I p',(d,b) }

.

'mln

'I - l/P I } {II,v -l/p'lIp',(a,c)' lv p',(d,b) ,

{~k} C R+

a k = c - !;k ' x

•

uk (x) =

we obtain in view of (8.28), (8.29) that

I[x a

such that

~k t

t;

and denote for

k E "N

bk = d + !;k '

(ak' c) (t) - X(d ,b ) (t) ] d t k

98 •,ltJ{~",'

99

Then obviously

uk E AC LR (a, b)

u (x) k

c - a

=

=

k

Sk for

uk instead of

Using (8.35) for

Obviously,

and x E (c,d) ,

kE:N

xk _ 1 ~ x k '

Yk;;; Yk-1

(a,b)

[x k _ 1 ,x k ] u [xn,Yn]U

and

u , we obtain

U kE Zn

Ilwl/qll and letting

k

(c d)' sk;;; Cess sup lu{(x)I , a<x
q,

-+

00

Consequently, xk

b

(8.40)

Il w1/ q I q,(c,d) • S ;;; C ,

flu(x)

q

k~

B;;; C .

Thus, we have proved the implication C <

00,

~

00

2-l/ P B ;;; C •

1 ;;; p ;;; q <

(8.38)

1 ;;; p ;;; q =

00

B <

,

Xk-1

w(x) dx +

I kE Zn

lu(x)

f

Iq

w(x) dx ;;;

Yk

Yn

;;; f s(n+1)q w(x) dx +

I kEZ

Xn

-~ C ;;; h(q) B ,

00

lu(x)[q w(x) dx +

Yk-1 q

Xn

It remains to prove the following two implications:

(8.37)

Zn

Y n

+ f lu(x)l

1 ;;; p ;;; q ;;;

f

I

w(x) dx

I

a

which together with (8.36) and (8.9) implies that (v)

[Y k ,Y k - 1 ]

C

=

we arrive at the estimate

,

U k E Zn

f

skq w(x) dx

n [xk-1,xJu [Yk'Yk-1J

Yn

The function

00

=0

u

B <

,

~ C ;;; 4B

00

f s (n+l)q w(x) dx· + X

belongs to

ACLR(a,b)

(8.6) holds for it with an arbitrary constant

and the Hardy inequality

C. Consequently, to verify

the implications (8.37), (8.38), it suffices to show that (8.6) holds for u E ACLR(a,b) ,

every function

u

to,

with the constants

+ kJZ

I <

to.

u

u E ACLR(a,b)

1 ;;; p ;;; q <

00

00

•

Fix

s E (1, 00)

Then there is a number

n E 2

and take

[xk-1'Yk-1J

skq w(x) dX]

[xk,YJ Yk

f s (k+1)q w(x) dx -

k E 2n X k

such that

f

skq w(x) dx -

Yk

B

Let

f

[ n

h(q)B, and

4B , respectively. (vi)

n

I kE 2'n

f s kq w(x) dx X

k

Yk

(8.39)

s

n

;;;

(sq - 1) kEZ I n

sup /u(x) I ;;; sn+l a<x
f s kq w(x)

dx .

Xk

Denote Further, we have

Z = {k E Z; k ;;; n} n

For

k €

'L

n

let

x

k

(Yk)

be the smallest (greatest) number from

such that lu(x k ) I

100

sk

( !u(Yk)

I

sk ).

(a,b)

xk

f

xk

lu'(x)ldx~

u' (x)

dx I

!u(x ) - u(x _ )1 ~ k k 1

Xk-1

Xk-1 ~ lu(x k )

f

I -

lu(x k _ 1)

I

s k-1 (s -

1) 101

_

~ ~ _ - = = - - - ._-""'!=~~!'.

_

_

_

~~=~=,=~==--_,"=~",~~~,~,=--,~-;;-~."~,-"-=-~~~".",,,,"",:.,,,..:o-.,.-.,_.-:_"..~

..

~~.=:-.,'~c--,--'--'~-",

_::...,":-:--:-::~_

.......

".~~_

.....

-c.".,_~.~,_~,.~ft~

__

__..._~~"~_"-,,,,,._,=~~.

._.~--=--_;-.,-..,.",_

..

....,.-=,...,....,,-~._--,

~

and consequently,

Yk Xk

S

k .<. =

S

s

--

-

1

(8.46)

I

Iu' (x)

I

skq

dx .

~

~ (s _s-

1)

q

q 1/p I q ) p, (Yk'Yk-1 B Ilu'v

The formulas (8.44) and (8.46) yield

Holder's inequality yields

sk

dx

Xk

Xk-1

(8.41)

I w(x)

Yk

_s_ II u' v 1/ p II

s - 1

p, (xk-1 ,xk)

• I v-I I p II , • p , (xk-1 "'k)

skq

f w(x)

dx

~

Xk

and analogously

(8.42)

~

s k ~ _s_ I u ' v 1/p I • I v-1/ p II , s - 1 P'(Yk'Yk-1) p '(Yk'Yk-1) for every

If

(8.43)

Ilv

-lip

II p ',(Xk_1,Xk

)

~ ~v

-l/ P II

q 1/P q q (y Y ( ) + Ilu'v1/ P l (-S-l)q B [ilu,v l p ' k' k-1 s p, Xk-1'Xk

k E 2 n • Using this estimate in (8.40), we obtain X b k q f1u(x)lq w(x) dx ~ gq(s) B I [[ lu'(x) IP vex) dx k E 2n a X~l

J

)

P"(Yk'Yk-1

then from (8.41) we have

I w(x)

dx

~

+

Xk

with the function

g

[I

q ~ (_s.) s - 1

~ (s-=-l s

min

+

lu'(x) IP vex) dX]q/P]

Yk

from (8.13). As

Yk

(8.44)

riP

Yk-1

Yk skq

)J

I

q w(x) dx' I v - 1I p I p',(x - ,x ) k 1 k xk

~u

, 1/Pllq ~ v P'(Xk_1'Xk)

f lu(x) I

q

w(x) dx

~

q gq(s) B [kJ2

a

{~v

+

~p"(Xk-1'Xk)'

v

I

p "(Yk'Yk_1)

[I

lu' (x)

IP

vex) dx +

n Xk-1 Yk-1

~ -l/p~q

P

xk

b

) q I 1I q I q w q, (Xk' Yk)

-lip q

~ ~ 1 , we have

}

I

IU'(x)IPv(x)dx)r/P~

Yk b

, , 1/Pllq ) ~ IIU Vip, (xk-1,xk

~ gq(s) Bq [Jlu'(x)I P vex) dx )

q/p

a

-~

q Bq lu 'I ' 1/PII q (_S_) v P,(xk_1,xk )

s - 1

Taking here the (l/q)-th power, we obtain the Hardy inequality (8.6) with the constant

If

(8.45)

Ii v-I I P I

, P ,(x k_ 1 ,xk)

> I v -1/ P I

g(s)B

where

s E (1,00)

is arbitrary. Consequently, for the

best possible constant we obtain the estimate

, P '(Yk'Yk-1)

C

then we proceed analogously using (8.42) and obtain

~

inf g(s) s>l

!l

=

inf g(s) B 1<s<2

=

h(q) B .

Thus, we have proved the implication (8.37).

102

l

103

;m -~----

--=

--=----------

~=- ==c~=~= ~-==~-==-="

(vii)

Let

u E: ACLR(a,b) , (8.47)

-~-----=-----:--------------------

B <

q =

00

and

ufO

-;~~~--~='.:.-~'-=:=----- - -----,--------,,-----=~-~~~=

I ;;; p ;;;

00

00

s £ (1,00)

Fix

•

-~---

r...::=;=-~:~,=--:---:::""::~~~

1 ;;; p,q ;;; (ii)

;;; sn+l

II u I 00, (a, b )

the case

(8.48)

~

~u'v I/PI!'p,(x n _ , x n ) l

s -

,II v -l/PIII P I , (x • n­ I'x n )

,

(8.49)

s

~

I < P

While the foregoing proof for the case

~

P

=

If

(iii)

s

, lip I

s-=-l

Iu

" -11 P I'

Vip,

(Yn'Yn-l)

II v

ii

p' , (y n' yn-I

1

is completely different and the proof for the case

p

q

00

then the number

,

:;

:2

', - 1I p ",

min { 'I v ,

s -

, U , ,II p II k

v

" p ,xn_I'x ( )' n

' -lip)

'' v

" p ( , Yn'Yn-1 )

•

sup min (c - a, b - d)

a
ACLR(a,b)

n+1

2 <

S

- -----=-1 s

q

00

).

If (8.45) holds, then we

p = q =

for

if

(a,b)

and unbounded intervals

(a,b) , we should

Now, we will return to the examples considered in Section 6.

ACLR(a,b).

Naturally, we will use Theorem 8.2. Let

1;;; p ;;; q <

00,

In view of (8.47), we have shown that

~u~oo,(a,b)

00

We will give the necessary and sufficient conditions for the validity of

(i)

=

00

the corresponding Hardy inequality, but this time on the class

B II ' IIp'l u v I p, ( a, b)

~uwl/q~q,(a.b)

=

proceed in accordance with Remark 5.5.

8.6. Examples. Ilwl/ q II q, ( c, d) = 1 since obtain analogously from (8.49) that

B

is unbounded. Therefore, if we wished to investigate the Hardy inequality on

2 lip, x ) < s_ B Ii u I v II p , (a, b) = s I P,(x n -l' n

(recall that

s

q

from (8.9) is given by the

B

(cf. Subsection 8.4 (iv), formula (8.36». Consequently, s

was a

formula

k = n ), then (8.48) impljes

If (8.43) holds (with s

00

)

B

n+ I .

q <

B. OPIC.

'1

n+1

p ~ q . Therefore, (8.50) holds for

00

is new and published here for the first time. Both proofs are due to

2

s

<

-"""~'2iiin~

slight modification of the proof given by P. GURKA [2J , the proof for

and analogously as in (8.41), (8.42) we obtain s n+1

·~~~.,,;r;m;ri"-=~:~.=-':;;:;:-ig~~~~~~ang-::=-~~-~~---i*~~----:::::=:=:-=~~~---=::,:,---=-=;:::::::",---=-__ ----o~i~~-=-=~,~·=

we have not used the assumption

and take

such that (8.39) holds. Then

n E 1

-"'-C"_·=

;;;

s

2

s -

I B

II u

(8.51)

' lip"~ v

[f lu(x) I q x

a

cR.

Then the Hardy inequality

l/q dx )

;;; C [flu'(x)!P x

o

I, p, ( a, b)

a, 8

8

dx

) lip

o

2

which is the Hardy inequality (8.6) with the constant

~I B, s -

s

>

holds for every, u E ACLR(O,oo)

arbitrary. Consequently, for the best possible constant we obtain the

(8.52) estimate C ;;; inf s>1

s

2

s -

B

8 '" p - 1 ,

with a finite constant

C

if and only if

a=8.9.-~-1 p p' •

Thus, the condition (8.52) combines the conditions (6.18), (6.19) from

4B .

Example 6.7.

Thus we have proved the implication (8.38).

o

(ii)

Let

I

~

P ;;; q <

00,

0 < b <

00,

a,

8 E R . Then the Hardy

inequality 8.5. Remarks. (8.50) 104

C

(i) <

00

Note that in the proof of the implication

> 2- 1/p B ~

C 105

._~-_._.

.

--

-".'.

....._. __.._-

~,

_.

..

~-.l.~~_~~:.,;~""-";"";~:-~=::':.~~;-;;:,:'~~"'",:,~-j~~~---,,,_:~

.,-~-

__ ::=:'~~;~~:"-:",'-~~2'~-:::~~';;;;:=:~:;':-:~:: __ :C;:"':::"';;';;::;;;';~~_-:_~'::;-~-_~_

.::;::::=-::':::~~,:,::-'=,:: ;.:;~~~",;,;:--,,-;';'~:

b

(8.53)

b

[JIU(X)

q

I

x

0.

dx

]l/q

~ C

o holds for every

(8.54)

(f Iu / (x) IP o

x

8

(vi)

with a finite constant

u E ACLR(O,b)

(8.60)

]l/P dx

for

8 '" P - 1 ,

0. > - 1

for

8 = P ­

(8.62)

Let 1

P

~

q < 00,

~

a, 8 E R . Then the Hardy

0 < a < 00,

~

1

~

P

q

<

[ J lu(x) Iq e ax dx

8

a, 8 E R . Then the Hardy inequality

f/q

~ C

( Jlu/(x) IP e Sx dx riP

with a finite constant

C if and only if

a = 8 .9.

0 ,

~

00 ,

u E ACLR(-oo,00)

holds for every

Cf. Example 6.8. (iii)

Let

a=s.9.-.9--1. P P'

C i f and only i f (8.61)

0.;;;8.9._.9--1 P p/

S>p-l,

P

Cf. Example 6.12.

inequality

~

[f Iu (x) Iq x a dX) 1/ q

(8.55)

C[flu/(x)I

P

x

8.7. Remark.

8 dx ) 1/p

a

a

J

holds for every

(8.56)

E ACLR(a,oo)

u

with a finite constant

0. ~ B .9. - .9- - 1 P p/

for

0.

for

(iv) Let

1

8 ~ P ­ 8 = P - 1

P

~

~

q

<

00,

a, 8

1

[Jlu(x)l

q

o

(8.58)

B

~

E~ample

l/q

! lIn xio. dx ) x

u E AC

P

1

p 1 ~ c[Jlu/(x)I P x lIn xlB dX)

with a finite constant

P

P

J ~

(8.59)

1

~

(Jlu(x)l

q

~

q < 00,

x lIn xl

a

+

holds for every 106

a, S ER . Then the Hardy inequality

dx

P p 1 lIn xl 8 dxf/P ~ c[flu/(X) I x -

u E AC

LR

(0,00)

with a finite constant

xl

a

dx = J 1 + J 2 '

1 J

J:

[ [) lu/(x) I P x P- 1 lIn xl S dx riP +

U

lu/(x) I P x P- 1 lIn xl 8 dx riP] ~

1 00

(

1 '"

C

;;-'

tl

o

o

lIn

1'1'::

p/

]l/ q

Iq ~

0

~

P

cq

C if and only if

6.10.

Let

dx + J1U(X)

II

a = B.9._.9-- 1 .

1

a

u E AC (I,oo) R

we obtain the following estimate for

E R . Then the Hardy inequality

o (O,l) LR

dx

for u E AC (O,I) as well as J 2 for L 1 (cf. Example 6.10 and Remark 6.11, formulas (6.30), (6.33»,

;':i

(v)

lIn xl

a

0

;'&,;' .

Cf.

Iq ~

and we are able to estimate

1

holds for every

lIn xl

q

0 Jlu(X)

1

< -

~

= J1u(x)l

C if and only if

Cf. Example 6.9.

(8.57)

Let us return to the inequality (8.59). Since

if and only if

j

I

~

cq [J lu/(x) IP

x P- 1 lIn xl S dx riP

0

taking into account that

P

~

q ,i.e.

obtained the inequality (8.59) for

q/p;;; 1 . Consequently, we have

u E ACLR(O,oo)

provided the corres­

ponding Hardy inequalities hold respectively on the subintervals

(0,1), 107

D=

AC (l,oo) . According to Example R 6.10 and Remark 6.11, the conditions which ensure the validity of these

(8.68)

Hardy inequalities are given by (6.31) and coincide with the conditions

is also sufficient.

(1.

00

and for the classes

)

AC L (O,l) ,

(8.60). However, the approach used in this remark guarantees only the

inf max (BL(a,c), BR(C,b)) < a
00

This approach forms the basis of the proof of the following theorem,

sufficiency of the conditions (8.60), while their necessity follows from

in which it will be shown that a slightly modified version of the condition

Theorem 8.2.

(8.68) is not only sufficient, but, moreover, necessary for the validity of the Hardy inequality (8.63) on the class

Therefore, a natural question arises whether this coincidence is accidental or if the 'trick' of splitting the interval subintervals ACLR(a,b) ACR(c,d)

(a,c) ,

(c,b)

(a,b)

tion reads

into two

and investigating the Hardy inequality on

via the investigation of the Hardy inequality on

ACL(a,c)

ACLR(a,b) . This modified condi­

U3

63 and

(a,b,w,v,q,p) <

00

where now

could be used also generally. Let us describe the general situa­

rf3

(8.69)

tion.

inf

max (BL(a,c), BR(c,b))

a~c~b

We investigate the Hardy inequality (8.63)

~UW1/q~q,(a,b) ~

For this purpose choose (8.64)

~ uw l/q~

C

with

~u'v1/P~p,(a,b)

c E (a,b)

on

ACLR(a,b) .

BL(a,c)

0,

from (8.66) and (8.67), respectively, and with

B (b, b) = 0 R

The advantage of the condition (8.69) consists among other in its covering also the cases when the Hardy inequality (8.6) holds on the broader class

P ) -< CL ~' u v l/ Ij p, ( a,c ) q, ( a,c

on

ACL(a,c)

ACL(a,b)

and (8.65)

BR(c,b)

BL(a,a)

(8.70)

and investigate the Hardy inequalities

and

or

ACR(a,b) .

Moreover, the approach mentioned can be extended to the case

~ uw

l/ q l

I q, ( c, b)

<

~' 1/ P 1 I p, ( c, b)

on

~ CR u v

The inequality (8.64) or (8.65) holds for

~

p

~

ACR(c,b) ~

q

~

00

if and only if

q < p

~

see Theorem 8.17.

00

8.8. Theorem.

Let

1

~

p

~

q

~

v, w E W(a,b) •

00

Let the number

03

be

defined by the formuLa (8.69). (8.66)

BL(a,c,w,v,q,p) <

BL(a,c)

00

Then there exists a finite constant

or (8.67)

equaLity (8.6) hoLds for every BR(c,b)

=

BR(c,b,w,v,q,p) <

00

(8.71)

,

respectively. Consequently, the inequality (8.63) holds for

~

p

~

q

~

<

00

E AcLR(a,b)

such that the Hardy in­

if and only if

•

Moreover, the best possible constant

00

C in (8.6) satisfies the esti­

mate

if max (BL(a,c), BR(c,b)) <

00

been arbitrary, the condition

(8.72)

•

This condition is sufficient for (8.63); moreover, since

108

8

u

C > 0

c £ (a,b)

has

with

2-l!P .{3

k(q,p)

from

~ C ~ k(q,p) fJ (1.24).

,J~ The proof of Theorem 8.8 is a consequence of the following two lemmas 109

_.____

___=

--=-..-;____

_

,,_

=--=-=-

_-=__

------

-:::--=:;====

o-=-

-'----==- - - -===-,=:

_,,;=---=- - - -_-__==_:: -'Co_ceo.

-o:="------~:- ..:~_=_.:_----

._:...-- -

---==-----o-==-~_'=:_

.o-=--=-­

_--­

_=.,_

------ - -----_ .. - - - --='--=---~

q/p ~ 1 , we immediately obtain from (8.75), (8.76)

Using the fact that

(and Theorem 8.2).

::=

--:;;:-

and (8.74)

constant

1 ~ p ~ q ~

Let

8.9. Lemma.

b

v, w 6 W(a,b) . Then the best possible

00,

C in (8.63) satisfies C ~ k(q,p)

(8.73)

o<

n=

If

00

•

Let

00

£ >

,

0

[b lu' (x) I P vex)

£ -+

J

dx J

q/P

.

a

0+ , we get (8.73).

then (8.73) holds trivially. Therefore, suppose be fixed. Then according to the definition of

there exists a number

see (8.69)

[k(q,p) ( () + £)J q

w(x) dx

a

d3 . Letting

Proof.

Iq

JIU/(X)

max (BL(a,e), BR(e,b») <

53

such that

c E [a,b] +

£

(ii)

6

q =

Let

00

•

Then

luw1/qlq,(a,b) =

(8.77)

~u~oo,(a,b) = max (~uloo,(a,e)' ~uloo,(e,b») (luw1/q~ q, ( a,c-)' ~uw1/q~ q, (-c, b»)

,

= max

and consequently BL(a,e) <

(8.74) If

c = a

6

+

E ACL(a,b)

u

<

{3 +

£

<

Analogously as in part (i) we arrive at the inequalities 00

k(q,p)(

(or

u €

C+

£)

C ~ k(q,p)(

If and

£ >

c

E

0

8

luw

,

;;; (V3

ACR(a,b) ) and a fortiori for

I UW

1

~

q <

C

00

q E [1,00)

•

Then

AC L (a,e)

Letting (or

C

£

1/q c q (flu(x) I w(x) dxJ

110

(a,e)

(or

c

~

P k(q,p) BL(a,e) [Jlu/(x)I vex) dx

IU/v1/p~p,(a,b)

b

[Jlu(x)!q w(x) dx

1/ P

~

q =

00

•

The last two inequalities together with

-+

II q II

, I lip/I Ip (a b) 'q,(a,b) ~ ( OJ + E) IIu v , ,

0+, we immediately obtain (8.73).

o

k(q,p) BR(e,b)

[J lu' (x) IP vex) c

1 ~ p,q ~

Let

00,

v, w E W(a,b) . Let

If

B =

00

(f3

then (8.78) holds trivially. Therefore, suppose

,

Obviously, it suffices to show that there exists a point

II dX)

Band

be

6~B

(8.78)

Proof.

J

b

l/q

J

for

defined by (8.9) and (8.69), respectively. Then

a

a

c

1/PII p,(e,b) ~

ACR(e,b)

(c,b) ):

(8.76)

+ £)

l/PI I p, (a, b) ,

(8.77) imply

8.10. Lemma.

[or

II u I v

I

1

k(q,p)

II u w

Let

E)

q, (e , b) ~ BR(e,b) I u v

~ (0 since

(a,b) , then we will consider two different cases:

and (8.74) implies the validity of the Hardy inequality on

(8.75)

II

+

+ £)

has been arbitrary, we have (8.73).

ACLR(a,b)

on

1 Iq I

C satisfies

q = 00 (i)

~

'I I l/PII p, (a,e) 1/qll -) Iq,(a,c ;;; BL (a,e) I u v

~u/v1/Plp,(a,b)

u E ACLR(a,b) . Consequently, the best possible constant

and, since

•

c = b ), then (8.74) implies that the Hardy inequality

(or

iuw1/q~q,(a,b) ~ holds for

B (e, b) R

<

£

eE

[a,b]

B
that p ] .

(8.79)

BL(a,c)

~

B,

BR(c,b)

~

B . III

For

c E (a,b] denote

Q =

c

~

for

E

[a,b)

{

2

-

= BL(a,b) =

denote

QC =

ll ( c ; sup d) E Q iiw q II q, (c, d) min {
B

(8.82)

_

(c;d) E: R ; c E (a,c), dE (c,c)} ,

{(c;d)

­ E R 2 ; c E (c,b), dE (c,b)} ,


define functions

xE(a,b],

I p I , ( x, b) ,

x

E

Co

sup {c E [a,b);
oo}

dO

inf {d E (a,b]; 'JI(d) < co}

'!' (x) = I v

-li P

< B , we have again arrived at

(8.79). (iii)

-lip

o

BR(c,b) = BR(b,b)

Since, moreover

by II p I , ( a,x ) '


BL(a,~)

[a,b) ,

'!'(b) = 0 ,

supE Qa Ilw (c;d)

B

(8.83)

c = a

a , then we take

Co

dO

If


ll

and obtain analogously

q'l q, (c,d) min {(c), '!'(d)} 1

and put

Necessarily we have (i)

dO =

If

~

dO

Co E

Co '

BR(a,b) = BR(~,b) Since, moreover,

since otherwise we would have

(a,b) , then we take

c =

Co

B

= co •

< B , we have again arrived at

(8.79). (iv)

(8.84) ~

o

BL(a,a)

If

dO <

Co '

then it can be easily seen that

and obtain easily

that B

BL(a,c)

sup ijwl/qij . ( c ; d) E Q_ q, (c, d ) mn { (c), '!' (d) }

'JI(a)

Consequently,

=

a

I,Iv -liPII P , a, b) = (b) < I

= dO

(

Co

<

00

•

b .

c

(8.80)

sup a
sup c
Il wl/q~ q, (c,d )

sup II l/q'l Iw I q, (c , c) a
(8.81)

~

sup Ilw (c;d)EQC

llq

1'1

v

(iv-l)

(c)

is increasing and

-l/PI'

Ip', (a, c)

BL(a,c) ,

dO =

as in part (i) that

Co

'!'

~

00

,

then the functions

is decreasing on

onto the interval

(8.85)

(O,L)

~

are continuous,

(a,b) , and they map L

= iv-l/Pi p I

(a,b)

such that

with c

¢, 'JI

(

, a,

(8.86)

(c) = 'JI(c) ,

{(c), 'JI(d)} = (c) ,

min a
min {
II -l/P'1 sup I, l/qll Iw q,(c,d) v Ip',(d,b) c
(ii)

(a,b)

sup Ilw ll q I c
_sup c
If

B (c, b) R

112 113

(iii)

which is (8.79). i.e. (iv-Z)

If

P = 1 , then the number

S

I'I v -1 II oo,(a,b)

. 1S

We have derived the estimate (8.87) using the inequality (8.78),

the number

B

from (8.9). As will be seen later (see Remark 8.20 (i»,

directly, but only for

this estimate can be derived also

finite

P ;;; q ;;;

<

00

•

according to (8.84). Denoting

we have

Xl

inf {x E (a,bJ; ~(x) = S}

X

sup {x

z

E

c = b

we have

w

Xl = x

=

2

1

Z

n

If

¢ ,

foY'

min {Hc), S} a
~

(c)

,

min {~(c), ~(d)} c
min {S, ~(d)} c
~(d)

.

(L)

X

holds not only for

u

E

ill ;;;

Proof· BL(a,c)

and

q < p

~

Lemma 8.10 as well as in Remark 8.5 (i) we have supposed only

114

I

C

=

k(q,p)].

see (8.66), (8.67) and (8.70)

BR(c,d)

8.13. Remark.

(8.90)

~

P- 1+P / q

dt

It can be easily verified that under our assumptions the numbers

, since in

v

BR(c,b) ~ 1 ,

weight function

i

o

by the formula

(L)

v(X)

q

with an arbitrary

l;;;p,q$oo.

fulfil

The assertion of Lemma 8.12 remains true if we express the

C

1

If w(t)

and (8.89) follows immediately from (8.69).

The assertion of Theorem 8.8 now immediately follows

but also for

E (a,b) .

1

BL(a,c) ;;; 1 ,

00

X

X

w1- p (x)

with the constant

ACLR(a,b)

o

1 ;;; p ;;; q ;;;

for every

(a,b)

IE

weight function

C3 ;;;

1-p

q

Moreover, the estimate 2- 1 / p

and

[and, consequently, the Hardy inequality (8.6) holds for every

from the estimates (8.73) (Lemma 8.9), (8.78) (Lemma 8.10) and the 1 p inequality 2- / B ~ C in Theorem 8.2 [see also Remark 8.5 (i)J.

(8.87)

c~[a,bJ

wE W(a,b) ,

,

Then

then we choose

Thus, we have arrived again at the relations (8.86) and proceed as at the

(ii)

00

c

(8.89)

(i)

00

v(X)

(8.88)

min {~(c), ~(d)} a
8.11. Remarks.

w(t) dtl <

q <

Put

c E M and have

end of part (iv-I).

~

(6.11), (6.11*».

c

= a , we have

(a,b) ~

1 ;;; P

v

X

~(c)

min {S, ~(d)} = ~(d) a
M = [x 'x ]

Let

8.12. Lemma.

and proceeding as in part (iii), we arrive at (8.83) and,

consequently, at (8.79). If

with formulas connecting the weight functions

(cf. the formulas (2.13), (2.15),

and proceeding as in part (ii), we arrive at (8.82) and, con­

min {~(c), ~(d)} a
b

min {~(c), S} a
sequently, at (8.79). In the case

taking

and

Xl = x 2

min {~(c), ~(d)} a
Z.6 and Theorem 6.4, i.e.

[a,b); ~(x) = S}

In the case

Xl :;; Xz

The next subsections deal with the approach described in Subsection

1-p

X

w 1 - p (x)

11

wit) dt+

dIP~l+P/q

dE R. The formula (8.90) sometimes enables us to give the v

in a more convenient form.

115

d ER is such that

The proof of this modified version is simple: if

J wet) dt

0 < d <

or

0 > d

c

cE

then it can be shown that there exists a number

provided or

(a,c)

~

a , and

min {~(x), ~(x)} c<x
J wet) dt ,

> -

a

~(x)

c ~ b . This together with (B.92) implies that

C E (c,b) (8.93)

such that

w(x)

*'

x

1

'

,

v -p (x) [¢(x)]-P -q

J wet)

J wet) dt ,

dt + d

c

,

x t: (a,~)

for

and b

instead of

c

other cases it can be shown that either x

If

f

wet) dt

x

c. In the

or

f

(8.94) wet) dt

x E (~,b)

v

from

provided

BL(a,c) ~ 1,

the formulas (2.15) or (6.11*), respectively. This fact implies that or

BR(a,b) ~ 1 and (8.89) follows immediately.

c E (a,b) . Comparing these formulas with the

BR(~,b) ~ 1 ,

which implies (8.89). If

c

= b , then (B.93) holds for x E (a,b)

BL(a,b) ~ 1 , which together with < p

Let

8.14. Lemma.

~

q <

00

v E W( a , b )

and If

{llv-lIPII p',(a,c)'

min c,dE(a,b)

]-l-q!P'

formulas (2.13) and (6.11), we immediately obtain that

from (8.90) estimates from above the weight function

BL(a,b) ~ 1

dt

x for

c

v

lrJ v l-p' (t)

can be

a

and, consequently, the weight

wet) dt + dl

1_' -'1_' w(x) =~v p (x)[~(x)J p q =*,v p (x) n

x

b

estimated from above by

)

a

c

and thus the proof reduces to Lemma 8.12 with

(B.91)

fJ v 1- p , (t) dt i-i 1-q!P'

.5L 1-p' p' v (x)

x

x

function

c

b

c

either

provided

II

v

-lIPII p',(d,b) }

<

BR(b,b) = 0

~ = a , then (8.94) holds for

BR(a,b) ~ 1 , which together with

implies (8.B9).

x E (a,b)

BL(a,a) = 0

and consequently

and consequently

implies (8.89).

0

00

c < d

8.15. Remark.

Put w(x) =;, v

(B.92)

1-' [ p (x) min

{II v -lip:lip'

I'

,(a,x)' Iv

-lIPII p' ,(x,b) }]

-p'-q

(8.93) and (B.94) are meaningful for

x

Proof.

.{1~1.

Let

~, f

(8.95) ,

Co '

dO

be the functions and numbers introduced in

the proof of Lemma B.I0. Take ~ E (a,b) and

c

= Co

if

Co

according to (8.85) if

dO <

Co

= ¢(x)

x E (a,c) w(x)

instead of

and

x E (c,b) , respec­

from (B.92) by the function c I), i.e.

[~(x)rp'-q

for

x E (a,c)

,

[~(x)rp'-q

for

x E (c,b)

,

The formula (8.95) is meaningful if we assume that there is a number

= dO . It can be easily seen that

min i¢(x), ~(x)} a<x
' { w(x) = ~ v 1 p (x)

c

such that the formulas

and (8.B9) again holds.

c

116

from (8.93), (8.94) (with

E (a,b) . Then

(B.B9)

c E [a,bJ

tively, then we can replace the function w(x)

for

If there is a number

E

[a,bJ

such that

¢(x) <

course, the weight function function

w

(a,c)

and

~(x) <

00

on

w

from (8.95) can be a little worse than the

00

on

(c,b). Of

from (8.92), which actually is the one from (8.95) with the

117

'optimal' choice

c = c

[ f lu(y) Iq

(8.97) 8.16. Example.

1 $ p;;; q < 00,

Let

x

fo e for

x

E

x f[ 00

t2 dt

00

2k)

k~O ~!

=

0

(a,b)

dt =

k~O

(_00,00)

2 e t . We have

wet)

00

2k+l k!72k + 1) ;;;

2k

k~O ~!

with

2

C(l

=

[a- 1 / 2 (l

e

ay 2

1

q dy J / ;;; C a

e

ClPY 2/

q dy J1/P

+ q/pl)Jl/q +l/p'. Consequently, we have shown that u E ACLR(-oo,oo)

the inequality (6.39) holds for

= eX

(oof lu'(y) IP

with

B = ap/q .

a > 0,

[O,l]

Compare this condition with the (necessary) condition (6.40) when con­ 1

x

r

J e

t2

x

; ; Je t

dt

o

2

r

dt + J t e t

o et

c = 0

dt $~x

sidering the inequality on ACL(_oo,oo) or ACR(-ro,ro) . for

2

x

E (l,oo)

we obtain the inequality (8.2) of

TREVES mentioned in Example 6.13 with the constant

2

C =

2/10.

is even, we have verified the condition from

and we conclude that the inequality

2 dx )l/ q

q [f lu(x) I eX

p = q = 2

In particular, for

1

and since the function Lemma 8.12 with

2

2

;;; C

r

J lu'(x) I P

;(x) dx

Now, we will consider the case $q
J l/p

Our considerations will be based on the approach described in Remark 8.7. holds for every

ACLR(_oo,ro)

from (8.69), we introduce the number

with the weight function

x 2p/q e

/ 2 '" e x (l-p) [e +2ex2r-l+P/q

vex)

~

Analogously to the number

e

u

(8.98)

for

Ixl

$

for

Ix I

>

inf max {AL(a,c), AR(C,b)} a$c$b

AL(a,c) = AL(a,c,w,v,q,p) ,

where

1 ,

..It; =."e;(a,b,w,v,q,p) =

AR(c,b) = AR(c,b,w,v,q,p)

are defined

by (1.19) and (6.7). Again, we define with the constant

C=

(8.99)

(P' -l/p' -)

k(q ,p )

q

8.17. Theorem.

Using the estimate

1 (

2

e + e

x

) < eX

2

f or

Ix I

> 1 ,

2

J Iu(x) Iq eX dx

holds for every

u

E AC

Let

q <

<'

P ;;; ro

v, wE

Then there exists a finite constant equality (8.6) holds for every (8.100)

r

AR(b,b) = 0 .

W(a,b) . Let the number

.It

be defined by (8.98).

2

we immediately obtain that the Hardy inequality

(8.96)

~(a,a) =

~::;

LR

Jl/q

;;; k(q,p)

Cf-), -lip' [ I u' OOf

I

(x) p e Px

2/

q dx

)l/ P

u E ACLR(a,b)

(8.101)

'

>

0

such that the Hardy in­

if and only if

ro .

tft, <

The best possible constant

(_00 00)

From (8.96) we obtain for

II

C

2-1/p ql/q

(~) r

1/ '

~

Cin (8.6) satisfies the estimates ~ C ;;; 2 1 / r ql/ q (p,)l/qIJh

uJ'i th a > 0

the inequality

.~!

l r

=

1 _1 q

p

'. ,7':­

118

\

.

I

.

"

119

JS = AL(a,b)

and mopeovep, fop

Jt

OP

'¥(c)

(compare with (8.85». If we show that the following two Hardy inequalities

C ~ q1/ q (pl)l/ql~ .

(8.102)


(8.108)

= AR(a,b) , also the estimate

on

Theorem 8.17 will be proved again via two lemmas. The proof of the

(a,~)

(~,b)

and

(8.109)

IIgw1/q l

(8.110)

q Jlgw1/ ll

(

hold:

_) ~ 21/P C f1g ' v1/ P I

q, a,c

first is omitted, since it is an analogue of the proof of Lemma 8.9. [Note that we obtain analogues of the inequalities (8.75) and (8.76) on and

(c,b)

(a,c)

and the passage to the whole interval (a,b) is made possible q (a q/p + Bq/p ) l/q ~ 2l/ -l/p (a + B) lip, a, B ~ 0 ,

by the inequality since now

(8.103)

1 ~ q

Let

8.18. Lemma.

constant

.J

q/p < 1

p ~

<

v, w

00,

C

2

~

1/r

q

A

and, mopeovep, fop

~

= A (a, b)

L

21/P C f1g l v1/ P I

for every

gEACL(a,c),

(- b)

for every

gE ACR(e,b) ,

p, c,

(8.111)

2-1/p q 1/q [~) r

1/' q ~(a,c,w,v,q,p) -

< ~

C ,

(8.112)

2-1/p q 1/q (~) r

1/' q AR(c,b,w,v,q,p) ­

~

C ,

which immediately implies (8.104).

1/q ( I)l/ q l ~ p

~

_)

then due to Theorems 5.10 and 6.3 we have

E W(a,b) . Then the best possible

C in (8.63) satisfies the estimate

(_ b)

q, c,

(

p, a,c


OP

Therefore, we will prove that (8.109) really holds (the proof for AR(a,b) , the estimate (8.102).

since

(8.110) is analogous). We will use the fact that

C

<

00

the

Hardy inequality

Let

8.19. Lemma. (8.104)

1

~

q < p

2-1/P q l/q [~)

~

1/ q '

.It

r

v, wE W(a,b)

00

~

•

Then

c .

(8.113)

~uw 1/qll . q,(a,b) -$

holds for

u E ACLR(a,b)

Take

Pmof·

(i)

If C

=

00

,

then (8.104) holds trivially. Therefore, suppose

C<

00

(8.105)

L

=

J

1-p I

(t) dt

<

00

Let

(a,b)

n


J vI-p' (t)

For

Cx;v)

n E:N

I

1/ P 11

n

t c

and

for x

dt ,

(8.115)

a

f (t) = n

E

n

'¥(x)

'¥(x;v)

J vI-p' (t)

dt .

~

= c(v) E (a,b)

such that

,then

(8.109) holds

-+

00

define

{ Ig'(t)1 0

such that

for

t E (a,c ) , n

for

t E [c ,~) n

and x

x

Analogously as in the proof of Lemma 8.10, part (iv-I), there exists a point

00

00

(a,~)

(a,~J

b

(8.107)

<

p,(a,e)

be a sequence in c

define functions x

(8.106)

I

{c}

a

and on

~g v

(8.114) v

P ~ g I v 1/ II, p , (a, e)

g E ACL(a,c) . If

trivially. Therefore, assume

In addition, assume that b

C I u I v 1/ p I p , (a, b)

(8.116)

gn (x) =

J f n (t)

dt .

a 121

120

due to (8.117). Consequently, in view of (8.121),

Then it follows from (8.115) and (8.114) that I f v 1/ p I

n

II g , v 1 / P I

_;S

p, (a,c)

(8.122)

_

<

(a,c)

p,

un E ACLR(a,b) .

Obviously and Holder's inequality together with (8.105) yields (8.123) (8.117)

f

n

If gn

(a,c]

is absolutely continuous on

lim gn(x) = 0 . x+a+ p(x)

for

x

E

(c,b)

P <

DO

then from (8.123) we have

,

decreasing, maps the interval lim x+b-

a,

p (x)

1

v(y) = v[p(y»)

b

b

by

flu~(Y)IP

flg~[p(y»)

v(y) dy

p'(y)I P v(y) dy

c

c b

~

The continuity and monotonicity of

and

(c,b)

'l'

onto

imply that

(a,c)

p

flg~[p(Y»)IP

is continuous,

and satisfies

Ip'(y)I P-

1

v(y) p'(y) dy

c c

b

lim p (x) = c x+c+

flg~[p(y»)

IP v(p(y») p'(y) dy

c

Now define g (x)

un (x)

=

{

g: (p (x»)

f- (a,c]

for

x

for

x E (c,b)

The function

un

o . (a,c] . Moreover,

[c,b)

x E [c,b)

since for

f

p'(y) dy

u

n

vex) dx

2

P =

flg~(x)

,

2 JIg'(x) /P vex) dx . a

a

DO

~

then we have from (8.123) that

p'(y) = - 1

for

is

~U~Vl/P~p,(C,b) = ~u~~DO,(c,b) = ~g~(p(x») p'(x)~oo,(c,b)

we have ,

p(x)

g~(p(y»)

flu~(x)IP

P vex) dx I

y E (c,b) , and in view of (8.120), (8.116), (8.115) we conclude

b

J

If

c

c

a

is absolutely continuous on

absolutely continuous also on

vex) dx .

a

b

From (8.118), (8.119) we have lim u (x) = lim u (x) x+a+ n x+b- n

JIg~(x) /P

This together with (8.120), (8.116), (8.115) implies

(8.124)

(8.121)

E (c,b)

y

and therefore

p(x) = ~-1 ['l'(x») .

(8.120)

for

and, moreover, Ip'(y)I P-

Define the function

(8.119)

l'

1 ­ E L (a,c)

Consequently, (8.118)

, 1

p'(y) = - v P - (p(y»).v -p (y)

g~(t)

I

gn[p(x»)

dt

u

n

(x)

Ig , [p (x») n

I

ess sup 19~(y) a
I

~ ess sup Ig'(y)l= a
= ~g'vl/P~p,(a,c)

a

x

e s s sup c<x
(8.125)

\

due to (8.120) and

flg~[p(y») c

122

c

c

b

p'(y) I dy

Jlg~(x) I a

dx

J

Now, it follows from (8.120), (8.122), (8.113) for fn(x) dx <

un ' and (8.124) or

(8.125) that

DO

a

,M

ia

123

-- ---=----=--==--=-.-:~_:-.:::=.:::::~::=..=~:;;;;::.~.=.~~

(8.128)

Ilgnwl/qll • (a,c) = Ilunwl/q\l q , (a,c) :;; \lunwl/q1I q , (a,b) ;;; q (8. 1 26)

S C Iu'v l / P I

n

o

Since

:;;

p, (a,b)

1 p 2 / C Ig'vi/PI

1/

P q

1/

' l/q' A (c - ,b,w,v ,q,p ) ­< C q (~) r

R

{c}, n

From the sequence ­ p, (a,c)

n

n

n EN, we can choose without loss of

generality a convergent monotone subsequence {y }, y

n

n

--+

y E [a, b ] .

Using

Fatou's lemma, we obtain from (8.127) or (8.128) the estimates (8.111),

and

~ gn(x) ~ gn+l(x)

2-

(8.112) with

y

instead of

C, and arrive consequently at (8.104).

x

for

x

E

lim g (x)

flg'(t)! dt

n+ oo

a

n

Let

(iii)

be a generaZ function from

v

n

obtain from (8.126), letting (a,c) , we x

Ilwl/q(x) flg'(t)\ dt\: q, ( a,c_):;; 2l/

P

-+ '" ,

the inequality

b n t b , and define for ¢ (x) n

C Ilg'vl/Pt' p, ( a,c

-)

a

x

x

(f

l

(il)

Let

n

1 < P < rn . For

e:N

n

n

Then

n

and suppose

holds for every is satisfied on

x

J dt

J

an

an

1 p v - '(t) dt ,

~

n

J

(x)

b

n

dt

x

( b) q, an' n

~

C

~u'wl/P~

p=rn. an

-I­

a,

n

Jv

I-p' (t)

dt .

x

C <

00

( b) p, an' n

u E ACLR(an,b ) . Since the additional assumption (8.105) n (a.b), we finally arrive as in part (i) at the n

n

following analogues of (8.111), (8.112): x E (a,b) .

satisfies the additional condition (8.105)

v E W(a,b)

Analogously as in

W(a,b)

define a function

vex) + -1 ( 1 + x 2/(p'-1») '

v (x)

a1 < b1 ,

- ­

luw1/ql

a

be a general function from

v

such that

b

x

1

c

dt .

g' (t) dt\ :;; f1g'(t)\

a

and suppose

= -2 (a + b ) , we have ¢ (c ) = ~ (c ) . The assumption n n n nn nn implies the validity of the inequality (8.113), and consequently

For

This implies the inequality (8.109) because

\ g (x)

{a n } , {b n }C (a,b) x E (a ,b ) n n

We choose two sequences

W(a,b)

part (i) of the proof, we obtain a number

-

cn E

l/'

(8.129)

2-l/ P q l/q

(~)

(8.130)

2-l/ P ql/ q

(~) l/q'

q

cf. (3.13). (a,b)

A (a -L n

,cn ,w,v,q,oo)

c ,

AR(C ,b ,w,v,q,"') ;;; C • n

r

~

n

such that

¢(c

'v )

n' n

= ~(c n''v n )

Choosing a subsequence proof by letting

n-+

{Ynl

and

oo

C {enl

Yn-+Y

as in part (ii), we complete the in (8.129), (8.130).

0

(cf. (8.106), (8.107) and (8.108». Since the inequality (8.113) implies that luw 1 / q l q, (a, b) holds for

on

(a,b)

(8.112):

(8.127)

u E AcLR(a.b)

~

B.20. Remarks.

l P C lu/v n / p, a, b)

i (

in view of the obvious estimate

The method used in the proof of Lemma 8.19 enables us

to derive the estimate vex)

~

vn(X) 1 , we finally arrive at the following analogues of (8.11 ),

, l/q' _ -1/P l/q (P~) A (a,c ,w,v ,q,p) ~ C , 2 q r L n n

(i)

(8.131) with

OJ

2-l/ P {J

~ C

from (8.69) provided 1 < P

~

q ;;;

00

•

125

---....,......,...~,-

0 - - -.. -

.""~~_~~~~..,..~~~~'..: __~=~=~~;,.~~~~~~~?~

Indeed: Using Theorems 5.9 and 6.2 instead of Theorems 5.10 and 6.3, we can

conditions is fulfilled:

derive from (8.109), (8.110) the inequalities

2-1/~ BL(a,c) ~ C

(*)

(8.135)

2- 1/p B (c,b) ~ C R

~

p

q <

~

00

,

a;;'6.'l-.9.--1 p' P

for

6 '" p ­

a > - 1

for

13 = p

we immedi­

(*)

which now replace the inequalities (8.111), (8.112). From

1

(8.136)

1

~

q < p <

00

a > 6£_.9.--1

6E R ,

,

q

ately obtain (8.131). In the case

p = 1

the method fails, and therefore we have to proceed

via Lemma 8.10 and Theorem 8.2. (ii)

(iii)

~

Let

p,q <

°< a

00,

<

p'

a, 6 E R . Then the Hardy

00,

inequality

The results mentioned in Theorem 8.8, Lemmas 8.12 and 8.14 and

8.17 are due to B. OPIC and the proofs are published here for the first

(8.137)

[Ilu(x) \q x

a

dXJ1/q

~

C

[[lu'(x)I' x' dX('

a

time. holds for every 8.21. Examples.

For the convenience of the reader

we will give here not

u E ACLR(a,oo)

conditions is fulfilled:

only the results concerning (for particular inequalities) the case 1

~

q < p

~

00

we will summarize all cases including those handled in

;

~

(8.138)

p

q <

~

00

,

Subsection 8.6. . (i)

~

Let

(8.132)

p,q <

a, 6

00,

[f lu(x) Iq

x

a

dx

f/q

~ C

[f

lu'(x) I P x 6 dx

riP

(iv)

0

~

~

p

q <

~

00

,

This is caused by the fact that

C

00

for

6 '" p -

a < - 1

for

6 = p

6E R ,

,

p,q <

a~6.9.-.9.--1 P p'

a<S.
a, 6 E R . Then the Hardy inequality

00,

1

lJ Iu (x) Iq

]l/q ~ [J lu' (x) I dx

a

1

~ lIn x I

C

=

max {AL(a,c), AR(c,b)} =

00

00

)

if

q < P .

for every

u E AC

holds for every (8.141)

1

~

P x P- 1 lIn xl 13 dx Jlip

o

o

a=6.
Note that the inequality (8.132) does not hold (i.e.

~

Let (I

(8.140)

6"'p-l,

q < P <

1

C ) if and only if

(with a finite constant

u E ACLR(O,oo)

holds for every 1

R . Then the Hardy inequality (8.139)

0

(8.133)

~

if and only if one of the following two

LR

p

~

q <

1

~

p,q <

00

i f and only i f

(0, 1)

13 '" p - 1

,

,

a=69.-.9.--1 p p'

c E [a,b] (see Examples 6.7, 6.8, 6.9). (ii)

Let

1

~

p,q <

00,

0 < b <

00,

a, 6E R . Then the Hardy

(v)

inequality

b

(8.134)

[f lu(x) I q

x

a

dx

J II q ~

C [flul(x)I P x

13 dx J lip

o

o holds for every

(8.142)

b

u E ACLR(O,b)

Let

lJrr!u(x) Iq

1

00

~ lIn

a, 6 E R .

,

xl

a

Jl/q

~ C

[J lu' (x) IP x P-

1

lIn xl

6

J1/ p

dx

o

o holds for every

dx

Then the Hardy inequality

u E ACLR(O,oo)

if and only if

if and only if one of the following two 127

126

;i;p;i;q
(8.143)

B>p-1,

then (8.147) holds.

a=S.1-.9.--1 p

p'

If

(iv) (vi)

Let

1;i; p,q <

[ JI u (x) I q e ax

(8.144)

holds for every

a, BE R.

00,

1 I q

dx

J

u E ACLR(_oo,oo)

1 < q < p <

and (8.149) is satisfied, then (8.148) holds.

00

Then the Hardy inequality

;i; C [ Ilu'(x)I P e Sx dx J lip

9. THE HARDY INEQUALITY FOR

if and only if

9.1. Introduction.

o

< q

< 1

In Subsection 2.10 we have mentioned that the Hardy

inequality

(8.145)

1

$

$ q <

P

B '" 0 ,

00

a = B .9.

p (9. 1)

8.22. Remark.

on

(a,c)

the case

(c,b)

for

c E (a,b) , and Remark 7.14 (i) allow us to for­

o

(9.2)

Let

vo' vI' w ~ W(a,b),

v

O

' vI

In

If

1 $ p $ q $

P '"

00,

dx J1 I p

or

<

q

ACR(a,b)

1

<

has been investigated by G. SINNAMON [1 J for

1 < P <

00

Here we will follow, again with some modifications, the ideas of SINNAMON.

satisfy (7.40), (7.41),

c E (a,b)

(i)

$

C lrJ lu' (x) I P vex) a

ACL(a,b)

mulate the following theorem which is in fact a complement of Theorem 7.13. 8.23. Theorem.

Jl/ q dx

a

The methods used in this section (see Remark 8.7), the re­

sults mentioned in Lemma 7.12, used now separately for the intervals and

b

b

l(J lu(x) I q w(x)

the

case

some special methods are used, based on a result

(9.2)

of I. HALPERIN [lJ. Therefore, let us first formulate this result; the 00

,

proof, which is rather technical, will be given in Subsection 22. (8.146)

BL(a,c,w,v,q,p) <

00,

BR(c,b,w,v,q,p) <

00

,

9.2. Theorem.

then (8.147)

w~'P(a,b;vO'Vl) ~ Lq(a,b;w) .

b

(9.3) (ii)

If

1 < P

$

q <

, (8.146) is satisfied and

00

lim FL(x;a,c,w,v,q,p) = lim x+a+ x+c­

FL(x;a,c,w,v,q,p)

lim FR(x;c,b,w,v,q,p) = lim x+c+ x+b­

FR(x;c,b,w,v,q,p)

J fer)

o,

dt

x

(9.4)

J f(t)

(iii) (8.149)

w~'P(a,b;vO'Vl) If

1

$

<; C;

q < P <

~(a,c,w,v,q,p)

Lq(a,b;w) .

I ~

I:""

>,,: . , ,~..: •

00

<

',I· ;',

,

" t,,;

be such that

00

J pet)

,

dt <

00

•

a f

O

e

~(a,b)

satisfying

x

dt ;i;

a

(8.148)

<

Then there exists a function

o,

P E W(a,b)

b

a

then

128

f E M+(a,b)

Let

J fO(t)

dt

for

x E (a,b)

a no~-increasing on

(9.S)

fO/p is

(9.6)

IlfO/p I p, (b) a"p

$

(a,b)

I f/p II p, (a, b) ,p

for every

p E [1,(0) .

i;~~:;;-' 00,

AR(c,b,w,v,q,p) <

00

~~~t'

,

.s.~j

'j,~I;

:

''

r

"

In the main result of this section, which will be now formulated, the 129

__

._.

~

.

.

~

b

b

[J [J a

q wet) dtr/

= q/(q -

9.3. Theorem.

holds ~

(9.8)

[J v -p

l'

(t)

1r / q ' v 1-p ' (x) dX] dtJ

~r = 1q - ~p

Let

0 < q < 1 < P <

for

=

every

proof. Obviously, it suffices to prove the following two implications

~(a,b,w,v,q,p)

<

00

ql/q(p,)I/

* '-) Ai
(9.13)

A < L

0 < q < 1

(i)

, l/r

(1)-..)

~

A

L

w, v

1-

I

-:7

00

(L) q

~ <

Assume

(9.14)

•

l/r

A* ~ ~ L

and suppose in addition that

00

I 1 PEL (a,b)

in Remark 7.8.

ql

If

~ ~ C ~ ql/ q ~.

v, ware general functions from

Theorem 9.3 is a consequence of several lemmas, which will be

(9.15)

vn(x)

(9.16)

w (x)

vex) + ~ (1 + x 2 /(p'-I»)

proved later (see Lemmas 9.7 and 9.8). First, let us go through some auxi­ liary considerations. 1 < q < P <

In the case ACL(a,b) ,

00

~ <

00

(9.10)

Ai*

A~

,where

x

{ f

[f

a

a

1 p v - ' (t) dtr p

is meaningful also for the case

'

9.5. Lemma.

A{

<

Let

00

•

wet) dt

r/p

J

w(x) dx

}1/r

v

instead of

n

-L

w (x) ~ w(x)

and

n

v

(L) q

n

w , v

l/r A*(a,b,w ,v ,q,p) . n n

-L

1- ,

p (x) ~ v

I

x

0 < q < 1,

00,

n

1- ,

p (x)

A..

-L

1 < p <

00

•

v, w E:W(a,b) . Let

Now we will show

AL ,

A{

,

(~) q

for l/r

x E (a,b) , we can A~(a,b,w,v,q,p)

obtain-

ing

,

This will follow from the next lemma.

0 < q < 1 < P <

w

A (a,b,w ,v ,q,p) = n n

(9.17)

For

that,also in this latter case, the condition (9.8) can be replaced by the condition

satisfy the conditions (9.14) and,consequently, the identity

estimate the right-hand side in (9.17) by

b

[f

n

I

~

Since /

,v

n

was introduced by

= At~( a,b,w,v,q,p ) b

w

2

min (w(x),n/x ,n) .

=

(9.11) holds for

was characterized not only by the condition (1.25), i.e.

but also by the condition

n

Then

the validity of the Hardy inequality (9.1)

,

the formula (7.33) (see Remark 7.8). The number

130

W(a,b) , we introduce auxiliary

functions

9.4. Remark.

00

A*

-L

q

Then we obtain the identity (9.11) by integration by parts analogously as

C in (9.1) the following

estimate is satisfied: (9.9)

(9.12)

if and only if

u E ACL(a,b)

A{(a,b,w,v,q,p)

l/r

v, w S W(a, b). Then the inequality

00,

Moreover, for the best possible constant

~ <

'-~

and that the number

is now negative due to the condition

1)

~(a,b,w,v,q,p) = (~ )

(9.1 1)

a

x

from (1.19) appears. Note that

on

.=.;;=--=----==,"';;~-==--'===-==~""__=:=

l/r

I

AL(a,b,w,v,q,p) =

~

(9.7)

(9.1)

_.

__ "".~~-'=-~_=-;'=;;'-=:::-'=:;'-----o;=--=-~_.---=_~-;;:::==,,"=

defined by (9.7) and (9.10), respectively. Then

number

q'

--

---=-=--==-~---==c=::;.-=======:::.=:;====_~='==::=,; .:;;:.,--;:;;.-_:='_-=-===;:-;;=~.:--=---=:;;:-=--=o._-_._-;:--:;;:::;;;:=---==7:=.=':~_._--=.:=:~---===-'-.=_"';=~_._=._.-==;='

be

I

'I";:

! .

)

.

n

-+

00

,

(a,b,w ,v ,q,p) ~ (~) n

n

l/r

A'~(a,b,w,v,q,p).

q-L

Fatou's lemma implies that the implication (9.12) holds.

A < 00 • Using the functions F from (1.17) and ~L L L from (7.34), we can easily verify that the assertions (ii) and (iv) of (ii)

Assume

Lemma 7.7 hold also for

0 < q < 1 < p <

00

,i.e.

that 131

___

.._-- '"_ --...-

"__......

444 _

--"--y

-

-~:---

1/r

I

(?-) lim

~+a+

-E!!!L"-'

gwmlgllL -E-K

'="

_

;-.;

9.6. Remark. x

b

J wet)

dt

<

and

00

J

V

1

<

FL(~)

lim ~+a+

A n,k

(a,b)

for

C

{a }, {b } n

n

[J wet)

r/q dtJ

such that

(a,b)

a

n

~

a ,

b

n

t

b,

1

r

[J v 1- p' (t)

/

i r /q

I

(t) dtJ

b

I

[J wet)

dt Jr p

l ak a

I

1

1/r I

v -p (x) dx

n,k

=

Lr

/

n

Ar

-L

Letting

k -+

~ 00

~ ~ since

L r ,

L r

lim F~(ak) k+ oo

A

L

~

number f

) 1/r

E

F~(ak)J

+

Lq

o instead of the in-

with the inequality 1/

1/

b

q;,;; C

(f fP(x)

vex) dXJ

p

a

from (9.2). The reader can easily realize that the proof q

>

1

by the

q > 0 . The same is true as concerns Remark 3.7 and Lemma 3.4.

Let

~(a,b)

(9.20)

0 < q < 1 < P < 00,

v, w

c W(a,b)

. Assume that the

with the constant

C

Proof.

(i)

= ql/ q ~

•

Let

f

E

~(a,b)

and assume in addition that

1

fE L (a,b)

and

(A* )r n,k

p =

p

1- ' v E L 1 (a,b) .

A k ' we have n,

L

Consequently, the assumptions (9.3) of Theorem 9.2 are satisfied and there (A* )r

n,k

exists a function

we obtain Fr(b ) + ~ lim (A* )r L n q k+oo n,k

~

L q I

due to (9.18) and

(9.23)

lim (A* )r k+ oo n,k r

E...:... F (b ) > 0 • Letting r L n

n -+

f

O

satisfying (9.4) - (9.6). From (9.4) we have

b 1/ bx 1/ U(HLOq(X) w(x) dX) q = U f(t) dt)q w(x) dXJ q;,;;

(I

a

a

a

b

x

;' ; (f (J f 0 ( t)

00,

we have

a and since

132

q

A from (9.7) is finite. Then the inequality (9.19) holds for every L

(9.21)

[Fr(b ) _ Fr(a )J + L n L k q

o

k+oo'

r

of Lemma 1.10 remains valid if we replace the assumption

(9.22) and since obviously

(A~ k)r) = E...:... (A~)

dtJr p w(x) dx

x

r [F (b )

L

n+oo

".,",,,,,~

In the next assertions we will deal

9.7. Lemma.

)

Integration by parts yields r A

q

",,"

",-,,-~_=_

I

E...:... lim (lim

p, q

assumption

a

n x

J

x

[J v -p

ak x

A~,k =

I

l"!'_

f E M+(a,b) . This is possible since the assertion of Lemma 1.10 holds

also for

n b b

J

--"'"---'---;:',--"'''''--'i.'''''-

a

define the numbers

=

~-----

•

[f(HLOq(X) w(x) dXJ

and

F (0 = 0 • L

E~

--.0,

b

is continuous on

Choose two sequences

-

xE(a,b),

for

00

(9.19)

the function

~"'---L

equality (9.1) I

-p (t) dt

a

x

and for n, k

..dE-~

and the implication (9.13) is proved.

L

Consequently,

(9.18)

""'-~--

A~ ~

= 0 .

(~)

:;::=--- -----,,--;;-

~ E (a,b) ,

FL(O;';; iPLCO ;';; AL , iP

-'fl a

73IM--

-¥4i44

118/pfI p, (a, b) ,p = I gv 1/ plip, (a, b)

1/ d t) q w(x) dx J

q

a due to the definition of

p

133

n.,__

u_-_

'""

~

b

we can rewrite (9.6) in the form

see (9.22)

U

x

f/

(f fa ( t) d t) q w(x) dx

f~(X)

b [f

vex) dx

]l/P

(b

f

;;;

fP(x) vex) dx

J1/ P

a

.

b a

(9.19) for

b

instead of x

b

[f

(9.24)

f , i.e.

the inequality

b

1/ q

q

~

(f fO(t) dt) w(x) dx J

a

q

l/q

A

L

[f

f~(x)

ql/q {

1/ vex) dX)

q-l

[J

pet) dt)

b

U

fO(y)

Y

b w(x) dXJ

G

p(x) dx

a

y

l/q

w(x) dX) d Y}

r-

1 •

l/q q pq/r (y) f6(y) p1- - q/r (y) d Y}

a

a

Y a

f (J a

p.

f (y) q-l

J [p~y»)

;;; ql/q {

This together with (9.23) implies that it suffices to verify the inequality fa

~

a

a

a

q

Since x

b

and Holder's inequality with exponents

f [f fO(t) dtJq wCx) dx a

Y

x

a

a

b

1

f [f [f fO(t) dtJq-

q

[f

fO(y) d Y] w(x) dx ,

a

a

x

b

x

f [f a

fa (t)

dtr

b

{ f

a b

q

J [1 a

-=

fO(t) dt ]

fO(y)

[f w(x)

a

dX] dy .

y

ql

p(y) dy

}l/r •

a

y

b

pl-P(y) d y

}l/ P = ql/q AL (f

f~(x)

1/ vex) dX)

p .

a

However, this is the inequality (9.24).

be fixed. The condition (9.5) implies

~ pet) ;;;

q 1

dt] -

-

f

is a genera2 function from

~(a,b)

and that

2 f (x) = min (f(x),n/x ,n) n

(9.21) is satisfied with f n instead of Therefore, the inequality (9.19) holds with f n instead of f :

Then

O
~

Assume that

(9.22) holds. Define

t E: (a,y)

for every

fO(t)

and consequently (note that

[f fO(t)

f~(Y)

q

y

fOCY)

y

b

a

(ii) (a,b)

;;;

{f [f w(x) dx )r/ ff,p(x) dx )r/ a

w(x) dx

w(x) dx ]

a

;;; q 1/ q

q-1

y

yields

l/q

q

(f fO(t) dt) b

the Fubini theorem Ylelds

Let

p/q

a

b

(9.25)

p/(p - q) ,

r/q

f

(y)

_0_ [p

(y)]

q-l

a

Y

[f

f

n

q-l p (t) d t)

E L1(a,b) , Le.

b. 1/ [f(HLfn)q(X) w(x) dxJ q

.

a

a

Using this estimate in (9.25), we obtain with

C

ql/ q A . Letting L

b

;;; c

U

f~(x) vex) dx )

f

.

1/ P

a

n ~ ~ , we obtain (9.19) by the monotone 135

134

...

__

_ _,

__~~._.__

_ . ~

-

.-

:.

-

--_..

o;,.m._

_.__ -_--__

instead of

...... -

a~

.-

..~._

C';. .'...,..·

V

v . The number

AL(a,b,w,vn,q,p)

b

, l/r

, l/r

~ (~]

~(a,b,w,v,q,p)

is finite since

<

--.~'-'.

-''-",:_

......::

_

.,..... _ .••• '.-.".,.

_

"~.

'''''~ ~-""-'"

v

I-p'

[f

(I

a

a

._..__

.....,.....

~_.,.c

l'

get) v -p (t) dt)

q

w(x) dx

)l/ q

......._::::"_..... _"'"''-':-._ .. :~;;;o

-;:

,-_.._-

."~;:;;;r,,-.:~"'

. .-... -

.­

~

a

0:>

g E ~(a,b) . Taking

which holds for arbitrary

immediately obtain the inequality (9.19).

Let

9.8. Lemma.

is due to the assumption of our lemma.

<

.- .._:"" ... _-

p 1- , ) lip g (x) v p (x) dx

[I

0 < q < 1 < p <

0:>,

v, w E

equality (9.19) holds (with a finite constant

Consequently, part (ii) of the proof implies that the inequality

, 1 f(x) v P - (x) , we

g(x)

o

posit~ve

integral) and the last sign

-""

"-,-..:-~.....

, the monotone convergence theorem yields

x

~ ql/ q ~

Indeed, the identities in (9.26) are due to Lemma 9.5 (cf. the formula 1 p (9.11», the sign ~ follows from the inequality v - ' ~ v 1-p ' (cL the 1 p definition of where v - ' appears in a power of an

~,

'.~._

b

A~(a,b,w,vn,q,p) ~ ~(a,b,w,v,q,p)

,

~

I-p' t

vn

_

""

n

n

n

~(a,b,w,vn,q,p) = (~) (9.26)

~,,~

Since

Assume that v is a general function from W(a,b) . Define 1 v 1-p ' E. L (a,b) , i.e. (9.22) is satisfied with v

(iii)

_

-,

convergence theorem.

by (9.15). Then

~~

,~-

W(a,b) . Assume that the in­ C) for every

f E ~(a,b) .

Then l/

b

[I(HLf)q(X) w(x) dXJ

q

~

(9.28)

q

l/q( ,)l/q' A P

L

~ C .

a

Proof. According to Remark 9.6, Lemma 3.4 implies that the inequality

b

if

~ q 1/ q AL(a,b,w,vn,q,p) ,

P

B

fP(x) vn(x) dx Jl/

a holds for

E

f

hand side by

~(a,b)

ql/

q

~

. Moreover, we can replace the constant on the rightq = ql/ ~(a,b,w,v,q,p) which is independent of n

due to (9.26):

[f

1/ q (H f) q (x) w(x) dx L

a Let

g

E

C

B from (1.18) holds also for L it follows that

with

C <

0:>

b i \

~(a,b)

and

f(x)

J

i

b

~

g(x) v

ql/

q

l/p

J wet) dt

b

~[f

fP(x) vn(x) dx

a

n

J

Choose two sequences

~

.

f v 1- p ' (t)

p (x) . Obviously,

f

E ~(a,b)

and

get) v - P (t) dt] q w(x) dx n

(

f/

q

ql/ q

A

gP(x)

v~-p' (x)

n

b

[f [f

n

~ (9.31)

~[f

dt <

0:>

x E (a,b)

for

.

(a,b)

such that

a

n

+

a ,

b

n

t

b , and

/

wet) dtr q

x

[f

x

r q vl -p' (t) dt J / ' v 1-p ' (x) dX]

l/r

an

and auxiliary functions

a b

n

an

x

C

{a }, {b }

n

(9.30)

fn(x)

rJb

~ wet) dt x

)r / p

x

[f v l ' Jr -p (t) dt

/

'

q

1 ' v -p (x) X(an,bn)(x)

an

dX] l/p . Obviously

a

136

,

n E N define numbers b

1- ,

If I I ' a

0:>

a

we have from (9.27) that

,

<

X

for

1, and, consequently,

0 < q <

x

(

(9.29)

I

b

(9.27)

~

L from

rl

f

n

E ~(a,b)

and from (9.30) we have

137

~

An

b

II

b

[f wet) dt]

n

[J [rJ

q

an

an (9.32)

b

U ,w (t)

(~ , ) 1/r

x

r

/q

dt

vl -p' (t) dt

r

I ' q

l/r

t

an dt rip'.

<

co

q

[J(HLgn)q(X) w(x) dX]

n

and the inequality (9.19) yields

[f wet)

fn(x)

dt J dx

}l/r

=

An

q A: I q ~

x

a and since

f (x) n

(9.33)

A

0

>

(a , b ) n n

a.e. in

A:

= {

a

bn

1/

f fn(x) [f wet) dt J dX}

C

q

{f

f~ (x)

vP

b

II t { f wet) [J fn(x) dXJ dt} q ,

(x)

[f

1 p

v -

' (s) ds

X

[f

fn(s) ds

-P/q'[f J

v

I-p'

(s) ds

f fn(t)

x

1/

[f

dt = [

a

fn(t) dtJ

q

a

x

[{ f a

r

Using Holder's inequality with exponents of

=

q A

r/q

n

a

t

a

-

-1/ql

ds

J

fn(t) dt

]q w(x)

dx

}l/

q

.

J

JP (q-l)

v(l-p')(l-q)(x) dx

I/P }

an bn

{J an

[j an

1 q

1/(l-q)

.

and the definition

we obtain

{f

a

{f[f [f fn(s) a

If we denote

x

cf. (9.31)

q Ar/q ~

n b n ~ C f~ I q(x) v P ' I q(x)

1/ ' [f fn(s) dSJ- q fn(t) dt]q ,

(9.35) b

138

n

t

we have (9.34)

f

P(l-q) v(l-p')q(x) .

an

an

and since x

vex) dX]

an

x

a

f~(x)

l/p

x

,

an

x

a

pI ' fn(s) dSr q

a

a

The Fubini theorem yields

/q

x

[f [f

c

b

g~(x) vex) dx riP

a

> 0 •

b

c []

w E W(a,b) , we conclude

and

b

n

q

a

b

{ f

1I

b

r/q

A

due to (9.29). Moreover, b

fn(t)

then we can rewrite (9.34) in the form

bn

[f v 1-'P (t)

l/q'

]

ds

a

an

an

[f fn(s)

gn(t)

vl -p' (x) dx ]

[fx v -P

l ' (s) ds r

r [f

r

I P •

an

/q

fn(s) ds

pI ' q vl -P' (x) dx

vl -P' (s) ds r

p

vl -P' (x) dx } (l-q)/p

an

139

_-== . . .. .: :.- -.=:::,.-=-'!'~':;;::;'~~~~~~~;;;;~':=::::::::~ib~~;;;;,:~_~,~~~~~

....:::.---===---=-=....__ ~~~~:..::~~~~~~:'5:.~~~:t::::~~~:~~-.".=_.-:::

~,~~~~~~~~~~~~~!'?:.!f(t,!~m~i~~l,~4.~,?;c~~~~~~:~~~~~~~~~~\'5,.~,~'?~~~~~ _..__ ._..:.._=-""~=',.""",,,,-~-~=~=,~,,,"';;',,=_ •..-.,,-,=,,.."~~,,~.,~-~'_--"'''","==",=.= .. -~->=.=,,,".= . ~~.,,,,",,,=..,,".,,,.--=->,,,~.~,,,-,,,,-,,;:,,,,,,~=,-- .....~"'.=.:;.== .._=.--='-'--=."""-..""".,,'=.... - ;;,,.-=~,---''''<====''''',-'''. ,''-'''-=-~''---=--""""'="'-'-'--'~-""'~'."''''';;''''''

b

C

A~q/p { (

[f

an

P/q fn(s) ds

J

~(x) dx

bn

q

(9.36)

x

~(x)

[

• v

f

1-' p (s)

ds

]-P

v

1-' P (x)

x

,

E (a ,b ) . n

Since, in view of (9.29),

(9.37)

I

x

- I rr 1-' wet) dt ~ - - - - J v p (s) ds ] p - 1 I,

x

1- P

[f an

(

I

fn(s) ds) p/q

~(x)

dx

t

P

bn

~(x) = (~ - 1)

[~(x)]

r

[f -p/q

1

and with

p/q

w

instead of

instead of p

~(t)

dt

A

n

=

II q

q ( !) p

r

9.9. Examples.

consider ,

x E (a ,b ) n n

o<

(a ,b) n

n

ds

k (E., E.) q q

]-P/q'

[f ju(x) Iq x

an

}q/P P/q (x) vex) dx

n

remain true if we

o

, [v 1- P (x)] 1- p/q

<

b < co

6

C), while the inequality

ct

dX]l/q ;'; C [flu,(x)I P x 6 dx ] IIp

o

<

holds for

P - 1,

U

u

E AC (0, b) L

if and only i f

> 6.9._.9--1 P p'

and the inequality co

[flu(x) f

q < P < co

b

b

n

{I

~

(with a finite constant

This estimate together with (9.31) yields b

~ C

n'

satisfying (9.2). More precisely, if we suppose that

u, 6 E~

instead of

with

[xI' v 1- p (s)

A

A comparison with the particular Hardy inequalities in­

p, q

q]. Using (9.36) and (9.37)

an

r/p C A n

q < 1 < P < co , then the inequalities (6.16), (6.30) and (6.36) hold

for no

~(x) ~ (p - 1) -p/q

-l/q'

o

o

-1

q

and Fatou's lemma yields (9.28).

in (9.39), we have

~ (~ _ 1) p/q

II '

6.8, 6.9, 6.10 and 6.12 for the case

/q

w ,with

and of

n

(L) q

vestigated in Section 6 shows that the results derived in Examples 6.7,

x

(2.16) and (2.15) with

q

Arq/p) O-q)/q

p

dx

an

with

(9.39)

n

, l/q'

[j" f~/q(x) ~(x)

~ k(~,~)

an

p/q -1

(L

.E...:...) q (q

we obtain the estimate

(9.38)

C A rq/p

In view of (9.32) and (9.33) we have < co ,

an

bn

A~/q ~

q

bn

n

and using this inequality in (9.38), we obtain from (9.35) that

n

an

(a,b)

an

= L Arq/p

where

[cf.

J

x

an

an

x Iq ' I wet) dt r / q u v 1- p' (t) dt lr v 1-' p (x) dx f P

b

[f

f {f

~

}O-q) Ip

q

I

xU dXJ l/q ;'; C

with

o

< a < co

I

u' (x) p x

6

dx

] l/p

a

a

~

[f I

holds for

u E ACL(a,co)

if and only if

141 140

B
9.10. Remarks.

(i)

a < 13.9._.9..-- 1 p p'

or

i3

~

1 ,

p -

We have dealt only with functions

be defined as the set of all

a < - 1 .

ACR(a~b)

A L

A in Theorem 9.3, where R

by

u

from

the condition

A

<

00

~

(10.4)

1

•

~

ACLR(a,b)

for

r=pq/(p-q) ,andfor A=AL(a,b)

or A=AR(a,b)

we have

(k-l)(b)

o

(and consequently

m = 0

n

n = k ). Thus we have

m conditions at the

conditions at the right endpoint of the interval

We will investigate conditions on given

(10.6)

1/

[Jlu(x)l

k

E W(a,b)

under which (for

w(x) dxJ

q

~

b

C [Jlu(k)(x)I

10.2. Remark.

P vk(x) dx

l/ P J

a

u

E AC(k-l)(a,b)

m,n

The problem just formulated can be solved using the Hardy -

which corresponds to the case

k = 1

-

successively

u E ACL(a,b) , u ' E AC (a,b), ... ,u(m-l) E ACL(a,b) , L (m) (k-l) . E ACR(a,b), ... ,u E ACR(a,b) . For example, taklng k = 2 ,

u

for the functions

n = 1 , we investigate the inequality

b

Let

k

for functions (10.8)

m + n

are non-negative integers. Denote

[flu(x)l

u

(k)

dku/dx k

and let

q

l/ q w(x) dx )

b

~

C

[Jlu

l/p

1l

P (x)I v

2

(x) dx

J

a

a

be a positive integer which we

will write in the form

AC(k-l) (a,b) m,n

q

a

(l0.7)

10.1. Formulation of the problem.

w, v

p, q ) the (Hardy) inequality b

10. HIGHER ORDER DERIVATIVES

=

u (k-O(b) = 0

if

m = 1,

142

u

inequality (1.11)

c~q1/qA.

(l0.2)

u(m)(b) = u(m-l)(b)

tit

the estimate

m, n

o

holds for every

2-1/ p q 1/ q (p') 1/ q'A ~ C ~ 21 / r q 1/ q

where

(m-l)(a)

u(b) = u' (b)

C in (9.1) we obtain in

this case the estimates

k

u

m < k , or

The proof follows word by word the arguments used in

slightly changed: For the best possible constant

(10.1)

and

n = 0 ), or

u(a) = u' (a) =

(10.5)

is given by (8.98), again characterizes the

the proof of Theorem 8.17, only the estimates (8.101) and (8.102) are

with

(k-l) E AC(a,b)

(a,b) .

00

validity of the Hardy inequality (9.1) on the class q < 1 < P <

(and consequently

m= k

left endpoint and

from (8.100), where

o<

if

if

ACLR(a,b) , then

u(a) = u'(a) = .. , = u(k-l)(a) = 0

AR

is given by the formula (6.7). (ii) If we consider functions

(10.3)

to the foregoing

one. The formulation of the corresponding result is left to the reader; in fact, one has only to replace

u

u EACL(a,b) .

Obviously, using the substitutions indicated in Remark 1.8, we can reduce the investigation of the Hardy inequality on

such that

u

UEAC~~i(a,b)

u(a) = 0,

u'(b) =

,Le. such that

a ,

and we can solve this problem considering two inequalities

143

__=~

_

_:;,,"":-~:"'

~'.'::

=

,_,__ ~"'--::'

~:;'~

----.::;=~__'"__

b l/q q [flu(x) I w(x) dX] a

(10.9)

_

__':""_"""::"""-::"';;~-:::

__ =~ -,-=--- ,:

.~~.-_,!",_-----,~

_

--=~~

__.;;,,.,__--­

"~"',,"',,",,'::'-:'

~.;. ~~

",;;L_,~=~:~,

b l/r r C1 [flu'(x) I v (x) dX] ,

1 a

~

",_~,

.='~__:'-

~_--.,,;o;-,;;::::,c::",:,,",;;.

-

v (x) = x Y . Then the inequality (10.9) holds (with 1 00 ) for u E ACL(O,oo)

b

=

for

u

~

]l/r b [flu,(x)!r v (x) dx 1 a

c

ACL(a,b)

and

u'

E

1

<

p

2

b ]1/ P [flu f1 (x)I P v (x) dx 2

a

~

vI q

<

r . For instance, if

r €

[p,q] , we know (cf.

and choose

Theorems 1.14 and 6.2) that the inequality (10.9) holds for

uE ACL(a,b)

if and only if (10.11)

B (a,b,v ,v ,r,p) R 1 2

<

= 0 and

b

00 ) for

=

y

r

B .::

P'

p

_ 1 .

[Cf. Example 6.7, the formula (6.18) with the formula (6.19) with

y

and

y

r

,

y, r

instead of

a, q

instead of

.J

B, p , and

From (10.14), (10.15)

r , and we finally obtain that the inequality

(10.13) holds for

00 ,

<

while the inequality (10.10) holds for (10.12)

B>p-1,

(10.15)

we can eliminate

B (a,b,w,v ,q,r) L 1

a

u E ACR(O,oo)

and a parameter 00

and

= y.'l_3.-_1 r r' ,

a

and the inequality (10.10) holds (again with

ACR(a,b) , respectively, with a certain

'intermediate' weight function we consider the case

C

1 ,

y < r -

__-"'--"'''''''''-'''_''"'"''~'c=,;;,,;,":'',;;,;0,,""""'"

a = 0

function

(10.14) (10.10)

-----,-­

._------,._-~._--------------

__ :-:;;';;:'~i.,..":",;';'£-"''''''",·-:",--''':~::"_-:':'¥,.'''-_-:~:E_,,~~~~-;;;,:;;=.;:-&,;.-;;;;,,,,~,,:~~,""~,_","-~',;$j?::~~':,,,:;;;:r"~~~~?_"~;:;;:4c"';'f.~.i;F""¥;:;;~1;;:~~~~~.;;'~"'''''''~~-F"";_""',m:;:;E;~)!C';'-~~_~""'~~='~""_~'""""'_--

u'

E AC R (a, b)

(1) u E AC 1, 1 (0,00)

i f and only i f if

00 .

p -

(10.16)

1

<

B

<

2p - 1

a

=

B.'l p - ~ p' - q - 1 .

Consequently, the pair of conditions (10.11), (10.12) is at least sufficient for the inequality (10.7) to be valid on

Aci~i(a,b) .

(Compare with Example 10.16.)

This approach has one disadvantage: the presence of rather un­ Let us just formulate the main result; for simplicity, we will not

determined and (from the point of view of the original inequality (10.7)) redundant parameters

r, v

Moreover, the number of these additional 1 parameters will rapidly grow for big numbers k.

deal with the limit values (0,00)

the initial parameters

p, q , w , v

k

(and

m, n ) would appear.

10.4. Theorem.

Let

1

<

p

q < 00 ,

~

there exists a finite constant 00

10.3. Example.

Let

1

< p ~ q <

00 ,

a, B r::' R ,

and consider the in­

equality

(10.13)

(10.17)

[J !u(x) I q w(x)

dXJ l/q

o

q

x

a

dXJ l/q

~

C [flu f1 (x)I P

x

B

dx

J l/p

o

ho Ld s for every

C > 0

~

C

m, n E R ,

.

funct~on

u

w, v

m,n

~ W(O,oo)

•

Then

such that the inequality

[J Iu (m+n) (x) I p vm, n (x) o

o [flu(x) I

= 1, 00 and will consider the interval

only.

Therefore, it is our aim to derive (necessary and sufficient) conditions for the validity of the Hardy inequality (10.6), in which only

p, q

(m+n-1) E AC m,n (0,00)

l/p dx

J

if and only if the following

two conditions are satisfied:

on the class

AC;li (0,00) , i.e. for u such that u(O) = 0 and u'(oo) = O. a Here w(x) = x " v (x) = x B ; if we use the approach just described, we 2 introduce an auxiliary parameter r , p ~ r ~ q , and an auxiliary weight 144

145

(10.18)

[( (m-1)q J 1/q sup J wet) t dt O<x
x x

(10.19)

[I

sup O<x
wet) t

mq

1/ dt J

q

[J vm~~ l' (t) x

have

I '

np , ] 1 p t dt

<

00

x

,

t(n-1)p' dtf/P' <

00

•

J(x

x

x

[J (s -

- t) m-1

The necessary and sufficient conditions (10.18) ,

(10.19) can be rewritten in terms of the constants

B , B L R

J

(10.23)

introduced in

BL(O,

00,

w(t)t (m-l)q, v

m,n

q, p) < '" ,

+

f(s)

(J

Similarly as in the foregoing sections, we will deal with a

little different problem analogous to Problem 1.7: For

o

J

- t) m-1 (s - t) n - 1 dt] ds

o

f)(x) + (J f)(x) 2 1

If we denote s

(10.24)

I(X

K (x,s) 1

- t)

m,n

x

f(X

_ t)m-1

o

[J(s

- t) n-1 f(s) ds ] dt

00.25)

r

K (x,s) 2

J (x - t)

dt,

o

dt,

o<

< s

< x

,

m-1

(s - t)

n-1

x < S

,

o -m+1

o

[J o

m+ n - 1

satisfies the conditions (10.4) [With

a = 0,

b =

00 ]

u = H f m,n and we have

s

m + n - 1 ;:£ x

fP(x) vm,n(x) dXJ lip

instead of the inequality (10.17). Clearly, the function f

n-1

then

[f(Hm,nf)q(X) w(x) dxJ 1/q ;:£ C

= u Ck ) .

(s - t)

x

t

and deal with the inequality

00.22)

m-1

o

f) (x)

(m - 1)!(n - 1)!

[J(x

f(s)

f E W(O,oo) , we

denote

00.21)

x

- t) m-1 (s - t) n - 1 dt] ds +

[I(X

x

BR(O, "', w(t)tmq, vm,n (t)t -(n-1)p, q, p) < '"

(H

- t) n-1 f(s) ds ] dt

x

(t)t -n p ,

(10.20)

(ii)

I (s

s

o

(1.18) and (6.2) as follows

t) n-1 f(s) ds +

t

x

(i)

t)n-1 f(s) dS] dt

t

x

0 10.5. Remarks.

[I(s -

t)m-1

0

[J v~~~' (t)

0

I(X -

c(Hf)(x) =

0

< =

x

-m

Indeed, in (10.24) we have t

t

x

s

s

-n

-n+1

K

1

(x,s);:£n

K 2 (x,s)

0 < t < s <

;:£ m x

therefore,

;

0 < x - t < x

and

1 - - > 1 - - , and consequently s

10.6. Proof of Theorem 10.4.

Thus, we shall show that the conditions

(10.18), (10.19) are necessary and sufficient for (10.22) to hold. Let us fix write 146

H, v , c

m, n E R

instead of

and denote

K (x,s) 1

~ J xm-1 (s o

- t)

n-1

dt

n

x

m-1

s

n

c m,n = (m - I)! (n - 1)! . Further,

Hffi,n ,v m,n , c ffi,n

for simplicity. Then we 147

_

-

-;;;,-

""

---y-

s x

-m+ 1

s

-n

J KJ (x,s) = s

I(1

-

n-J t m-l - ~) (1 - !) dt s

....,;

(

~

0 s

~-=-I(J-!) s s

) fP(x) vex) dx

f (xnf(x»)P

o

o

x-

np

vex) dx

f fP(x)

~(x)

dx .

o

But then, according to Theorem 1.J4, the inequality (10.27) holds if and

m+n-2

dt

only if

m+ n - 1

0

BL(O,oo,~,~,q,p) <

(10.28) K , where we use the fact that in (10.25) we have 2 ! > J - ! < t < x < S ; therefore, 0 < s - t < sand

00

similarly for

o

s

which is the first condition in (10.20), i.e. the condition (10.18).

x

Analogously, the function

Thus, the function

x

f o

(J f)(x)

1

is equivalent to the function

x

I

m-l

s

n

f

f(s) ds ,

x

m

s

n-1

f(s) ds ,

x

o

and the inequality

and the inequality

(l0.29) (10.26)

f(s) ds

x

is equivalent to the function

x

J K2 (x,S)

(J 2f) (x)

K (x,s) f(s) ds 1

II J 1 f II q , (0,00) ,w ;;;;

c

II J 2 f I q, (0,00) ,w '"

c

II f II p , (0, 00) , v

I f I p, (0,00) ,v

will hold if and only if the inequality will hold if and only if the inequality x

(10.27)

[J (J x o

m-l

n

s

1/ q

q

f(s) ds)

w(x) dx

0

J

.

; ; c [J

fP(x) vex) dx

] lip

o

holds. However, the last inequality is nothing else than the inequality (1.12) with f (s) replaced by f(s) = s n f(s) and with the weight functions w(x) , v (x) replaced by w(x) Indeed, = x (m-l) qw(x) , ~(x) = x-npv(x) x

J [J

o

x

m-l

fo [Jx

(l0.30)

x

ill

s

n- J

f(s) ds

]q

w(x) dx

l'f

~ c

o

holds. This last inequality is nothing else than the inequality (1.13) with n-l mq f , w ,v replaced by f(s) = s f(s), w(x) = x w(x) , -

vex)

=

x

-(n-l)p

,

v (x) , and, according to Theorem 6.2, the necessary and

sufficient condition for its validity reads as follows: (l0.31)

BR(O,oo,w,v,q,p) <

00

s n f(s) ds ) q w(x) dx This is the latter condition in (10.20), i.e. the

0 x

x

I

[J

o

0

fP(x) vex) dXJjl/p

sn f(s) dS]q x(m-l)q w(x) dx

Io Jr' 0

f(s) dS)q

~(x)

condit~on

(10.19).

Thus we have shown that the conditions (10.18), (10.19) are necessary dx

and sufficient for the validity of the inequalities (10.26) and (10.29), respectively. However, according to (JO.23) we have

and 148

149

10.9. Remarks. Hf = c

-1

(J.f)(x)

The pair of necessary and sufficient conditions from Theorem 10.8 can be rewritten in terms of the constants A • A L R from (1.19), (6.7) as follows:

(J f + J 2f) , 1

and since the operators $

H, J

1

c(Hf) (x)

' J

are positive, we obtain the estimate

2

x E (0,00),

for

(i)

AL(O, 00, wet) t(m-l)q, v

i = 1,2 .

(10.32)

A ( 0, 00, w( t) t mq , v

.1

R

This implies that

(t) t- np , q,

m,n

p)

<

00

(t) t - ( n - l , )q,pp) < = m,n

(compare with (10.20».

I\Jif\lq,(o,oo) ,w

~ c I!Hfllq,(O,oo),w ~

(ii)

~ IJ 1 f! q, (0 " W + I

J f 2 q" (0 00) ,w

00)

l

If we use Convention 5.1 and consider the necessary and sufficien

conditions in the form (10.20) or (10.32), we can formulate analogues of Theorems 10.4 and 10.8 also for the limit values of ~

q

~

(Theorem 10.4) and for

p , q , i.e.

and in view of 00.26), (10.29), 00.22), the conditions (10.18), (10.19)

are necessary and sufficient for the validity of the inequality

suffices to use Theorems 5.9, 5.10 instead of Theorems 1.14, 1.15.

II

Hf II q, (0,00) ,w

~ c

\1

00

p

q

= 00

10.10. The case 10.7. Remark.

In the foregoing proof we did not exploit the particular

form of the necessary and sufficient conditions which guarantee the validity

1

~

q < p

~

00

for

(Theorem 10.8). It

(a,b) = (0,00) , the case is in fact excluded due to Remark 5.5. ]

[Note that due to

o

f 1\ p , (0,00) , v •

P

~

1

OUI:"

= 0

n

special interval

m

01:"

=

(i)

O.

Let us now consider the Hardy 1n­

equality (10.6) [for (a,b) = (0,00) ] on the class of functions e (k-l) 00 • u _ ACk,O (0,) , .t.e. satisfying the condition (10.3):

of the inequalities (10.27) and (10.30). Therefore we can repeat all

(10.33)

P > q , replacing the constants B , B by the constants A ' AR from (1.19), (6.7). According L R L to Theorems 1.15, 6.3 and 9.3 we immediately obtain

If we introduce the operatol:"

u(O)

=

u'(O)

u

(k-l) (0) = 0 .

arguments used in the proof also for the case

o

~TheoreIl1' Let 1 < p < =

m, n E: N, w, v E W(O,oo) • Then

m, n

< q <

00

q '" 1,

p > q,

fo~lowing

=

pql (p - q),

(Hkf) (x)

(k

1)J

(x - t)

k-1

J

f(t) dt

o

> 0

( ) uEAC m,n m+n-1 (0,00)

for

f E: W(O,oo) , then the function

(10.33) and we have

two conditions are satisfied: x

q 1r/q r i v 1- P ' (t) t np' dt )r/ ' v 1- p' (x) x np' dx < 00 m,n

wet) t(m-1)q dt l J m,n o

x J

00.35)

J U o

x

(10.34)

there exists a finite conDtant C

such that the inequal,,,ty (10.17) holds for' ever1J function if and only if the

r

by the formula

H k

u = Hkf

satisfies the conditions

f = u(k) . Thus we can again deal with the inequality

/q [J(Hkf)q(x) w(x)

dXJl

~

o

00

C

[f

lip

fP(x) vk(x) dx

0

J

instead of the inequality (10.6). Since obviously 00

x

f U o

0

t mq dt lr/ q J wet)

fJi v 1- p ' (t) l m,n x

t(n-1)p' dt ir/q'v1-P' (x) x(n-1)p' dx < ) m,n

x

00.

(Hkf)(X) ~ (k _1 I)!

f x k-l

f(t) dt

o

151

150

-~~=~~~~~=~~===--==~--=~"-~-=-"--~~~,===~==~===~~~===-===~-=c~-

for

f E ~(O,oo) , the inequality (10.35) will hold if the following in­

holds. However, this is the inequality (10.30) for

equality is fulfilled:

(f

o

0

x

k-1

f(t) dt)

11 q

q

w(x) dx ]

;'; c

[J

fP(x) vk(x)

v

dxf/P .

(10.39)

o

m = k,

o

n

= v k ), and it holds if and only if

(10.36)

=k ,

n

BL(O,oo,~,vk,q,P) <

00

l;'ip;';q;';oo,

,

00

BR(O,oo,W,;:;,q,p) < 00 ,

1

(10.40)

A (O,oo,w,v,q,p) < 00

1 ;'; q < P ;';

with

= x-(k-1)p vex) . These last two conditions, which coincide with

~

p ;'; q ;';

R

vex)

or AL(O,oo,w,vk,q,p) <

00

:s

1

,

q < P ;'; 00

or

o

< q < 1 < P < 00

the second of the two conditions in (10.20) and (10.32), respectively, are now suff~cient for the validity of (10.35) (With

(10.37)

= 0,

or

However, this last inequality is the inequality (10.27) for (and

m

""~c===~~~~~~~_~~~~

v = v k ' and it holds if and only if

x

[J

~_"_ -~,_-_~=__=~_=__'c._co~~~_=~~ ~=,c-=~~__==_~_===="'=~~~~

i--_....,--------~---=--~~~:c

or

i.e.

o < q < 1 < P < 00

H~

instead of

for the validity of the Hardy inequality (10.6) on

H

k

),

AC6~~1)(0,00) .

Let us summarize these results: with

(k-1)q w(x) = x w(x) . Therefore, anyone of the conditions (10.36) or

(10.37) is sufficient for the validity of (10.35). The conditions (10.36)

Let

10.11. Theorem.

k

E R,

k

~

2

w, v k E [v(O,oo). Then the inequality

and (10.37) coincide with the first of the two conditions in (10.20) and (10.32), respectively. (ii)

o

If we consider the Hardy inequality (10.6) on the class

AC6~~1)(0,00) (10.38)

fflu(x) i q w(x) dxl 1/q ;'; C [Jlu(k)(x)I P vk(x) dxf/P , )

(10.41)

,i.e.

for functions

u(oo) = u'(oo)

u

= ... = u (k-1)

f (t

= (k _1 I)!

(i)

= 0 ,

we can proceed analogously using the operator 00

~ (Hkf)(x)

holds

satisfying the conditions (00)

if

sup O<x
- x) k-1 f(t) dt

or if

f

1)1

k 1 t - f(t) dt

x

with

Consequently, the inequality (10.35)

-

with

H":

instead of

k

H

k

-

if the inequality

rr l'f LJ

0

x

t

k-1

f(t) dt)

q

i

w(x) dXJ

1/q

_ ;'; C

[J

wet) t (k - 1) q dt r/q

x

[f

0

x

[f

0

or

l ldt / p ' < 00 v 1- P '(t) k )

o < q < 1 < P < 00 and

wet) t (k - 1) q dt r/q

[f

v 1- P I (t) dt r k

/ql

v 1- P I (x) dx < k

00

0

(ii)

for

1 P fP(x) vk(x) dxJ /

o~~~oo

0

AC6~~1) (0,00)

u E

l
[i .e.

satisfying the conditions 00.38)J

and

[f

wet) dt] 1/q

o or if

152

x

[f

r = pq/(p - q) ;

holds

if

f

satisfying the conditions (10.33)J

and

1 < q < P < 00

00

~

[i.e.

00

and the estimate

;'; (k

(k-1) u E ACk,O (0,00)

for

1 < P ;'; q < 00

H* k

x

(H~f)(x)

o

1 < q < P < 00

[J V~_pl (t)

t (k-1)p' dt J l/p' <

00

x

or

O
and

153

X

00

[f wet) dt

f

o

r

-­

(f V~_pl (t) t (k-l)p I dt

/q

0

r

/q

in terms of the numbers I vt- p I

(X) X(k-l)p I dx

<

~

00 00

•

Al =

X

[J (t - x) (k-1)q wet) dt

[J o

x

~

~

r

/q

X

(f v~-p I (t) dt

r

1

/q I

v~-p I (x) dXy

0

(10.46) 10.12. Remarks.

The conditions just mentioned are only sufficient for

(i)

the Hardy inequality (10.41) to be valid on AC~~~l)(O,oo) or on . AC (k-l) (O,~). Necessary an d su ff"&c&ent cond'&t&ons have been derived by O,k V. D. STEPANOV [lJ. More precisely, it follows from STEPANOV's results that the inequaLity (10.41) hoLds for

AC~~~l)(O,oo) (10.42)

1 < p ~ q <

00

and

k ~ 1

on

if and only if

i=1,2

where F1 (x) = [f (t - x) x 00.43)

(k 1)

-

[f

[J vk-p 1 I dt j (t) illq

dt

JlIpl

(ii)

[f(X - t)(k-l)p' V~_pl (t)

Moreover, he has given estimates of the best constant

with

B

~

(k - 1)1 C

~

(f

(x ­

0

we have

= F (x) = FL(x;O,<Xl,W,vk,q,p) 2

with dtJlIPI

C

in (10.41)

F from (1.17), and STEPANOV's condition (10.42) coincides with the L condition B < ~ from Theorem 1.14. Analogously for the case 1 < q < p < ~ L if k = 1 , then Al = (p l /q)l/r A = A (cf. Remark 7.8) and STEPANOV's L 2 condition (10.47) coincides with the condition ~ < 00 from Theorem 1.15. For

B

x

k = 1

F (x) 1

of the type 00.44)

For

,

0

x

dt riP

Section 7, Theorem 7.5.)

x

wet) dtf/q

o

w( t)

1

]­ w(x) dx r

A = max (A ,A ) < <Xl 1 2

0

00

F (x) = 2

q w(t)

[f [J

r/p'

= pq/(p - q) . The condition

00.47)

X

00

r

with

=

t) (k-1)p I v~-p I (t) dt J

is necessary and sufficient not only for the continuity of the operator Hk from (10.34), but also for its compactness. (Compare with the situation in

sup F.(x) < ~

O<x
B = max

A 2

x

or

o(p,q,k) B

k

>

1 , STEPANOV derived a pair of conditions ( F.(x) 1

H k

defined by the

formula 00.34),

C.1 <

00

i = 1,2 ) while in Theorem 10.11 only one condition appears

A. < 0 0 , 1

from (10.42), and has shown that the operator

$

[FL(x;O'~'~'Vk,q,P) ;;;(x) = x(k-1)q w(x)

Co <

$

-

00

AL(O,~,~,Vk,q,P) <

or

00

with

cf. 00.36), 00.37)J. This single condition is

simpler than the pair of STEPANOV's conditions and can be more easily verified but, on the other hand, it is only sufficient. This is confirmed

00.45)

by the following example.

H : LP(O,~;v) -+ Lq(O,oo;w) , k

10.13. Example.

is compact if and only if

B

<

00

and

lim F.(x) = lim F.(x) = 0 x+O+ 1 x+oo 1

i

00.48)

1, 2 .

Further, analogous results have been derived for the case

J u(x) I

0 1 < q < P < ~

for

u

such that

Consider the inequality

1

2 eax dx

$

C2

J u"(x) I

1

2

eax dx

0

u(O) = u'(O) = 0 . This is the Hardy inequality (10.41) 155

154

for

2

k

= q

p

2,

w(x)

v 2 (x)

e

ax

a ~ F . If we suppose

with

(10.42) is satisfied.

a < 0 , (ii) then we can easily show that the functions

F.(x)

from (10.43) are bounded.

1

Consequently, the condition (10.42) is fulfilled and the inequality (10.48) (1) holds for u E AC ,0(0,00) . On the other hand, we have 2 2 ax 2 ax -4, FL(x;O,oo,e x,e ,2,2) = a L(ax -

1)

2

] ax + 1 (1 - e )

numbers

The same is true also in the case Ai

ciA i ;'; ~(O,oo,~,vk,q,P)

(10.50)

follows from the inequality

less than infinity, is

10.14. Remarks.

= AL ,

with appropriate positive numbers

the sufficient condition (10.36), BL(O,oo; eaxx2,eax,2,2)

and consequeutly

a5 in part

(i) above. For

Since STEPANOV's necessary and sufficient conditions have

;'; J

[f wet)

o

x

J

[J

o

x

(!)

i = 1,2

c i . Indeed, the estimate

(t - x)(n-1)q ;'; t(k-1)q A 2

not fulfilled.

A~

1 < q < p < 00 , since for the

from (10.46) we have

(k-l)q dt J

AI;'; A L by the same argument

we have similarly as in part (i) x ,rip' [J x (k-1)p' v 1-p (t) dtJ w(x) dx k

r/P

0

x

been mentioned without proof, let us show that his conditions follow from our sufficient condition. (i) (10.49)

<

Let

P ;';

q < 00

Then the following estimates hold:

F. (x) ;'; FL(x;O,oo,~,vk,q,P)

= FL(x)

1

i

= 1,2

, J

with

(k-l) q w(x) = x w(x)

in the first integral in consequently

x E (0,00)

for F 1 (x)

and

F. (x) 1

o

from (10.43). Indeed,

we use the fact that

o

< t

- x < t

and

;'; [J

1/q

x

x

v 1-p' (t) dt ] k

FL(x) . In order to estimate

F (x) we use 2 in the first integral and the inequality

oof

wet)

(~)

(k-l) q

dt

] 1/q

[xf .

x

[f x

wet) t(k-l)q dtJ1/

x

x

r/P

w(x) x(k-1)q

k-1

[fx v~-p' (t) dt J

dx

o

x

(i)

r/P '

r

­

~ A (O,oo,w,vk,q,p) p' L i

2

with

c

2

= (p' Iq) 1/r .

In Remarks 10.12 and 10.14 we dealt only with functions

from

AC~~~l)(O,oo)

case

AC~~;l)(O,oo) can be handled analogously; conditions analogous to

, i.e.

satisfying the conditions (10.33). Clearly, the

v~-p' (t)

x(k-l)p' dt

f

conditions analogous to (10.39) and (10.40).

/P

'

(ii)

The results of V. D. STEPANOV [lJ mentioned above without proof

are in fact particular cases of a more general investigation. He studied the Riemann-Liouville operator

x

1-k

[f wet) t(k-l)q dt J

(10.42) and (10.47) can be stated and compared with the sufficient

0 q

x(1-k) qr / p +(k-1)r w (x)dx

o

10.15. Remarks.

the inequality 1;'; (t/x)(k-1)q (k-1)p' (k-1)p' (x - t) ;:; x in the second, obtaining

F (x) ;'; [ 2

(t)dt]

1/ P '

0

here the right-hand side is

rip'

,

(see Remark 7.8), and this is (10.50) for

[J

t(k-l)q wet) dtJ

[Jv~-p

* ]r [Ai(O,oo,w,vk,q,p)

(t - x) (k-l)q ;:; t(k-1)q, which immediately yields

F (x) 1

rip x wet) t(k-l)qdt]

[f

o

V

1-p' J1/P' (t) dt k

F (x) . L

x

(l0.51)

(H f) (x) a

r~a) f(x

- t)

a-I

f(t) dt

o

Consequently, it follows from (10.49) that if the (sufficient) condition (10.36) is fulfilled, then also the (necessary and sufficient) condition

for

a ~ 1

and derived necessary and sufficient conditions for the validity

of the inequality 156

157

.

---~'=-==------------_._...,=---._--------,----_:-.::-:=..;,_ .:.=----:-~~~~~_._----

------

~.-

-

~H o f'w 1 / q ! q,(O,~) ~ C!f v a1 / p ! p,(O,oo)

00

(or, more precisely, for the continuity and compactness of the operator H : LP(O,oo;v ) -+ Lq(O,oo;w) ). Here we have mentioned the results a a concerning the special values a = k, k E ~ . (See Subsection 10.10,

t o +(m-1)q dt

f/

q

x

[J t B(l-p')+np' dt J l/p'

< 00

0

(10.53)

[J t°t-rnq dt f/ rJ tB(l-p')+(n-l)p' dt f/ P ' < x

sup O<x
formulas (10.34) and (10.35).) (iii)

[f

sup O<x
0=

~

g

o

00

x

The results mentioned in Theorems 10.4 and 10.8 are due to

H. P. HEINIG and A. KUFNER (see A. KUFNER, H. P. HEINIG [lJ). Although only

and an easy calculation shows that these conditions are satisfied

the Hardy inequality (10.17)

consequently

from (l0.21)

n ~ 1

i.e. the continuity of the operator H m,n is considered, it is clear how to obtain conditions for

the compactness of the corresponding imbedding and estimates for the best

the inequality (10.52) holds for

AC(k-1)(O,oo) m,n

if and only if

-

(10.54)

and

with m

~

1,

B E (np - I, (n + l)p - 1)

possible constant in (10.17). Further, it is obvious that also in this case the results can be extended from

E~

m, n

instead of the operator

to general H m,n

0,

S ER

a~l,

S~l,if

we consider the more general operator H , S ' o x

(H

0,

Bf) (x)

f(a) f(B)

I(X

t)

o-l[I (s

o

- t) B-1 f(s) ds ] dt ,

and

(10.55)

a

p

(m

p

+ n)q

B g + g - 1 - kq .

P

(10.55), we can also write Let

= m,

k

0

= Sg P

~ - (k - l)q - 1 .) P

n = 0 , i.e. let us consider the inequality (10.52)

for functions

u satisfying the condition condition (10.36) has the form

and investigate the inequality

~H a, Sf.w 1 / q l q"oo (0 ) ~

C If

v1/PS~ a, p, (0

,00

P

(Compare these formulas with the formula (10.16) from Example 10.3; in

(ii)

t

= Bg + g - 1 -

(10.33). Then the sufficient

) sup O<x
arguments from the proof of Theorem 10.4 can be used almost literally.

[I

t o +(k-1)q

x

Let us conclude this section with some examples.

) l/q dt

[Ix

t

SO - p ') dt J l/p'

~

<

a

first condition from (10.53) for

k

m,

n

=

0

J,

and it is

if 10.16. Example.

10.52)

Let

[flu(x)l

q

1 < P

x

a

dXJl/q

o On the class

(i)

Let

$

q < 00,

$

C

a, S

E~ ,

and let us consider the

[f Iu (m+n) (x) Ip x B dx Jl/p

B a

<

P -

1

satisfies (10.55). STEPANOV's necessary and sufficient conditions

(10.42), (10.43)J have the more complicated form

o sup O<x
AC(k-1) (0,00) m,n m

~

1,

n

~

[f x

l/q taCt - x) (k-1)q dt J

[J t B(l-p' ) dt J 1/ p , x

<

~

,

0

1 . The conditions (10.18), (10.19) have the

form 158

159

with sup O<x
[f to dt) l/q Gt S (l-p') x

lip'

a E (0,00)

mentioned in Definition 1.4, but the foregoing results can be extended also

(x _ t)(k-l)p' dt]

to such modified weights). Using Theorem 10.8 we can show that the inequality

0

a

[f I

and lead to exactly the same result (10.56), (10.55). (iii)

(10.52) for functions

u

u (x) I q dx

k = n , i.e. consid~r the inequality

m = 0,

If we take

o holds for

S

again with (iv) k

>

0

0 < q < 00,

AC(k-l) (0,00)

~C

Iq

[Jlu(m+n) (x)

IP

x S dx

riP

o

1 < P < 00,

q

~

1,

P

>

q

on the class

if

m,n

kp ­

S E (np - 1, (n + l)p - 1) ,

from (10.55).

Le. for the same values of

Let us summarize the results for the special case

= 2 . According to (10.55), we have

0

= S - 2p

p

S 2p dx flu(x) IP x -

~

as in the case

p < q

[cf. (l0.54)J.

and the inequality 11. SOME REMARKS

S C f1uU(X) IP x dx

o

S

=q ,

00 (l0.58)

r

satisfying the conditions (10.38), then we can

show that it holds for (l0.57)

(this weight function does not fulfil all conditions

0

11.1. A modified Hardy inequality.

holds

The inequality (1.2), which has

initiated the investigation of Hardy-type inequalities, can be rewritten

S< p - 1 S e (p - 1, 2p - 1) S > 2p

for for for

if

u(O)

u'(O)

if

u(O)

u'(oo)

if

u(oo)

u'(oo)

o o o

in the form 00 (l1.1)

u

and

u'

f[~ r f(t)

f E W(O,"')

(cf. Example 10.3). It can be easily

p

,

E

u

(HL f)

particular classical Hardy inequalities, i.e. the intervals (for

u) and

(x)

u'(O) = 0 ,

since the corresponding intervals of admissible values of (_00, p - 1)

S

C

E f fP(x) x dx ,

o HL '

x

f o

f(t) dt ,

i.e. a certain integral mean value, then we can rewrite (11.1) in the form

for the

00

(2p - 1, 00)

~

E

f(HLf)P(x) x dx

u') have no common point.

(for

dx

x

such that

u(oo) = 0,

x

< P - 1 . If we introduce the operator

seen that it is impossible to obtain the inequality (10.58) by this approach for

~

E

dt )

0

o

These results could be obtained also by using the classical Hardy inequality (0.2) successively for

x

o

Consequently, the ordering of the 'boundary conditions' in (10.4)

C

f

fP(x) x

E

dx .

0

This notation represents only a formal change, but it can be substantially

has been reasonable.

extended if we introduce more general 'integral means' 10.17.

Example. w(x)

160

x

Let us consider the modified weight function

{ 0

for

0

for

a < x < 00

<

x

<

(11.2)

a

(H L f)

(x)

R~X)

J f(t)

ret) dt

a

.j ,..,.;~;1 . •. ..

,

'

;;­

(

161

""".

for

...

_.... _

f ~ ~(a.b)

with

~-

_"""""",,.-;or.

"777

."'.....

"'..,

5llii-m

.....

..t:tt::ili~.

~

~;,;

''''_.........''''".,''''''''''''",.... ~./'''''\

R. r E W(a.b) • and consider inequalities of the

b

type

(11.6)

b [J(HLOq(X)

(11. 3)

~(x)

II dX]

b

q;O C

[J fP(x)

a

1/p

1

~

p.q

~

v. w E W(a.b) . This approach. in which in fact four

00

= w(x) = 1 ). K. C. LEE. G. S. YANG [lJ ( w(x) = vex) = x [1 J and others. Mostly

<

00

•

a

a

).

(ll.7) B. G.

they derive sufficient conditions in

b

x

[J

(J

a

a

f ( t) ::; (t) d t )

and

1 • we obtain the in­

R(t)

q

~ (x) dx

] 1/q

b

~

C [J [p(x) ::;(x) dx )

l/ P

a

b sup [J a<x
x

Let us show that the investigation of inequalities of the type (11.3) can be reduced to the investigation of our fundamental inequality (1.12).

vet)

ret)

and the condition (11.6) has the following simple form:

p = q .

the form of differential inequalities for the case

If we take

equality

little more general situation. Let us mention N. LEVINSON [lJ (for PACHPATTE

I

~l-p I (t) r P I (t) dt ) 1 P

a

weight functions occur. has been used by various authors. sometimes in a vex)

[f

::;(x) dx ) 11.2. Example.

with

I

x

.) sup lrJ wet) R-q(t) dt 1 I q a<x
I:

~(t)

11 dt]

1I

x

q

[J ::;(t)

dt)

I

p

<

a

i. e.

b

(11.4)

[J

(J

a

a

f(t) dt) q w(x) dX]

Let us note that G. SINNAMON [2 J

b

1/

x

q

~

1/p

C [J fP(x) vex) dx ]

of the inequality (11.7):

.

x

a

[J (J

For this purpose, rewrite (11.3) in the form

b (11.5)

[J

x

a

with

1

(R(x)

J ret)

ret) dt)

q

~(x) dx

)l/ q ~

C [J [p(x) ::;(x) dx ]

1/ P

a

l/q

q

dv(x) )

~

C

[J

fP(x)

d~(X»)l/P

with general Borel measures

~

, v .

Some imbedding theorems. In Subsection 7.11 we have introduced the 1 weighted Sobolev space w ,P(a,b;v 'V ) and its subspace w~'P(a.b;vO,vl)' O 1 emphasiZing the connection between the Hardy inequality 11~3.

f ~ ~(a,b) • denote f(x) = [(x) rex)

01.8)

and introduce new weight functions

Obviously,

f(t) d~(t»)

_00

b

a

w(x) = ~(x) R-q(x) ;

investigated the following generalization

vex) = ::;(x) r-p(x)

~uw1/q~ q, ( a. b) ~

C

~u'vl/p~ p, ( a. b)

and the continuity and compactness of the imbedding

f E ~(a,b) • and we obtain (11.4) from (ll.5). Moreover, (11.5)

01.9)

w~'P(a.b;vO'Vl) C

Lq(a,b;w)

will hold if and only if (11.4) holds (with the same best possible constant C ). Since the necessary and sufficient condition for the validity of (11. 4 )

(see Theorem 7.13). Here only the weight functions

is well-known (see Theorems 1.14, 1.15 and 9.3), we immediately obtain the

important. the weigh\. function I

necessary and sufficient conditions for the validity of (11.5). For instance, for the case

1 < p ~ q <

00

the inequality 01.5) holds for every

fE:- ~(a.b)

V

o

v

' w have been 1 has not played any role.

The Soviet authors M. O. OTELBAEV. K. T. MYNBAEV and

R.OINAROV have

investigated the inequality

i f and only i f

(11.10)

162

~uwl/q~q.(a,b) ~ c(~uv~/p~p.(a,b) + lu'v~/Plp,(a.b») 163

=-~~~=~=.-.~=.=-=-~.=~~'="

-::-._,-~~

for

w~'P(a,b;vO,vl)

u E

u E

as well as for

w1 ,P(a,b;v O'v 1 )

However, then the numbers

establish necessary and sufficient conditions for its validity, i.e. for ,l~

wl,P(a,b;v

O

'V

1

)

C

Lq(a,b;w)

,

1

~

q < P <

v

o '

v

' w . The cases

1 are considered.

00

, 0'

'

v

~"'''--''::''''"" ~.","'::..~

(see

which is finite due to (11.15). Consequently, the Hardy inequality

and also for the compactness of these imbeddings in terms of weight functions

i

••

1/ P II I' -1 / p I IIIV O I p ,(a,b)'lv 1 p',(a,b)

the continuity of the imbedding (11.9) as well as of the imbedding 01.11)

B (a b v

R

p p) i = Land i 1" , (1.18) and (6.2)) can be estimated from above by the number

. They

"._':C,"_

1 < P ~ q ~

00,

l/ p lp,(a,b) 'I < II' 1 / p II II,uv O ~ C u v 1 ip,(a,b)

aLL three p ~

00

,

and holds for every

u E C~(a,b)

w~'P(a,b;vO,vl)' the

and on the space

norm

(7.39) is equivalent to the norm given by

The results mentioned are rather complicated and some of them appear

without proofs. Here we will reproduce some recent results mentioned in the

iu'vt/Plp,(a,b)

.

paper by R. OINAROV [lJ. Thus, the inequality (11.10) reduces to the Hardy inequality (11.8) It is supposed that w, YO' v 1 E

(11.12)

v

L~oc(a,b)

,

-lip E

v1

LP '

(a, b) loc

(compare with our conditions (7.40), (7.41)). Further, the following numbers

c

c

=

J vO(t)

f

dt +

c

E

h

(ii)

h

(iii)

h

(iv)

h

a a a a

o

q E L (a,b) ,

w~'P(a,b;vO,V1)

u E

-

i.e. the continuity of the

b

and sufficient for the inequality (11.10) to be valid for u E W1 ,P(a,b;v 'V ) , i.e. for the imbedding (11.11) to be continuous. o 1

J v 1-1 P ' (t)

dt +

dt

c

<

00

,

h

<

00

,

h

=

00

=

h

h

b

b

o'

V <

00

In the cases (ii) -

(iv), the integrability of the weight

VI1-p ,

. h e~t er at one

(iii)

b b <

-

. ~s

. 1 d v~o ate ,

or at both ends

E

v~/p E

h d ' teen po~nts

-

functions

.. ) t h e cases ( ~~,

the case (iv). Then it is necessary to

introduce some new notation and define some modifications of the numbers A ' A ' B ' B ' which have been used to characterize the validity of L R L R the Hardy inequality in the foregoing sections. Following R. OINAROV [lJ

=

1

L (a,b) ,

LP(a,b)

-

0f

00

we define some functions, intervals and numbers, which have been introduced 1-p' 1 v E L (a,b) , 1

i.e. (11.15)

l/q

b

and extensively used (11.14)

w

imbedding (11.9). OINAROV has stated that the condition (11.16) is necessary

In the case (i) we have

V

. , Le.

a

J vO(t)

=

equality (11.10) for

(a,b) , and four cases are distinguished:

(i)

1 wE L (a,b)

a

c with

condition

guarantees the validity of (11.8) and consequently, the validity of the in­ p v 1- ' (t) dt , 1

(11.13) hb

instead of v), and the conditions of its validity have been studied 1 in detail in the foregoing sections. The reader can easily see that the

(11.16)

are introduced:

ha

(with

,

For

x E (a,b)

by and

M. O. OTELBAEV for the case y ~ 0

such that

[x, x

+ y)

v (x) 1

C

(a,b)

=

1 .

define

v~ 1/ p E LP ' (a,b) 165

164

x o(x,y)

f V~_pl (s)

sup {d > 0;

ds

f vi-pI (s)

$

x-d

ds The last assertion concerns the case (iv). In the cases (ii), (iii),

x (x - d, x]

C

similar assertions hold with the following change in (11.19): in the case

(a,b))

B- (a ,b) ; in the case (iii), B+ (a,b) p,q 0 p,q

B- (a,b) is replaced by p,q + is replaced by B (a, b ) . p,q a (ii),

d+(x)

sup {d

>

0;

IP

x+d

[ f

Vo(S)dSf

1I

x+d

V~-pl(S)dSJ

[f

[x, x + d)

d-(x)

t:,+(x)

t:, - (x) \) t:, + (x) ,

a

inf {x E (a,b); x - d-(x) > a}

Further, for

sup {x (a,B)

C

(11.18)

B

p,q

(a, B)

E (a,b); x + d+(x)

B+ (a,B) p,q

x

a

[I

sup tEt:,-(x)

b}

<

t

[I

a

/q w(s) dSJ1

x

J V11- p ' (s)ds J1/pl

[

(11.10)

holds for every

p ~

u E

00

,

and

166

only the

[I

1 J

w(s) ds

r/P

J

wet) dt

]1/r

,

t

JUq

r

J

1 I J1/pl v ­ P (s) ds 1

B (a,B,w,v ,q,p) i 1

i

~

v -p I (s) ds

t:,+(t)

r = pq/(p - q) x + d+ (x) ,

x

[I

J


and -+

t

rip I

w(s) ds


r/P

J

wet) dt

]l/r

.

are functions inverse to the functions

x - d ­ (x) , respectively,

x E (aO,b ) . O

(11.11), i.e. the validity of the inequality (11.10) for every 1 u E W ,P(a,b;v 'v ) , the corresponding necessary and sufficient conditions O 1 are derived only for the cases (ii) and (iii) and have, roughly speaking, the form B + F(a,B) p,q

L,R ).

<

00

,

Where

a

= h

b

w~'P(a,b;vO,vl)

BUB

. Then the

i~equality

F(a, B)

[and consequently, the

B = max {B­ (a,b), B+ (a,b)} p,q p,q p,q

[f w(s) a

imbedding (11.9) is continuous] if and only if (11.19)

,

These results concern the imbedding (11.9). As concerns the imbedding

t:,+(t)

x

h

-+

t:,- (t)

w(s) ds

I

[f [

t

sup sup E (a,B) tE t:,+(x)

00,

v -p I (s) ds

t:,-(t)

B

Then the following assertion holds:

1 < p ~ q ~

00

¢,-( t)

rip I

~

J

[I [

Here

(compare with the definition of the numbers

Let

<

~

(a,b) , denote sup xE(a,B)

p

<

A+ (a,B) p,q

x

(11.17)

q +

(a,b)},

+ [x, x+d (x)],

t:,(x)

O

C

$

numbers B-p,q (a,B) are replaced by the numbers A-p,q (a,B) which correspond to our former numbers , A*R (d. (7.33» and are defined by

1 ,

o(x, d+(x») [x - d-(x), x]

b

$

1

+

A (a,B) p,q B

t:,-(x)

o

P

case

An analogous result holds for the

I

x-o(x,d)

x-o(x,d)

c in (11.10) is equivalent to Bp,q

Moreover, the best possible constant

x+y

dSJ

q

[f v 1-p 1

~~:

"jiJl"

"{

)­ I ,"'

'J~. i ';

i

(s) ds

J lip I

a

(a,B) = (a,a ) in the case (ii), O This concerns the case 1 < P $ q $ 00

and

<

I

(a,B) ,

p

~

(bO,b)

= 00

;

for

in the case (iii). 1

$

q

<

P

<

00

,

the

167

number

F(~,B)

has to be added to the corresponding number

For the case

= q = 2,

p

v 1 (x)

=1

A

p,q

, a necessary and sufficient

condition for the inequality (11.10) to be valld on

AC

(a,b)

36

J w(s) ds

:;;

C[

J v0 (s)

d s + 2 (f3 _

for every interval

(CL, (3)

such that

(3a,3B)

~) -1] C

=q

: he has shown that the inequality

(11.22)

J!u(x) Ip x-),p dx o

, holds for

3a

CL

p

has been

LR obtained by E. T. SAWYER [2] . His condition has the following form: 6

inequalities have been investigated by P. GRISVARD [IJ for the special case

u E C~(O,oo) < p <

(a,b)

00

)

~

C

JJ lu(x)

P

- u(y) I .1+Ap oolx-y,

dx dy

provided

o

<

A < 1 ,

A

;<

1.

p

We have mentioned here results concerning the continuity of the

{note that the right-hand side in (11.22) is the p-th power of the norm of

imbeddings (11.9) and (11.11). Results concerning the compactness can be

the 'derivative of order A of u ' which appears in the definition of Sobolev spaces of fractional order WA,p cf. R. A. ADAMS [IJ, Chapter VII, Theorem 7.48, or A. KUFNER, O. JOHN, S. FucfK [lJ, Definition 6.8.2J. A. KUFNER, H. TRIEBEL [lJ have shown that the inequality (11.22) holds also for 0 < p ~ 1 , and extended (11.22) for p > 1 to the case of two general weights w, v

found in the book of K. T. MYNBAEV, M. O. OTELBAEV [lJ.

11.4. Hardy inequalities with fractional derivatives. In Remark 10.15 we have mentioned the inequality

(11.20)

[J (Hex f) q (x)

l/q :'; C

w(x) dx )

o

[J

fP(x) v(x) dx)l/P The proofs of the results just mentioned are based on the theory of interpolation.

a ~(O,ro) , where

on the class

x

(H~f) (x)

r~ex~ J(X -

t)ex-l f(t) dt

o and

ex;: 1 . For

0 < a < 1,

Ha f

integration; conditions on

is the 50-called operator of •fractionaL w, v E W(O,oo) which guarantee the validity of

(11.20) have been investigated by several authors. Here, let us mention

K. F. ANDERSEN, H. P. HEINIG [lJ; H. P.

HEI~IG

[lJ (sufficient conditions);

K. F. ANDERSEN, E. T. SAWYER [IJ (necessary and sufficient conditions). If we formally denote by

a D

the operator inverse to

H

a

,then we

can rewrite the inequality (11.20) in the form

01.21)

[Jlu(x) Iq o

l/ q w(x) dx

J

~

c

[fID(XU(X)IPV(X) dxf/P

o

which represents a 'Hardy inequality for fractional derivatives'. Such 168 169

12.2. Formulation of the problem. Let 1 $ p.q < 00, w,v 1 ,v 2 , ... ,v N e W(Q) We will deal with conditions which guarantee the validity of the inequality

Chapter 2. The N-dimensional Hardy inequality

l/q [flu(x)!q w(x) dx )

(12.7)

$

N I

C [ i=l

Q

on a certain class

12. INTRODUCTION 12.1. Some definitions. Let For

u

u(x)

(12.1)

Q

be a domain in

defined (a.e.) on

R

N

aQ •

u , Le.

Me

For a subset (12.2)

Q = Qu

aQ

S

(12.9)

$

is a subset of the set

[flu(x)l

q

[.I

C

JI~~.(x)IP Q

l/ P

vi(x) dx ]

1

{1,2, ... ,N}, or the (equivalent) in­

l/q w(x) dx ]

Q

u

on

Q

$

C

I

dU

iE S

[Jl Q

~(x) I

p

vi(x) dx

]l!P

.

1

such that

12.3. Remark.

supp un M = f/J

00

w(x) dX]l/q

lE S

denote by

C~Ul)

CO(Q)

q

equality

We will deal with the inequalities mentioned above mainly

for the special case

Further, denote by

(12.4)

Me Q

with an appropriate set

Q

o}

7

the set of all infinitely differentiable functions (12.3)

[flu(x)l

the closure (in the Euclidean norm) where

u(x)

1

modifications, for example. the inequality

(12.8)

<.:: Q;

~ C~(Q)

]l/ P

denote by

Q

of the set {x

K

Xi

Q

dx

Instead of the inequality (12.7) we will also investigate some of its with a boundary

supp u

the support of the function

K.

f I~(x) d IP v.(x)

p = q

1

or

CO(Q)

The reason is in its simplicity which enables us to explain the fundamental the set of all infinitely differentiable or continuously differentiable functions

u

on

(12.5)

supp un

ideas more clearly and without disturbing technical details.

such that

Q

The passage from the case aQ

and, moreover, the set

bounded domains

= f/J supp u

p = q

to the case

p > q

is easy if we

content ourselves with estimates which are sometimes rather rough: we need 1-S

bounded. [Note that

C~(Q)

C;Q (m

for

only Holder's inequality. as will be described in the following example.

Q.J

The case

p < q

needs more sophisticated considerations

-

cf. Subsection

12.13. A general approach to both cases is made possible via imbedding

Finally, denote by

theorems for weighted Sobolev spaces investigated in Chapter 3 (cf. Lemma (12.6)

W(Q)

the set of all weight functions on

16.12) . Q, i.e.

measurable, positive and finite a.e. on

the set of all functions

A complete answer was given also by V. G. MAZ'JA

Q.

[1]

who used the

notion of capacity and investigated the inequality (12.7) on the class C~(Q) ; we will mention his results in Section 16.

170

I,a,

!• .

171

,-=------=- - - - - - .- - -

=------------

12.4. Example. for the case

---=..• - - - - . - .- - . - - - - .- - - -

-

... -- ---

Suppose that the inequality (12.7) is fully investigated p = q , i.e,

-.-.. -

---~.-.;,,-. -~.

(12.16)

_.,- .

dM(x) = dist (x,M)

for

-

..

'~;;_.-.~~.~;c;;;~;;;;;

x EO r2 •

that the inequality Investigating the inequalities from Subsection 12.2 for these special

~(x)

[Ilu(x) IP

(12.10)

dXJ lip

~

i=l

~

1

q

~

exploiting the results derived in Chapter 1. Let us illustrate this

~

xi

r.!

approach by some examples.

u E K .

holds for every Let

weight functions, we can use with advantage the 'one-dimensional' approach,

a ]l/P C [ NL f 1~(x)IPv.(x)dX

<

P

<

00

Using Holder's inequality with the exponents s = .P. q ,

p - q

Ilu(x)l q w(x) dx = Ilu(x)l r.!

(12.11)

q

~q/p(x) ~-q/p(x)

w(x) dx

~

[I lu(x) I

P ~(x) dx

J

(I(~,w,~,q,p))

(p-q)/p

(12.18)

M

Further, let

where

K = C;(Q)

I

I(r.!,w,~,q,p)

I

~q/(q-p)(x) wp/(p-q) (x) dx .

Since

The estimates (12.11), (12.10) imply

[flu(x)l

q

l/q w(x) dx

r.!

J

~

I

L I~U

(N

C I1/rl

i=l

r.!

(x) IP

Xi

~.(x) ~

dx

]

lip

l/r

1/q - lip. However, (12.13) is the inequality (12.7) for

q

with

v. =~., and consequently

p

~

P

(Le. for

uJe have derived (12.7) from (12.10)

~

provided the number

= q )

I

(12.19)

functions appearing most frequently are of the type

with

(12.15) with

vex) = [dist (X,M)r~

Me r.!

and

a

ER

notation

=

1

N-1

N 1

2

,

Q, Le.

O} . and consider the inequality (12.7) on

with the weight functions

(see Definition 1.4). Sometimes we will use the

N

for

x E Q , we obtain

w(x) = ~(xN) ,

vi(x) = ~i(xN)

Under the notation from (12.17) we have

Ilu(x)

I P w(x) dx

I

[fIU(x',xN)I

P

~(xN)

dX N] dx' .

M 0

r2

Using the one-dimensional Hardy inequality, we can estimate the inner integral on the right-hand side of (12.20) obtaining 1

IIU(X' ,x N) IP

(12.21)

or, more generally, of the form

vex) = ~(dist (x,M))

v = ~(t) E W(O,oo)

p = q

. For

( x ,x ' ... ,x _ ) E R

1

(12.20)

from (12.12) is finite.

12.5. Some special weights. In the theory as well as in applications, weight

(12.14)

with x'

~'~l""'~N E W(O,l)

for

N

, denote

{x E Q ; xN

dM(x) = x

(i) where <

N

(0,1)

Q

w(x) = ~(dM(x)) , vi(x) = ~i(dM(x))

r.!

(12.13)

be the cube

M be the 'basis' of

and let q/P

~

(12.12)

r.!

x = (x' ,x ) N

(12.17)

r.!

~

Let

x = (x ,x 2 '···,x N) E R 1

--p- , we have

s'

12.6. Example.

o

1

~(xN)

dX N

~

cP

II~(X' ,x N) IP ~N(xN) Ox o

N

dX N

provided

02.22) with

t3 = 8 (O,l'~'~N'P'P)

<

00

113 given by the formula (8.69). [Note that

u E Coo(Q)

o

and 173

172

-~~~,~~~~~.!,~~,f~~~~~~~~~·~*~~

u(x',x ) E. AC LR (O,l) N

consequently

x' E M

for every

.J

which is the inequality (12.8) for

X

J1U(x) IP w(x) dx

~

JI~~N(X) IP

cP

Q

{N}

S

and

q . This immediately

p

~

p <

00

]

(r

holds for every

<

(Ni~l

Q

00

•

u

E C~(Q)

IIa a~i

(x)

IP

Vi (x) dx

)l/P

Xi

Ilu(x)I P w(x) dx

I (J

Q

M1 0 00

,

=

~j(dM(x)) ,

1

instead of

Let

~

lu(x ,x") 1

IP

dX 1]

~(XN)

with

w,

v

= w(x) and c 1

Xl'

= 1.2, ... ,N-l)

i

and obtain immediately the following

p <

00

•

Let

w, v

N

E

W(Q)

be given by the formulas

o

Multiplying this inequality by

I

lu(x) IP w(x) dx :;;

Q

1

~(xN)

ci

P

Q

<

00

holds for every -

C=

we obtain

P (x) I w(x) dx ,

12.7. Remark.

I

l/ P

w(x) dx ]

r c(N-l E I~(x) IP i=l . dX Q

M1

~N(dM(X) )

~

Q

d Xl .

and integrating over

II~~

(0, l';:;'~N'P'P)

[J lu(x) I ~

fl~( dX x 1 'x ") I

VN(X)

Then the inequality

dx" .

(12.28)

P

P < cI =

o

(12.26)

vi(x)

E W(O.l) . Suppose that

N

U3

1

dX I

with

j ~ i

w(X) = ~(~(x))

(12.27)

we can estimate the inner integral on the

I

flu (xl ,x") I

u E C~(Q)

be fixed. Then thein­

Now we can combine the inequalities (12.23) and (12.26) (with

right-hand side of (12.25) using the ('non-weighted') Hardy inequality

P

it. {1,2, ... ,N-l}

i = 1,2, ... ,N-l

1

<

N , and we immediately obtain the following

assertion:

without any further assumptions on the weight

;i(dM(x)) ,

A (O,l,l,l,p,p)

l~p
(iii)

N in the form x = (x ,x") where (ii) Let us write x E R 1 be the face {x E Q; Xl = O} of our cube x~ = (x ' ... ,x ) , and let M 1 N 2 Q . Then

Since

~

i

equality (12.24) holds for every vj(x)

Then the inequality

Q

functions v.(x) 1

(12.25)

and with the

instead of C without any further assumptions on the weight functions

!3 (0, l,~'~N'P,P)

,

x. , 1

Let

l/P . lu(x) IP w(x) dx :;; C

(12.24)

q

assertion:

vN(x) dx ,

implies the following assertion: 1

1

being played by

Q

which is the inequality (12.8) for

Let

p

This procedure can be obviously repeateu with the role of the variable

(12.20) that (12.23)

and

VI (x) = w(x)

special weight function

M , we obtain in view of

Integrating the inequality (12.21) over

{ 1}

S

u

i

E C~(Q)

1 N max

w(x) dx +

I

d Id~N(X) IP vN(x) dx

]l/P

Q

with the constant

(C.C 1 )

.

In the last example we in fact derived three types of

.

estimates of the form (12.7) (i.e. (12.24)). providing rather-big

possibilities of choice for the admissible weights on the right-hand side:

The symbol

6

(O,I,l,l,p,p)

denotes the number

(8.69) for the particular weights

w = v

=

1 .

e (0,1.w,v,p,p)

from

(a)

V connected with N

i = 1,2 •... ,N-l ;

w via the condition (12.22).

Vi

arbitrary,

175 174

(13)

v. = w

(y)

v

1

-

1

= v

2

for some

{l, 2, ... , N-1} ,

i S

-

-N

- , = ... = v _ = w N 1

v

Clearly, the special form of the domain

arbitrary for

v, J

j

;<

inner integral on the right-hand side of (12.34) obtaining

i

satisfying (12.22) .

rl

12.8. Example.

Multiplying (12.35) by 1

>

0,

i

1,2, ... ,N} ,

dU ar I

M=

{oj p

q

on

C~(rl)

with weights

v. (x) = ::;, (d (x») , 1 1 M

w,::;. E W(O,oo) • If we introduce the spherical coordinates

xl = r cos 8

x

= r sin 8

(0,

i)

Here

3l/

vex) fl~~.(x)IP rl

pl

C [J1

lip dx ]

1

p

q

Obviously,

rl cos 8

00 TI/2 2

x

i) , and

(0,

TI/2

[f lu(r,8 ,8 ) I P d8 2] 1 2 000 f

Since

the Jacobian of the corresponding

f

0 (0,TI/2,1,1,p,p)

<

00

,

~(r)

r

2

sin 8 1 d8 1 dr .

we can estimate the inner integral with help

of the ('non-weighted') Hardy inequality J(r,8) = r

dM(x) = rand (i)

$

flu(x) I P w(x) dx

1 1

dX] lip

v (x) = v (x) = v (x) = vex) . 3 2 1 (ii)

mapping is (12.33)

i=l dX i

I

with

= r sin 8 sin 8 312

G is the square

P

!dU -(x)

vex) = ::;(dM(X») . However, (12.37) is the inequality (12.7) for

(r,8) = (r,8 ,8 , ... ,8 _ ) , then the image of rl is the infinite 1 2 N 1 'cylinder' Q = (0,00) x G, G = {8 = (8 , ... ,8 _ ); 0 < 8 < ~, i i

1 N 1 1,2, ... ,N-1j . For simplicity, we will consider the case N = 3 . Then

2

I' L

G, going back

and using the obvious estimate

with

1

x

' then integrating over

3

1

V E W(O,oo)

rl

w(x) = ~(dM(x») ,

(12.32)

::; 3 P -

[f lu(x) IP w(x)

(12.37)

of the type (12.15), i.e.

where

P

1 x

with

we obtain

and consider again the inequality (12.7) for

(12.31)

(x) I

r 2 , p , p ) < 00

sin 8

to the Cartesian coordinates

put (12.30)

J

o

(,l( 0,00, w(r) - r 2 , v- ( r) a)

(12.36)

x.

vCr) r 2 dr

provided

Let N

IP

dU r Idr(r,8)

cP

$

dr

o

by another example.

rl = {x = (xl ,x ' ... ,x N) E R 2

(12.29)

2

r

I

as well as our special weights

rl

(12.27) have played an important role. Let us illustrate the influence of the 'geometry' of

f !u(r,8) P ~(r)

(12.35)

2

sin 8

w(x)

1

.

:;\

~ (r)

v. (x) = ::;. (r) 1

!~.,'

1

Using the spherical coordinates (12.32), we have

TI/2

of

lu(r,8- ,8- ) Ip 1 2

TI/2 d8-

2

< ~

CP

2

f Iae-(r,8 dU - ,8- ) Ip 1 2 0

Multiplication by

w(r)r 2 sin 8

1

d8

2

•

2

' integration with respect to

r, 81

and

the obvious inequality (12.34)

J1U(X) I P w(x) dx rl

J [[IU(r,8) jP G

~(r)

r

2

drJ sin 8 1 d8 .

0

IP d8 1 ~(X) 2

$

1 2P-

rl~(x) liP l dX2

Ixl

P

P

+

I~(X) I dX3

P \xI ]

Making use of the one-dimensional Hardy inequality, we can estimate the 176

177

finally yield

12.10. Example.

[Jlu(x)I P w(x) dx

(12.38)

n

lip

;:;;

J

This is the inequality (12.8) for v (x) = v (x) = w(x) Ixl P 3 2

3 1/p ' C 2 [iL 2

p

q

and

fl

au ax. (x)

IP

rl

~

S

{2,3}

w(x) Ix Ip dx J1/

P

Since with

v

o= N

=

x

In the last example we in fact derived two types of estimates

v , where

v

is connected with

v

w via the

[Jlu(x)I P Ixl

(i)

a

11pl

~3T p

la

[ 3

it

a p fl~~.(X)IP Ixl + dX]1I n ~

u (x) I p Ix I0. dx 1I

holds for every

r sin 8

=

p

q

and

u

E

C~(Q)

We have IIU(X) IP w(x) dx

J I [J lu(x 1 ,r,8)I P ;(r)rdr] d8 dX

n

000

J

P ;:;; C [3 iL II

~~i (x)

1

'

c~(n)

and

a E R

lip'

min [3

p

(2p) -lip n

Io I

au ­ 3r(x ,r,8) I p vCr) r dr 1

;:; cP

o

8

(12.42) v

(O,"',~(r)r,~(r)r,p,p)

<

00

E W(O,oo) • Integration with respect to

xl' 8

and the obvious

estimate

Ix Ia+p dx J1I

I~~(X)IP;:;; 2P- 1 [I~~/X)IP

Q

u €

r dr

provided

P

P I

~(r)

with the constant

I~~/X)IP

+

leads to the inequality

::';W;~:

J .

,ill;

J i,'!:

,

,."

'

,\

lip

(12.43)

[flu(x)IP w(x) dx Q

J

;:;; 2

1I

p

I

[3 JIla~. a (x) I p C.L ~=2

Q

vex) dx

J lip

~

179

-

=

=

~

'

"

=

,

=

~

=

-

.

_

_

=

=

_

-

=

~

c

=

=

-

_

=

r •

Ilu(xl,r,G) IP

.

V.,ji

178

=

P

with

n

=

r cos 8

dxf/P;:;;

the inequality

c

3

=

and the one-dimensional Hardy inequality yields

3

[J I

Q, we can use with advantage the

'" n/2

On the other hand, for w(x) = Ixl a we obtain in (12.38) again v (x) = 2 = v 3 (x) = Ixl a + p , but now the inequality (12.38) holds for all a E R with the constant (2p)-1/p n • Combining these two results we show that

(12.40)

= .•• = xN = o} .

dM(x) = r

n

~

3

Again we consider the Hardy inequality (12.7) for

equality (12.37) has the form (12.39)

be defined by (12.29), but now put

with a weight of the type (12.15), where now

arbitrary,

1

="'==__

and

v (x) = v (x) = w(x) Ixl P .

2 3 a a

If we put w(x) = Ixl , Le. wet) = t , then (12.36) will be fulfilled _ a+p . a for vet) - t , ~.e. vex) = Ixl + p , provided a 7 - 3 , and the in­ (8)

2

J(x ,r,8) 1

= v2 = v 1 3 condition (12.36);

._"._.

(x ,r,8) with r = dist (x,M) and 1 (8 , ... ,8 N ) . For simplicity, we restrict ourselves to the case 1 -2

3 . Then

1

v

= x

."'_____ = _ _ _ = = _ _ .

xl = xl '

of the form (12.7): (a)

N

..

cylindrical coordinates

x 12.9. Remark.

2

R

.= -

M is the edge of our polyhedron

= q ) with the above mentioned weights v 2 , v 3 without any further

assumption on

nC

Let

M = {x E~; x

(12.41)

Consequently, we can assert that the inequality (12.7) holds (for p

= __

~

.. __._--_._--_.

_

._ _ =_. . . .__.__.. __ __. _ - - - -. - -... _------------_._-_ .. _---_.-_ .. __._._-_._---_._-_._-_.----_._-_._--_._-_._----_.-._--_._-----_._-------­

- - - _-=-=-=--==-=-=--=---=.

--=

-=:;;:- --

.,---~_:-::"~~"~~

with

vex)

. However, (12.43) is the inequality (12.8) for

p

=

the inner integral by means of the one-dimensional

q

n

v (x) = vJ(x) = vex) . Consequently, the inequality 2 p = q ) with these v ' v without any further assump­ 3 2

Hardy in­

S = {2,3 } with

and (12.

~[dH(x)J

=

holds (for

tion on (ii)

°1 dO 1­

f lu(r,01,8 2 ) IP sin

vI'

:S

o

On the other hand, we can write

'" '"

II

f lu(x) I P w(x) dx

n/2

n/2

;;; C

[f !u(x 1 ,r,0)I P dO] w(r) r dr dX 1

1

(I

au J W(r,Ol ,( 2 )

o

Ip v(Ol) ­

sin

°1 dO 1

°1 ,

p, p) <

1

000

Q

and since

ni2

[0, ~' 1, 1, p, p) <

£J

with help of the

00

(3

we can estimate the inner integral

,

[0, n/2, sin

°1 , ~(81)

sin

00

•

('non-weighted') Hardy inequality is approach is possible but leads to weight

n/2

n/2

f lu(x 1 ,r,0) I P dO

;;;

C~

o

[ 1~~(X1 ,r,O) I

functions

v,(x) 1

which are

necessarily of the form (12.15).

P

dO . (ii)

In Example 12.10 it was possible to express our integral also in

form

Multiplication by

w(r)r , integration with respect to

xl ' r

and the

'" n/2

obvious inequality

I~~(X)IP;;;

P 1 2 -

[I~~/X)IP

Ixl

P

+

I~~/X)IP

J

o

Ix1p]

f0

[jlu(x ,r,0) IP dX ] ;;;(r) rd8dr.

1 1

o

the inner integral cannot be estimated by some 'non-weighted' Hardy in­ ~i.;equality since the corresponding number $i~:,; .

finally yield the inequality (12.38).

f.

Consequently, we have shown that the inequality dip

[I [u(x) /P

w(x) dxJ

;;; C

[J1 II ~~. Q

Q

1

(x)

I

lip

vi(x) dx ]

r,'J,~

is infinite, while

weighted analogue leads again to weight functions which are not of the

~\';form

P

') (O,"',l,l,p,p)

(12.15). (iii) In the foregoing examples we have dealt with the inequality for the case

p = q

In the next two sections we will illustrate

modify the approach described if we omit the assumption holds for our special domain

Q

p

=

q .

M from (12.41)

from (12.29) and for

with an arbitrary weight v and with v (x) = v (x) = vex) , where either 2 3 l P vex) = w(x) Ixl or vex) is connected with w via the condition (12.42).

The case

~-_.~_.

1 S q

<

P

This case was mentioned in Remark 12.3.

< '"

we will illustrate how the one-dimensional Hardy inequality can be and deal with the special domains and weights from Examples 12.6

12.11. Remark.

(i)

express the integral

In Example 12.8 we have not used the possibility to f1u(X)

I

q

w(x) dx

in the form

12.8. (i)

Under the notation from Subsection 12.6 we can write

Q 00

f

n/2

I

[f lu(r,01,02) IP sin 01 dOl]

000 180

1

n/2

~(r)

r

2

d8 2 dr

(12.44)

fIU(x)[q w(x) dx

f [jlu(x' ,xN)!q ;;;(x N) dXr-;) dx'

Q

H

0

181

- - - - --=--==--=-=-==­

[compare with (12.20)J and estimate the inner integral by means of the

1

(12.45)

~(xN)

W(x N) , integrating over

dX N

1

~

q

C

o

ax x ,x N) I (JI~('

o

1

q/p

P vN(x N)

N

J1U(X) I

J

q

w(x) dx s

M

Q

provided

A,(O,l'~';N,q,P)

<

ci f Ult(X 1 ,X")I

00

[JI~~

ci J

1

(12.46)

M1

and using

Holder's inequality we obtain similarly as in (12.47) that

one-dimensional Hardy inequality fIU(X' ,xN)jq

Multiplying this inequality by

P

1

0

1

q/p P .; (x ) dx" (x 1 ,x") I dx 1 N J

/ )q/p ~p q(x ) dxd dx" ~ N

MOl

1

with

Ji

over

given by the formula (8.98). Integrating the inequality (12.45)

M and using Holder's inequality with the exponents

p/q,

p/(p-q)

1

(M)J (p-q)/p N-1 1

flu(x) I

w(x) dx

1

~

P ]q/P dx' ;;; (x' ,x N) I vN(x N) dX N

[fl~~

M

0

1

(12.47)

M

cq

0

Cq1

=

P [)rl~_( ax x )I wp/q (x)

Q

N

[fl~(X' x )I aX ' N I

[f

s Cq[m (M)J (p-q)/p N-1

[f [f I ~~1 (xl ,x") IP ~P/q(XN)

dx 1J dx

p =

M 0 1

q C f

Q

"r/

1

cq [m

S

we obtain in view of (12.44) that q

=

N

P

;N(X ) dX ] dx ,riP N

N

q P dx J / .

1

This is the inequality (12.8) for

S = {1}

and with the special weight

function every

v (x) = wp/q(x) . Consequently, the inequality (12.7) holds for 1 u E C~(Q) without any further assumptions on the weight functions

v 2 ""'YN .

au IP ]q/P dX (x) vN(x) dx , [fl N

(iii)

Q

In the case of the domain

~

and set

M from Example 12.8

we can proceed similarly if we use the expression which is the inequality (12.8) for equality (12.7) holds for every on the weight functions

v,(x),

S = {N} , and consequently

u E C~(Q) i

=

the in­

without any further assumption

1,2, ... ,N-1 . We only need that the

l

condition (12.46) be satisfied. (ii)

1

Since

[f !u(x 1 ,x") Iq dX 1J ~(xN)

f

Q

M1 0

Jt(O,l,l,l,q,p)

<

00

dx" .

we can estimate the inner integral on the

right-hand side by means of the ('non-weighted') Hardy inequality 1

1 flu(x1,x")lq dX 1 0

s

C~

[f\~~1 (x 1 ,x") I 0

P

r/

dX 1

f [flu(r,G) I

~

G 0

q

~(r)

r

2

dr J sin 8 1 d8

[compare with (12.34)J. If we use the expression

If we use the notation from 12.6 (ii), we have

f lu(x) lq w(x) dx

f1u(X)lq w(x) dx

f lu(x)

I

q w(x) dx

Q 00

TI/2

f

f

TI/2 q

[f lu(r,8 ,8 2 ) I d8 2 J 1

~(r)

r

2

sin 8 1 d8 1 dr

000

[see Subsection 12.8 (ii)J, we can estimate the inner integral by means of

p .

the ('non-weighted') Hardy inequality since Holder's inequality with the exponents

p/q,

J"t (O,TI/2,1,1,q,p) p/(p-q)

<

00

•

Then

yields

183 182

(12.48)

since otherwise the inequality (12.49) holds trivially. If we denote

~

J1U(X) jq w(x) dx ~

1/q -l/q ) U(x) = u(x) w (x) = u(x',x N) w (x N

(12.54)

~

[fl~~

ci

~

P (x)I 2

wap/q(x)

dXr

/P

[I(a,~,q,p)J(p-q)/p

then

'lau ~(x) IP dx = Jlau ax. (x) J

(12.55)

where

Q

I ~(1-a)p/(p-q)

I(a,~,q,p)

for

(r) r 2 dr .

such that

I(a,~,q,p) <

then (12.48) yields the inequality (12.8) for

S = {2,3}

with the special

weight functions v (x) = v (x) = wap/q(x) Ixl P 2 3 12.8 (ii) - cf. the formula (12.38)J.

00

12.13. The case

1

p < q

~

~

[similarly as in Subsection

from Example 12.6 with

M

P

~

dx

[f1~~N (x) I

1 2P-

P wp/q(x)

Q

I

given by (12.18). Then we have the following

I u (x) I P

I

,1/q ~C [N '~1 Iiaa~.(x) I

P

[f lu(x)lqw(X)dxJ

1-

N

N

q

p

E C~(Q)

Q

(12.57)

I Jl!P wPq(x)dx

A

~

(12.53)

-

0

N

provided

= C6

II~~N(X)

IP wp/q(x) dx ,

0 ,

E W(O,l)(" -p/q

(O,l,v 'w O

v0 ( t) =

AC(O.I)

,p,p)

<

such that the derivative

~'

~8

that

~~.

E LP(Q)

for i

1,2, .... N.

1

00

Moreover, since I

U E Lq(Q)

with

q

from (12.50) and

that

u E C;(Q) . We can assume that for

sUPP U C Q , we can use the classical imbedding theorems

for Sobolev spaces which imply that

~-p I q (t) I ~' (t) I p .

fl~~.(x)IP wP/q(X)dx
dx'~

and consequently we conclude from (12.55) and (12.56) in view of (12.53) ~

with

Indeed: Let

dxNJ

pf[fl~(1 ~< Co ax x 'X N)IP-p/q()d w XN XNJ dx I M

where (12.52)

~O(XN)

1

non-zero a.e. in (0,1) , and (12.51)

P

Q

- - - + 1

w(x) = ~(xN)

[flu(xr,xN)I

1

Q

(12.50)

1

M 0

Q

u

vO(x) dxJ

Moreover. in view of (12.51), we have

v0 (x) dx

The inequality

holds for every

P

Q

vO(x) = vO(x N)

assertion:

(12.49)

dx + (*)p Ilu(x) I

Q

Here we will deal only with the cube

00

1

Q

where <

fl~~N (x)I

(12.56)

a E R

P w I q(x) dx

i = 1,2, ... ,N-1 , and

o Consequently, if there exists a number

Q

1

P

I

(12.58)

[flu(x)l

Q

i=1,2, ... ,N {( f

1

i'if"

~::' ,

184

,

t";

'.'

.\

q

dXJl!q

~ c2 [Jl II~~i(X)IP dXJl!P Q

(see, e.g., A. KUFNER, O. JOHN, S. FucfK [lJ, Theorems 5.7.7, 5.7.8, or

185

.

~...:::...--::=~=-~:=::~:._----'-~-----~ ~-~

IR. A. ADAMS [1]. Theorem 5.4 ). Now, (12.49) follows immediately from

Further, let us denote by

(12.58) in view of (12.54), (12.55), (12.56) and (12.57).

(13.3)

was the one-dimensional Hardy inequality from Chapter 1. We have used here

let us put

the special weights (12.15) and the considerations have been relatively

(13.4)

simple due to the special choice of the domain

~

and of the set

Q and

C(Q;x~)

onto the hyperplane

(x~,t)E

{tER;

1

1

X.

1

o,

x ~ E P. (Q)

and for

1

1

Q}

M. ;r(the cut of

Let us mention that the approach described in Examples 12.4, 12.6, 12.8 can be used also for more general sets

~

the projection of

The main tool we have used in the foregoing subsections

12.14. Remark.

Pi (Q)

Q by a line orthogonal to the hyperplane

Xi

=

0 ; draw a

,picture!). Obviously, there exists a (finite or infinite) sequence of open

M using· local

and mutually disjoint intervals

coordinates, but the considerations are rather cumbersome and, sometimes, J.(x~) = (a.(x~),b.(x~») J 1 J 1 J 1

(13.5)

restrictive additional assumptions on the weight functions are needed. We will not deal here with this approach, which is described in A. KUFNER [2]. In the next section we will again deal with the approach via the

such that (13.6)

=

C(Q;x~)

one-dimensional Hardy inequality but for more general weights than those

UJ.(x~)

j

1

mentioned in Subsection 12.5.

For

u = u(x)

(13.7)

f(t) =

(13.8)

AC. (r2)

J

1.

defined on u(x~,t),

t

1

Q.

x

=

(x~,xi)

,

x~ 1

, denote

E p. (Q) 1

C(Q;x~)

€

1

13. SOME ELEMENTARY METHODS 13.1. Some useful notations. The weight functions from Subsection 12.5 have been very special and, in fact, have depended only on one variable (either

1

the set of all measurable functions

directly or after an appropriate change of variables). Here we will deal

corresponding

f

u

on

Q such that the

from (13.7) satisfies

mainly with weight functions independent of one variable (or depending on it in a special manner). Therefore, for the point

x = (x ,x ' •.• ,x ) ~ R 1 2 N

N

f E AC

(J . (x ~ ) ) J

we will use also the expression (13.1) where

I

(

xi = xI"'" xi -1 ' xi+ 1 ' ... , x N) E R For

Q E R

,

I.(~)

= S.(Q) = sup {x.; x = (x~,x.) E 1 111

D.

1

xi E

= inf {x.; x = (x~,x.) E Q} 111

S.

1

186

D

i

~_I-a.e.

x ~ E p. (Q) 1

1

lim

is the 'diameter of

r2

in the direction of the

J

x.-axis'). 1

AC

u(x~,t)

t+a.(x~)+

= D.(Q) = S.(Q) - I.(Q) 1 1 1

such that the corresponding function

u E ACi(Q) L

(J . (x ~ ) ) J

for every

1

and for

~_I-a.e.

Pi(Q) . Consequently,

~}

for (1. e.

and for

j

AC.1, L(Q)

{1,2, ... ,N}.

from (13.7) belongs to

=

1

i E:

denote

I.

1

(13.2)

N-l

the set of all

N

for every

Finally,

= (xi ,xi)

x

1.

~_I-a.e.

1

o

1

x~

E PieD)

and for every

j . Similarly, we can introduce

187

"f,;<;''''''3'''':;;~';:'f~,-~,~-~,_w7_.

the set (13.12)

AC.1, Run .

vi (x~)

=

Z j

The following lemma generalizes the approach described in Example 12.6, part (ii).

w(x:,x.)(x. f J/xi)

111

"-iit~~~~
- a . (x: ) ) p-1 dx.

<

00

- x.) p-1 dx. <

00

J

1

._-~"\;;

'-'.'MYIii

1

[or

(13.13)

f

v.(x:)=Z

11.

w(x~,x.)(b.(x:) 11 J1

1

]

1

J J. (x: )

Let

13.2. Lemma.

1 ;;; p

JIu (x) Ip

Di p-

< w(x) dx ~

Q

holds for every Let

such that

Jlau ax. (x)

Q

Proof·

Q be a domain in

Let

•

be independent of the variable

w E W(Q)

(13.10)

00

i E {1,2, ... ,N}

there exists a number let

<

1

N

and suppose that

= D. (Q) 1

<

00

Further,

•

C~ (Q)

J

jor ~_l-a.e.

(13.14)

w(x) dx

J1u(x) I P w(x) dx ;;;

holds for every

o

u(x)

for

xE R

N

\ Q . Then

P

Jl~~. (x) I vi(x~)

Q

Q

obviously

u

13.4. Corollary.

u(x)

u(x:,x.) 1

and consequently

w(x)

~~(x:,t) dt

J

1

ax.

Ii

1

1

and

E C;(Q)

In addition to the assumptions of Lemma 13.3 let us

!u(x)I P ;;; (x.1 - I i ) p-1

J I.

w (Xi) w2 (x~) 1

1~~.(x~,t)IP

x~

p

f w1(x . ) (S. >

- x.) p-1 dx. ;;; C

1 ). Multiplying (13.11) by

Q we immediately obtain (13.10) since

w(x)

The proof of the following assertions, which are simple generalizations

I.1 = I.(Q) > 1 W

E W(Q)

00

1;;; p

<

00

•

Let

[or S.1 = S.(Q) 1

and suppose that

Q be a domain in <

00

]

for some

<

00

R

]

•

fl~~.(X)IP w2(x~)

J1U(X)!P w(x) dx;;; C Q Q

holds for every Let

1

Then the inequality

of Lemma 13.2, are left to the reader. 13.3. Lemma.

00

Ii

0

.

<

Si

1

(we have used Holder's inequality if

Q

1

Si

111

depends only on

1

[or

dt

1

and integrating over

x = (x: ,x.) E

,

J w (x.)(x. - I.)p-1 dx. ;;; C 1

1 111 Ii

we have Si

w(x)

dx

1

suppose that

Xi

(13.11)

x~ e Pi (Q)

1

1

and define

1

Then the inequality

x . . Then the inequality

E C~(Q)

u

u E

P I

D.

R

dx

1

00

u 6 Co(Q)

N such that

i E {1,2, ... ,N} . Let

13.5. Remark.

In Lemma 13.3, the assumption

u E C;(Q)

can be obviously

weakened: it suffices to suppose that

u E AC. L(rl) 1,

188

189

~'="''''"~=.:-=-~-=-'--~====-'='''-=-:---=--'--:-~~--''----~=~---=:::::-:---~-~~-~---------==--~-

u E AC.l, R(~)

[or

J. j'" 1,2, ... ,N .

13.6. Examples. for every u E

(i)

Let

ACI,L(~)

N= 2

and

~

J)au ~(x) ~ I

~ I

C(a,p)

fe °

2 ax i

~

2

~

for every u E ACI,L(~)

x

IP e ax~

j

dx

2

~

dx ~ C(a,p) Jr!au ~(x) ~ I

a

< 0 .

Then we have

where

C(a,p)

Je

at2

ax~

J1u(x)I

e

<

dx

holds for every

C(a,p)

J [ I~~

~

C. v. (F(y)) , J

j

= 1,2, ... ,N •

J

~ cPo j=1 I II~(X) I aX j

P v.(x) dx J

u = u(x)

on

~ such that

2

aX2

+ j au aX

I

(x)

IP

u (F(y)) E

C~(Q) with

u

J

u(F(y)) E C~(Q) . The transformation of

be such that yields

coordinates x = F(y)

2

(13.19)

e aX I ] dx

2

w(F(y))

\DF(y)! dy ,

Q

is independent of

Yi '

w(F(y)) = ~(Yi) ,

we have in view of (13.15)

u E ACI,L(~)r: AC2,L(~) .

In Lemma 13.2 we have dealt with a weight

f1u(x)IP w(x) dx w

independent of

following assertion extends the corresponding result to functions

~

xi' The

~

C f1u(F(y))IP

i.~l, ~

J j..

"

.¥

.

-'.,. •. •: .

:.

~(Yi)

dF(Yi) dy.

Q

Using Lemma 13.2 with the weight function

w

independent of some curvilinear coordinate. 190

Yi

~

I

such that

~

Let

dt. Consequently,

~

(x) P e

R+

P [Di(Q)]P p-I C

C = N - max C.

pC. J

since

1.

-+

c, C ,

-the constant

~

- 2

d F : Pi(Q)

f1u(x)!P w(x) dx = flu(F(y))\? w(F(y)) dx

be

~ CdF(yi) ,

Ir(y) aF. Ip ~

f1u(X) IP w(x) dx

° alxl 2

I

°

-I

~

l

~

tP

VjE'W(~),

w,

,

inequality

IP e ax~ dx

P dx ~ C(a,p) Jlau ~(x) I e ~ 2

N

y=(y~,y.)EQ,

every function

the inequality P

and a measurable function

'" 1,2, ... ,N , l

(0,00),

R

00

and

e alxl

u E AC2,L(~)

(0,00)

be domains in

Q

be a regular one-to-one mapping of Q onto

cdF(yi) ;;; IDF(y)

and

~ ,

Let

•

D (Q) < for some i E {1,2, ... ,N} and let w(F(y)) i of Yi . Suppose that there exist positive constants

p-I dx .

xI I

Jlu(x) IP e alxl

for every

F

w(F(y))

Let N = 2

f lu(x)I P

Let

00

D(FI,···,F) D = N

F D(YI""'YN)

~

Jlu(x) Ip ea\xl 2 dx ~ C(a,p)

I;;; p <

with the Jacobian

a E R • Then

I Jjau dx ~ P ~(X) jP e aX2 dx,

~

(ii)

(0,=) ,

x

we have

f lu(x) IP e aX2

where

(0,1)

Let

~(y~) dF(Y~) l l

we obtain from

formula (13.10) that

191

(13.20)

P f1u(X) IP w(x) dx "C[Di(Q)]P flau(F(y))I w(yi) dF(Yi) dy. p ay i It Q

assumptions concerning the special structure of the weight functions. We

l

Let

Lemma.

Since obviously au(F(y)) IP ~ aYi

we obtain (13.17) with

NP-

Co

(13.15) and (13.16).

1

N

ax. (F(y)) L au i=l l J

IP laF. ayJ(y)

,

B .(x~) L ,J l

i

from (13.18) in view of (13.19), (13.20),

,here €

D

It

and

w

<

ItCR For

x E It

2

N =

2

and let

in such a way that Lemma 13.3 can

\ {(x ,0); xl 1

It

be such that

~ O} ,

(0,0) E alt •

l

J

w, v E W(r2).

,

l

l

l

00

•

the inequality

for every =

AC.

u E

l,L

~ C~ fl~~. (x) I r2

vex) dx

l

with

(r2)

p 1 /p (p,)l/ P'C

P

.

i

dx

~

(Z1T)P Zp-1 P

It

jt fl~~/X)IP

u E AC. L(It) . Fubini's theorem, the one-dimensional Hardy in­

Let

(Y1'Y2)'

x 2 = Y1 sin yz ' we obtain from Theorem 13.7 that the in­

f !u(x) IP w(x)

l,

and

condition

w(x) Ixl

P

P,(r2)

It

l

f [~ P.(It) J

1

S J lu(x) IP Ixl -P dx r

~

(ZIT)p Zp-1

It which holds for every

we obtain the inequality

BE R (and

P

.I

JI~~. (x) I H

J=l It

P

lu(x~,xi) I P w(x~,x.)

f

l

B dx

J

u E C~(It) ).

The following lemma is an application of the one-dimensional Hardy

l

~ f {~ P.(r2)

dX i ] dx'i

l

lu(x~,x.)IP l l

f

l

C(r2;x~)

b j (xi)

O

B p Ixl -

(cf. Theorem 1.14) yield

f [

=

dx

u E C (r2) . (Cf. Example 1Z.8.) w(x)

(13.Z3)

f1u(x)IPw(x) dx

r2

In particular, for (13.Z1)

N

l

flu(x) IP w(x) dx It

equality

holds for every

R

;(Ixl)

w E W(O,oo) . Introducing the polar coordinates

xl = Y1 cos Y2'

J

l

sup B .(x~) < L ,J l j

ess sup

x~EP.(r2) l

Co =

.. b e a d oma1.n 1.n

{1,2, ... ,N}

put w(x)

with

Let

It

let

,

BL(a.(x~), b.(x~), w(x~,·), v(x~,·), p, p)

=

be used. 13.9. Example.

00

B is given by the formula (1.18), and suppose that for some

L

Obviously, we can extend the assertion of Theorem 13.7 by

weakening the assumptions on

1 ;;; P

from (13.5).

P

I

C. = 13.8. Remark.

J. (x~) l J

will use the intervals

w(x; , x .) dx. ] dx; l

l

l

dX i }

dX~ ~

l

~

a (x~) j

[ Pl/p( P ,)l/p

,

B . (x;) ] L,J l

P

J

l

b

j

«) P

Ia;z-(x~,x.) au I v(x~,xi)

. J a. (x') J

i l l

i

inequality. We proceed similarly as in Section 12, but now without any

19Z

193

-~

I

f [

cP0

1

f I~(X) Ip VeX) dxi

oD

13.11. Remark.

1

1

1

C(D;x~)

Pi(rl)

cP

I~uXi (X',X.)!P v(x~,x.) dx .

dX i ]

basis of

1

(i)

w , v

from Lemma 13.10 have the

2,

N

C~

w(x~)

w(x.) 1

1

;(x.) w(x~) 1

1

E AC 2 ,L(D)

u

x~ t: P. (D)

,

1

x.

E(L,S.) .

u

111

1

E AC

2,LR

if

(Q)

C

Then obviously

=

. (x ~) L ,J 1

L (a.J (x 1~) ,b.J (x 1~) ,':;,;, p, p J

= B

~

a

if

(0,1)

D

I~~

I

(x) p e

x

aX2

B.1, L

<

00

;;; BL (1.1 , S.1 ,;,;, p, p) = B.1, L

implies (13.23). Consequently, we can formulate

(0,00) ,

a E

~

a

>

. Then the

dx

2

a < 0 , for

u

E AC 2 ,R(D)

if

0 , and for

0 . Here we have

~

A comparison with Example 13.6 (i), where and the condition

~ (xi)

Ia I

to 03.25).

o

B

f Q

Q

vex)

(13.26)

Let

P eaX2 dx ;;;

f1u(x)I

special form w(x)

G

from (6.2) or

B

R An analogous remark can be made concerning Corollary 13.12.

o

Assume that the weights

or

BR,J.(x~) 1 from (8.69).

from (13.22) by the corresponding numbers dx~

au/ax1

appears, leads to

the inequality

the following corollary of Lemma 13.10. flu(x)

Let

13.12. Corollary.

Suppose that for some

1;;; p <

ro,

let

E {l,2, ... ,N}

i

Q

be a domain in

N

R ,

w, vEW(Q).

; , ; : (I.,S.) 1

-+

~+

rl

and

1

1

(13.27)

aX2

dx ;;;

I

P au 1 J ---(x) ;;; -e aX 2 dx + -1 aX l 2p 2 rl

1

u E C~(rl)

for

w(x) ;;; K ':;(x.) w(x~)

e

D

there exist positive constants

k , K and positive measurable functions + W : P.(Q) -+ R such that

IP

with

nJP (-Llal

P au fl. ---(x) I dx D

e aX2 dx

2

a ~ 0 .

1

vex) ~ k ;(x.) w(x~) 1

Theorem 13.7 has extended Lemma 13.2 to the case of curvilinear

1

coordinates. Similarly, we can formulate the 'curvilinear extension' of If B. L = B (I . , S . , 1, 1 L 1

then

the inequality C

o

13.13. Remark. on the class

=

(~J lip k

03.24) P

Corollary 13.12 (and Remark 13.13). The proof, which is obvious, is left <

ro

to the reader.

holds for every

uEAc·L(n) 1,

with

P

i,L

1,

AC.

(Q)(') AC. R(n) l,L 1,

,

if

u E AC. R(Q) or 1, we replace the numbers

p <

ro.

Let

rl ,Q

be domains in ~N

,

= 1,2, ... ,N. Let F be a regular one-to-one mapping of Q onto rl with the Jacobian D . Suppose that for some F iE {1,2, ... ,N} there eX'ist positive constants K, c , C, C. , j J = l,2, ... ,N , and positive measurable functions w, ~F : Pi(rl) -+ ~+

W,

AC.1, L(D) . Obviously, we can deal with =

Let 1 ;;;

13.15. Theorem.

lip ( ') lip' B

In Lemma 13.10 we have dealt with the inequality (13.24)

even with uEAC. LR(n)

194

w, v, p, p)

and

v

j

€

W(Q),

~,~, d

j

F

:

(Ii(Q),Si(Q») -+~+

such that for

y = (yi'Yi) EQ

195

__- _c__ ~

~

~ __c

_~ __ ,_~-__

c-

w(F(y») ~

--

K

~(y.) w(y~) l

l

__ ~ __ c_~ ~

~c : : : : -

__: :

-__ ~

~_ ,~~-:~~~=~~~,~ _~o~ ~~ _~::~_

~

c~__ ._~

~_~~; •.,-'---.-----

--

I

Yj

P

I

,. _ •••..••.• -.,_ ••

- -- "'----'-'--"---'--'"

.. -- - _._

--'

'~"' .."'."--"-.",--".." " " " ' ' ' ' ' ' ' ' ' ' ' '

the inequalities (13.21) and (13.30), we arrive at the inequality

;;;(y.) w(y~) , l l aF -a-(y)

,

flu(x) IP IxlS-p dx ;;;

~ C.V.(F(y») J J

Q

cdF(y ) ~F(Yi) ~ IDF(Y) I ~ CdF(y i ) ~F(Yi) i

~

Zp-1

-

. {( 211 ) P ( --,

mln

p

P

Is

)P }

+ 2 -

pi

•

Let

oj.

=

l

dl(l.(Q), s.(Q), wd F , vd F , p, p) l l

<

Then the inequality (13.17) holds for every function that u(F(y») E C~(Q) with C

o

=

p1/ p (pl) 1/ p l

D.l

Jil'lau ~(x) I

co

= u(x)

u

on

1

j

(13.28) Denoting by

i3

(0, R, ;;;(t)t, ~(t)t, p, p)

y = (Y1'YZ)

, ~~.",.

Lemma.

Let

1;;; p

w(x)

o

j=l

Q

l

<

G a domain in

13.15 that the inequality

I lu(x) IP

JI~(X)IP ~(IxI)

<

co

AC. L(Q) . The next lemma shows that,for l,

and let

C > 0

l

Q be the cylinder

l

R N- 1 . Let

Ilu(x)I P w(x) dx;;; Q

dx

w, v£ W(Q) . Then there exists a finite

such that the inequality

cP

II~~.(x)IP Q

v(x) dx

l

dx j

Q

every function

u ~

holds with Co = 2(p-1)/p p lip ( p 1) 1/p1

(13.29)

SE R .

Q= {(x~,x.); x~E G, x. EO (a,b)}

the polar coordinates, we obtain from Theorem

PI dx ~ c

dx

Suppose

l

=

s

some special domains, this condition is also necessary.

be the domain from Example 13.9. Denote

13.16. Example. Let QC R ;;; E W(O,R) R = sup Ixl and put again w(x) = ;;;(Ixl) with xE Q that there exists a function ~ E W(O,R) such that

63

Ixl

The condition (13.23) was sufficient for the validity of the Hardy in­

J

(13.Z4) on the class 2

+ lau dx (x) IPJ 2

and for every

u (-=: CO(Q)

(KN P- 1 f max c.)l/p . c

n

such

Q

P

6

if and only if

ess sup BL(a,b,w(x~,.),v(x~,.),p,p) <

C.

x~EG

l

•

AC. (Q) l,L

l

00

l

l

In particular, for

w(x)

Ixl 8- p

with

8 ~ p - 2

we obtain the in-

the best possible constant

C in (13.32) satisfies the

equality (13.30)

Ilu(x) IP Ix\8-p dx

;;; cP

o

j=l

Q

where

~(t)

=

6

Co is given by (13.Z9) with t 8 . We can easily obtain Co :;; 2(p-1)/p

+ Z -

Q

Ixl 8 dx

i

l

dx j

from (13.28) where

;;; (t)

t

8-p ,

For our special domain and consequently

Q we have

P.(Q) = G, l

C.(Q;x~) = (a,b) l

l

the number

pi

197

196

,

C from (13.33) coincides with the number Ci i from (13.23). Therefore, according to Lemma 13.10 the condition (13.33) is

sufficient for (13.32) to be valid.

p

18

C ;;; C ;;; p1/p (pl)l/ p l c.

P

I2 Iia~(x) I

Suppose now that there exists a number every

u

such that (13.32) holds for

C

E AC.l , L(Q) . Assume that for such a function

u , the integral

then x~ ~ l

on the right-hand side of (13.32) is finite, and rewrite this inequality

f

F

is measurable on

I [c

(13.34)

p

J

[F(X~)rl/p l

=

x~

E M.

(x~) l

x~

E G

X M j

P v(xi,x i ) dX

for a.e.

<

v EO

W(Q)

since

J

l

l

F(xi)

for

00

and

f 0 . Put

¢(x~) l

fl~~. (xi,x i ) I

F(x~) > 0

M. . Moreover,

in the form b

and,by virtue of (13.36),

G

u(x~,x.) = ¢(x~)

i

l

l

f(x.) ,

l

l

(x~,x.)EQ.

x

l

l

l

Gal

Then

b

I IU(X~,x.)IP w(x ~ , x .) l

l

l

l

dX.] dx l~ l

u E AC. L(Q)

~ o.

JI~~. (x)

a

Q

We will show that (13.34) implies that the inequality b

(13.35)

v(x~,x.) II

P dx.l - f1f(X.)I l

a ~

f

w(x~,x.) II

~

dx.l

b

0

[c

f

x~ l

there exist a function > 0 ,

f E ACL(a,b)

f E ACL(a,b) ,

and a.e.

f f 0 , and a set

x ~ E G , then l

such that

Hf(x~) l

f [c

=

J

(13.36)

x~ EM.

p

If'(x.) IP l

v(x~,x.) II

-

If(x.) IP l

Consequently, there exist a number

l

o

<

m _ (M ) N 1

Hf(xi)

<

j

< -

1 j

-

next

two

00

,

such that

for every

-I f

!F(xi)

1- 1

dx ~ l

<

0 ,

M.

J

and 13.17 for the case

w(X~'x.)] II

x~ l

E M.

J

j

E

:N

Let

o

~

~

P

=

q . The

q <

P

<

P

00

,

00

q <

<

r

1

1

q

p'

let

Q

N . . b e a d oma~n ~n R ,

w, v E W(Q) . Denote

dx.l

< 0

and a set

(13.37)

~ (a.(x~),b.(x~),w(x~,.),v(x~,.),q,p)

A . (x ~) L, J l

--L

J

J

l

l

l

l

A is given by the fopmula (1.19), and suppose that fop some L iE {1,2, ... ,N}

whepe

(13.38)

Ci

J [ I.

= {

P. (Q)

,

AP . (x ~) L,]

]r/ P '

l

dx ~

}l/r

<

00

•

l

J

l

If we denote F (x ~) l

b

Then the inequality

Ilf'(x i ) IP v(xi,x i ) dX i

(13.39)

a

198

dX.] dx'.l = l

lemmas are the analogues of the assertions from Lemmas 13.10

13.18. Lemma.

a

M. eM,

II

MeG,

b

for every

l

II

a

In the foregoing Subsection we have dealt with the case

If (13.35) does not hold for every

N 1

l

P

flu(x~,x.) I w(x~,x.)

the condition (13.33) is satisfied.

The proof of the validity of (13.35) will proceed by contradiction.

m _ (M)

dx.l -

II

which contradicts (13.34).

l

E G , and consequently

l

•

b

JI~~. (xi,x i ) I v(x~,x.)

G l

00

P

l

BL(a,b,w(x~,.),v(x~,.),p,p) ~ C x~

p

J I¢(x~) ( Hf(x~) dx~ ~

E G . According to Theorem

1.14 we then have

for a.e.

m _ (M ) < N 1 j

vex) dx

Gal

and for a.e.

ACL(a,b)

P

Further,

a

holds for every

I

l

b

P cP ,rlf'(x.)I l

and

l,

l/q [flu(x)\q w(x) dx ] r2

,; - C0

P au [fl ax. (x) I vex) Q

dx ]l/P

l

199

~~~8~jf;;~~~~;;£t~~;~~~:0~i~~1Wi~~~~~1~~~~~~~~~~

holds for every C =

o

Proof.

Let

with

u E AC.1, 1(Q) q

l/q ( I)l/ql P

C

J [I.

q1/ q (pl)l/ql [

~

1, J

(x~)J (q/p') (p/(p-q»)

I] (p-q)/(pq) dx.

1

J

P.(Q)

i

API.

1

1

b j (x~)

1,

I

equality (see Theorem 1.15), the inequality

c~ ~ ( J

j

I

c.)q , Holder's in­

J a.(x~) J 1

Pi(I?)

J

j

lip

[ J [~ f I~~:~(X)IP

u EAC. L(Q) . Fubini's theorem, the one-dimensional Hardy in­

equalities for sums and for integrals and the condition (13.38) yield

[f1~~.(X)IP

Co

Q

vex) dX i ]

dx~

=

J

1

lip vex) dx

1

.

J

o

[Jlu(x)[q w(x) dXJl/q

Q

b j (x~)

{ f

[ L. Jr

P. (Q) 1

~

{ J [ Pi(lI)

J a.(x~) J 1

'v"-~ Lemma.

lu(x~,x.)lq 1 1

1/ q w(x~,x.) 1

1

dx. ] dx~ } 1 1

1)l/qIA

j

=

{

I'

1

1

1

l/P]q,

J dx:

1

}l/

1

i . Let

1

be such a domain

Q

C(Q;x~)

the cut x~ ~

for mN 1-a.e. -

1

Co

>

P.(Q) 1

•

consists Let

0 such that the

~n-

if and only if

u E AC. 1(Q) 1,

l/r dX~ ]

<

00

1

S

~ (x~) ~L 1

q ClU IP v (x) dX Jl/P]q dX~ }l/ ~ k(x) i I

. [~

~L

~L

1

1

1

(~) l/q'

1

1

.

Co in (13.39) satisfies the

C ~ Co ~ q1/ q (pl)l/ql C i

i

J q/p'

For our special domain, the number 'the number

J a. (x!) J 1

C.

1

from (13.40) coincides with

C.

1

from (13.38) and, therefore, the condition (13.40) is suffi­

cient according to 1emma 13.18. q P [k(x) IP vex) dX i J / dX~ }l/q ~

Suppose now that there exists a finite number

ClU

J

ql/q

(13.42) i

1)

,J

A (x~;w,v) = A (a(x~),b(x~),w(x~,.),v(x~,·),q,p)

1

f [~Ai'· (x: J

=

Moreover, the best possible constant

J

bj(xi)

every

u

Co

such that (13.39)

E AC.1, L(Q) .

1

(i) (13.43)

Assume in addition that

J w(x) Q

200

-

P. (lI)

(13.41)

j

Pi(Q)

v(x:,x.) dx.)

[ f A~ « )

C.

q

A .(x . ) . 1,J 1 '

a (x~)

{

IP

[I

[ J ql

~

=

E {1,2, ... ,N}

i

1

~

1

b j (x~)

ql/q(pl)l/

00,

1

~(x~,x.) ox. 1 1

P i'(lI)

~

<

equality (13.39) holds for every

.(x~)

L,J

Jf11Cl

a j ( Xi)

ql

P

(a(x~),b(x~»)

of only one interval

(13.40)

I

l/q( p l)l/ q

<

that for some (fixed)

~

b. (x ~)

<

~ q

1

w, v E W(lI) • Then there exists a finite number

'I [ l/q( L q p

• (

Let

dx

<

00,

f v 1- p' (x)

dx

<

00

•

Q

201

Let us fix

x~l E P.(~) l

real numbers such that

a

n

* a(x~)

(x~) l

,

l

Then there exist non-negative functions b(xi)

(13.44)

{an (x~)}, l

and choose two sequences

f la x. a(xi)

agn(X~,x·)IP v(x~,x.) l l l l

(x~)

b

n

g

n

t

l

E AC.

l,L

b(x~)

{b n (x~)} l

for

l

n

~

of

[fl

00

rl

such that

(~)

{ f <,k(X~)

l

a (x ~)

1::~(X~'Xi)IP

l/P v(x~,x.) l l

dx~ l

dx. ] l

}

l

l

[ f A~,k(x~) dx~J l/p

b(xi)

gq(x~ ,x.) l n l

[ f

[ J

Pi (rl)

and

(13.45)

l

b (x~)

dx.

l

p l/p au ·a::k(X)[ V(X)dX) =

l/ q w(x~,x.) l l

~

dx. ) l

q

l/q (~)

l/q' A

r

n

Pi (rl)

(x~) l

a(xi)

[!IUn,k(X)!q w(x) dXJ 1/q

where b (xi)

bn (xi) Ar(x~) =

n

[

I

l

an (xi)

J

w(x~,t)

dt

r/

[

1 I v-p(x~,t)dt

r

I-p'

(x~,xi)

{Qk}

b(xi)

{ f Pi (rl)

dX i

~-l(Qk) <

Arq/p (x~) n,k l

l/q

~

[f

gq (x ,x. ) w(x~,x.) l l n l l

dx~

dx. ] l

l

}

~

a(x~) l

1/ q

f

(U) l/q/[ r

Arq/p (x~) Aq(x~) dx~ n l l J n,k l

~

P. (rl) R N- 1

be a sequence of domains in

QkC Qk+1 C Pi(~) ,

~ ql/q

13.48)

(cf. Lemma 3.11, formulas (3.34), (3.35)). Further, let

ql •

an (x:f)

Xi v

r/

Xi

q

l

such that ~ q1/ q

co

l/q'

(¥)

A~,k (xP dX~

[f

) l/q

.

P. (rl)

and

l

U

Q

kE:N

k

= P.

(~)

u

Using now the inequality (13.39) for the function

l

n,

E

k

AC.1, L(~) , we

in view of (13.47), (13.48) that For

n, k E:N

(13.46)

define

u n, k(x~,x.) l l

' ,x. ) Aripk ( x.') g (x.

n,

l

n

l

l

,

x

=

q

(x~,x.)E~, l

l/q

(¥)

l/q'

l

[f

A~,k(X~) dX~

J l/q

P. (rl)

P. (rl)

l

l

where

f A~,k(x~) dxd1/P

o[

;;; C

since A

n,

k (x ~) l

{ min {An(x~),k}

o

for

xi E Qk '

for

x E Pi(rl) \ Qk .

The formulas (13.44), (13.45) and (13.46) yield

o

f A~,k(x~) dX~

<

<

k

r

mN_ 1 (Qk) <

co

,

S

Co .

P . (rl) l

we have ql/q

(~) l!q'

[ J

Ar (x~) n,k l

dX~]l/r l

Pi (~) 203

202

,

-

The monotone convergence theorem, applied first for n -+

00

,

k -+

00

and then for

~,-

(14.2)

yields

(13.49)

(P~q) 1/q'

q1/q

[ I A~ (x~;w,v)

dX~ ]

~

(14.3)

from (13.41) for

w, v

Ilu

_'"' ,<.,

__

"W

~~~

(au oX

grad u

-~-,

and

oX 2

w, v

IL'lg(x) I '

w(x)

__

"-""'-'<""""'-'==~.,"_,

) ... , au oX -~-

N

are chosen according to the formulas

vex)

2

Illg(x)

1

lL'lg(x)

I

satisfying (13.43).

be general functions from

w ,v

Let now

au

-~-,

l

with an appropriate function (ii)

._._.~~~""='===~

Co

1

~(x~;w,v)

; , _.... _.

if the weight functions

1/r

P. (Q)

with

--

W(Q)

, and for

xEQ

N

n E:N put

L

L'lg

i=l vn(x)

vex) + ~ (1 + Ixl (N+1)/(p'-1»)

wn(x)

min (w(x),n/lx!N+1,n) .

2 a-2 g aX i

g. (Recall that

.)

The following assertion is a direct (N-dimensional) extension of the result mentioned in Theorem 4.1 (but here for

Obviously,

wand v n n fulfil the additional assumptions (13.43) and consequently, the inequality (13.49) implies that

q1/q

(P~q) l/q'

[

f

A~(X~;Wn,vn)

Let w,v1, ...

1/r dx 1~ ]

~

-

Co .

(14.4)

p.(Q) 1- ,

wn (x) t w(x) and v n p (x) t v convergence theorem yields for n -+

1- ,

P (x)

for a.e.

x E Q, the monotone

I

(13.49) is the first inequality in (13.42).

holds for every

P

w(x) dx

c~ (Q)

u E

r, u

(14.5)

~

I

i=1

R

N

,

let

Jla~ (x) jP a i

vi(x) dx

Q

if there exists a solution

a [v. I-?a Ip-1

N L -0i=1 Xi

14. THE APPROACH VIA DIFFERENTIAL EQUATIONS AND FORMULAS

(14.6)

y(x)

t-

Xi

1

such that for a.e.

14.1. Introduction.

be a bounded domain in

Q

• Then the (Hardy) inequality

,VNE Wen)

J!u(x)

Let

•

y

of the (partiaL)

differentiaL equation

that (13.49) holds for all

00

w, v E W(Q) • However,

00

Q

1

Since

1 < P <

p = q ).

E

x

0

sgn

a] f-Xi

+ w !yIP- 1 sgn y = 0

in

Q

and ~(x) I 0 , dxi

i

1,2, ... ,N.

In this section we will extend to the N-dimensional

case the approach described in Subsections 2.2 and 2.4. Some attempts in

14.2. Remark.

this direction can be found e.g. in the paper by D. C. BENSON [1J, where

general Theorem 14.4. The precise formulation needs some assumptions

the special case

concerning the domain

p = q = 2

has been considered. For the same values, R. T.

LEWIS [lJ, [2J has shown that the inequality (14.1)

f

lu(x) I

2

w(x) dx ~ 4

204

u

J Ilu (x) I

r,

Q

holds for

E

C~(Q)

Q

2 1

v (x) dx

The assertion just formulated is a corollary of the more Q and the introduction of some function spaces.

This will be done in Subsection 14.3. Note that the equation (14.5) can be viewed as the Euler-Lagrange equation of the functional

with 205

J(y)

=

J[J1 1~IP vi

-

lylP

W]

dx

a domain

Q in

Wl,p(Q) . The domain

(i)

Q.

For

1::; p ::;

00

and

vex)

v

=

LP(Q)

{ [jlu(xJ[P

(ii)

(14.9)

llull p Q ,

=

Q ess sup xEQ

on

u

dX(P

for

Iu (x) I

for

1 ::; p <

p

=

ro

,

~uxi

Let 1

14.4. Theorem.

where

Q

ro

y

C n

= y(x)

Q

n+1

C

aX i

belong to

such that their distributional LP(Q) , i = 1,2, ..• ,N . The space W1 ,p(Q)

Ilu I1 1 ,p,rI

(1Iuli:,Q + i t

exist a.e. in

be a domain in

Q

R

N

Q = lim Q n+ oo

w,v " " ' v E W(Q) 1 N

n

n

and

fulfil the following conditions:

a

1

~'

'

1

ax.

1

ay

[Vi Ik l

p-1

ay ] sgn h~

1

1

,

i

1,2, ... ,N ,

1

Q

is a solution of the differential equation

y

(14.5) and satisfies the conditions (14.6).

Il~~.rP, rI )l/p

Further, let

In the sequel, we will use some properties of functions from

u

= u(x)

be such that

lul P v. y. E wl,l(Q),

(iii)

1

n

1

i

=

1,2, ... ,N,

nE" N ,

and

Sobolev spaces; in particular, the existence of the trace

lim sup

(iV)

n -+

J lulP

ro

;W

ul aQ

[,I

1=1

.J

Vi Yi v nl

dS

~

°

n

u

E

1 W ,p(rI)

assumptions about the domain

on the boundary

aQ. This notion needs some

where

Q. We will not go into details here; let us

1~IP-1

only note that we will use domains of the class (14.11)

Let

•

O Q E C ,l . Let

av,

The function

(ii)

1

of a function

rI

00

The derivatives

ay

u E LP(Q)

< P <

and

rI

defined on

(i)

is called the Sobolev space and is normed by

(iii)

x E aQ

m _ -a.e. N 1

with the finite norm

W1 ,p(Q)

the set of all functions

(14.10)

aQ

(Chapters II - V, VII), or A. KUFNER, 0. JOHN, S. FucfK [lJ (Chapters 5, 6).J Q

Further, denote by

derivatives

E

[For details, see, e.g., J. NECAS [lJ (Chapters 1. 2), R. A. ADAMS [lJ

the set of all measurable functions

(14.8)

x

(v ,v '· .. ,v N ) 1 2

=

R N , denote by

is well defined for (14.7)

Q at the point

the outer normal to

( S)

14.3. The Sobolev space

u

(we will write simply

Q

u E W1 ,p (Q) is meaningful u I aQ for instead of u IaQ in the sequel),

the notion of the trace

(a)

C O ,l

which are, roughly speaking, bounded domains whose boundary can be locally described by functions satisfying the Lipschitz condition. A precise defint"tion of this class will be given in Section 19. Here we will use the fact that for this class of domains Green's formula can be used, since

1

(14.12)

and

v

3y

sgn k

Yi

.

nl

,

i

outer normal to

1

\y\p-1 sgn y 1,2, ... ,N , al'e the components of the unit vector of the Q

n

Then the inequaUty (14.4) holds for

u

satisfying (iii), (iv). 207

206

--_._Proof.

If

u

~~= .......=-~~~ £~~=~~~~~

--;.:a7~ ~

is such that

fI~'I ax.

N

= i=1 I

J (u)

rl

~~,

P

/uI P- 1

a:-; [ lul P vi

=

sgn u vi Yi

P

~

1

P Vi dx

Yi] - lul

a:-:-(v i y i ) 1

the inequality (14.14) implies

1

P I laX i

ay

is infinite, then the inequality (14.4) holds trivially. Therefore, assume that

J(u)

N

<

00

i=1

The following well-known inequality

1 < p <

o

00

N

(14.13) for

s. , 1

~

s,t

<

00

and consequently

,

L (I s.1 p +

(p - 1) It. Ip - pis.

1

1

i=1

t.ER,

S.

au

-v

ax.1

1

lip

t.

i

u

1

y

[I u \

P Vi Y

1

a

a

1

-a-(v,y.) X, 1

1

1

a

aX i

~L

aX i

lylP

ax i

1

[[Vi

I~IP-l sgn~] 1

[vil~~.-IP-l

L

i=1

~~ .1 p

[ 1

~IP

P vi + (p - 1) jul

-

and consequently

L i=1

p

I~~J

p l lul -

P

[Iau ~I

vi + (p - 1) lujP

1

+(1-p)V

ly~P-l

[yiP

I I

1 u P - sgn u

1

from (14.12) holds a. e. in

xi

1

- w + (1 - p)

1

~

Vi]

P

L lau i=1 aX

0

I i

vi - lul

P

I

i=1

lylP

-­ 1

v.

lylP

1

N

w ~ i~1 a~i

Integrating this inequality over

rl

n

(lul

P

vi Yi)

and then applying Green's formula to

the right-hand side (which is possible by virtue of the assumption (iii»,

P

,

we have

vi v. y. ] 1

1

> =

I

[iLI~~JP

Vi -

lul

P

w] dx ~

Q n

0

I arl

lul

P

N

(i=LV. 1

1

y. v .) dS . 1

nl

n

Due to the assumption (iv), we finally obtain the inequality (14.4) from

(14.16) letting

rl •

n

-+

00

LJ

"

Since

;\,J

i"';"'j"­

j

...... J\." < \,.

/""

208

1

i

I~~.IP

N

a

-a-(v.y,)

N

I

l ~

I~~ .I

P

This formula together with (14.15) yields

(14.16)

auX. - P --a-Yi

1'=1

I ~~ .1 p-l

ay

sgn y ]]

1

\

1

with

L

v. ­

P

is a solution of the equation (14.5), it follows that

y

N

the inequality

N

(14.14)

lylP

1

1

1

\yIP-l sgn y and since

[ Iyj l-

ay ] sgn ~

1

l/p vi

I ~

1

i] .

then we have

N

~(v.Y.)] ~ xi

+ lul P

vi

Further, we obviously have

It. Ip- ) ~ 0

1

a~,

i=1

1,2, ... ,N. If we put

i

1

1

we have

lul P vi + (p - 1)

1

I

~

p 1

sP + (p - l)t P - pst - ~ 0 holds for

L [I~~.IP

(14.1 5)

,

!, .

,­

:~

14.5. Remark.

(1)

If the domain

rl

belongs to CO, 1 , then we can choose 209

n

n

n ~ ~ . Nonetheless, sometimes this choice need not be the best

= n for

one. For instance, the assumption

where

1 lul P v. y. E W ,1(n) 1

u

imposes conditions on the function w , vi ' while the assumption (iii),

Me n

small set taking

u

~(M)

such that

nn

or

w, v. =

M

by

M = 0 . Analogously, we can sometimes eliminate N n by taking n = n (J {x E R ; Ixl < n} . See also n

the unboundedness of

(14.18)

00

Let

•

be a domain in R

n

and n ECO,l. Let n

w, vEw(n)

N

and

n = lim n n-+oo

n

y = y(x)

The derivatives

, av

ax. 1

,

_a_[v ax. 1

II7YIP-2~], ax.

i

1,2, ... ,N

1

n , where

exist a.e. in

n ~I n

1 < P <

n fulfil the following conditions:

~ ax. 1

which exhibit 'bad behaviour' on some

° . Indeed, we can eliminate the set 1

(i)

as well as on the weight functions 1 jul P vi Yi E W ,1(nn) , allows to

Let

n en en n n+1

defined on

1

consider functions

14.6. Theorem.

IVy I =

[.

I

I

~_12] 1/ 2

1=1 ax.1

Example 14.13 and Remark 14.14. (ii)

If

u

E

(ii)

c~(n) , then the condition (iv) of Theorem 14.4 is

fulfilled automatically and the assertion from the end of Subsection 14.1 (14.19)

follows immediately. If

u

I L I~ ax.

P

N

i=l

n

E C O,l , and if we suppose that

1

-

av vi vi sgn ~

1

1

y

u

E coo(n)

provided there exists a solution

If we consider the inequality (14.4) on the class c:(n) = {u E COO (n); supp u (J M =

y

condition (14.17), but this time only on

o in n

+ w lylP-2 y

1

u = u(x)

(iii)

lul P v y. E

Civ)

lim sup

1

be such that

w1 ,1(nn )

i

1,2, ... ,N ,

=

nEJII,

and .n -+

co

J lulP v[ 1=. I1y. v .J 1

an

nl

dS

~

0

n

where Ivy1P-2 ~

0}

hi

an , then we can show that the inequality (14.4) is

satisfied if there exists a solution

f;~]

Ivy1P-2

an,

condition (14.17) is a non-linear Neumann-type condition.]

Me

is a solution of the differential equation

and satisfies the conditions (14.6).

of the boundary value problem (14.5), (14.17). [Note that the boundary

for some set

y

1

Further, let

° on

then again the condition (iv) is fulfilled and we can assert that the Hardy

inequality (14.4) holds for

a:.

.LN 'lv 1= 1

E Coo(n),

(14.17)

The function

of (14.5) satisfying the boundary an \ M .

Naturally, the condition (iii) is supposed to be satisfied.

(14.20)

Yi

Then for (14.21)

lylP-2 y u

satisfying (iii), (iv) the following inequality holds:

JIU(x) IP w(x) dx

n

~

P 1 N -

.I

JI~~. (x) I

1=1 n

P

vex) dx .

1

The result contained in Theorem 14.4 is due to B. OPIC, A. KUFNER [lJ, [2]; for

p

=

2 , see also A. KUFNER, B. OPIC

is due to V. P. STECYUK [lJ

[1~.

A modification which

will be described in the following theorem;

see also A. KUFNER, B. OPIe, I. V. SKRYPNIK, V. P. STECYUK [lJ.

210

Proof. Suppose that the right-hand side in (14.21) is finite (otherwise (14.21) holds trivially). Analogously as in the proof of Theorem 14.4, we start from the inequality (14.13) where we put

211

kaU

Nl/ P '

S. 1

v

lip

~

N- 1 /P u

t.

,

1

1

vl/p •

y

[vylP ly[P

- w + (I - p) - - v

~

We obtain

N

I~~ .1 P

P 1 [N -

L

i=1

v

and this formula together with (14.23) yields

+ ~N 1

Iu IP ! Vy Ip v -~

_ P I

~~ ./ /uIp-l

NP- 1

Iy Ip-l

- 0

(14.22)

P

L i=1

lau ~I

~

N

v +

1

au - P ax.

Yi

from (14.20) holds a.e. in

Iu Ip

w

~

a (I u IP I a---

N

i=1

v

Xi

y.) . 1

, Green's formula and the application of the

n

[J

lul P Ivy/P v _

[yiP

14.7. The approach via formulas.

~

jujP-l sgn u v Yi]

The foregoing theorems have shown that

the investigation of the Hardy inequality is closely connected with the

0

1

with

~

14.4 (integration over

the inequality

P 1

[N -

v -

The rest of the proof repeats the arguments used in the proof of Theorem

condition (iv». N

P

L la~ I

N

i=1 dX i

v] ~

Ivy Ip-l

1

and consequently

­

lylP

1

solution of a certain boundary value problem. Now we will prove a theorem which is due to B. OPIC [IJ

Q.

and represents an analogue of the approach

described in Subsection 2.4. Before formulating the general result, let Since

us illustrate it on a certain simpler case.

a

[u IP-1 sgn u v y.

au p -a-

X.

ax.1

1

1

(lul

P

lul P

v y.) 1

~ oX.

(v y.) 1

,

lip ;.)

the inequality (14.22) implies (14.23)

P- 1

N

I I~IP v +

(14.24)

lulP IVylP lylP

N

L lui

i=1

Using (14.20) and the fact that we have

N

L

i= 1

Xi

y

.I a:.

a

-a~(vy.) 1

1=1

v +

paN -a--(vy.) ~ L

holds fO!' every

a

a- (lul P v y.) . 1 Xi

(14.25)

w(x)

is a solution of the equation (14.19),

(14.26)

v.(x)

Xi

[rv/vy p-2

1

1

i=1

~ [v I i

Vy

IY IP

p-2

I

2 Y

~] ax . 1

+

N

L (1

IiIheY'e

p ~~.J [/y/l1

i=1

- p) Iy I- P

au I [fl ~(x)

IP

;;;) 1=1

u t:' C~(Q)

Q

Vi (x) dx

]l!P

1

if the i.Jeight j'unctiorts

w,v , ... ,v are given N 1

by the fOl'Trtlilas

1

g

div g(x) , P [div g(x)]I- p ~ p P Ig.(x)I 1

(gl' g2' ... , g:-;r)

sgn y ]]

:-;r

(14.27)

N aX L i=1

[f lu(x)IPw(x)dx ] Q

(p _ 1)

i=1 aX i

+

It can be shown that the Hardy inequality

1

(~ 2 I dX

)

div g(x)

L

i=1

I

Vy P

-2

v

i

1,2, ... ,]\ ,

is an approjJriate vector function such that ago

~(x)

>

Xi

° a. e.

in

n.

The formula (14.26) can be rewritten in the form

i

(14.28)

. P' !-p' dlVg-p v. 1

p' Ig.[ 1

=0,

.

1=1,2, ... ,N,

and the formulas (14.25) - (14.28) can be exploited in two ways: 212

213

(i) function

v ,v , ... ,v are given, then the weight 1 2 N

for which (14.24) should hold can be determined by solving

w = 6G ,

If we suppose that w

(14.29)

functions . (ii)

g. ) and using (14.25).

~

w

is given, then we have to solve the equation

div g - w =

~G(x)

[flu(x)I P

(14.30)

° v.

~

by (14.26).

n = (a,b) , write

N= 1

N

~ p

gl

g ,

vI

= v and assume

I

i=l [Il n

in addition that

1-p'

(t) dt <

ro

for every

14.9. Theorem.

= __1__

1 - p

(p')-p

v -p [XII'

au IP laG \P ax. (x) I ax. (x) ~

(~G(x»)l-p

,lip

dxJI

~

x E (a,b)

then the function g(x)

~

u ~ c~(n)

and holds for every

x

Ia v

dXr/P

Q

and then determine the weights If we take

~

and the inequality (14.24) assumes the form

~

If

I~r (~G)l-p ax.

v. = pP

the system of non-linear differential equations (14.28) (for the unknown

nen 1CQ, n n+

P dt )l-

(t)

Let

n

1 < p < E C O ,l

Let

00

Q

.. b e a d oma'Z-n 'Z-n

N

,

n = lim n-+oo

Let the functions gi' i

n

R

=

Q

n

1,2, ... ,N, satisfy

. E w1,l(Q )

(14.31)

g~

a

n

and is a solution of the ordinary differential equation (14.28). Moreover, g'(x) >

°

for

x E (a,b)

and consequently

the function

w

from (14.25)

is given by

div g(x)

(14.32)

where

g

>

°

( g 1 ' g 2 ' ••• , gN )

for a.e.

x E n

•

x

w(x)

(p')-p vI-pi (x)

g' (x)

[I

vI -p' (t) dt J-p .

(i)

Let us define the weight functions

w,v 1 ,···,v N by the formulas

(14.25), (14.26). Then the Hardy inequality (14.24) holds for every

a

u

= u(x)

defined on

Q such that

This formula coincides (except for a multiplicative constant) with the formula (2.6), and thus

the approach just described is a natural extension

of that in the one-dimensional case.

u E coo(~)

(14.33)

n

g

from (14.25) is connected with the function

A

(14.34)

I

lim inf n -+ 00

ann

from (2.11) by the formula

function Let a function

x E

g

214

Q

and put

=

grad G

G

= G(x)

be such that

in (14.25), (14.26). Then

~G(x) >

°

for

(14.35)

lul

P l(

I

g.V

.J

i=l ~ n~

dS

~

°

Let us define the weight function

(ii)

A=(1-p')lng

14.8. Example.

n E~

proJ)1:ded

The formulas (14.25), (14.26) are also extensions of the formulas (2.11): the function

for every

v

w by (14.25) and the weight

by vex)

I

P 1 v 1/ (p-1) i ­ 'j=l j (x)J

r

215

with

v.

J

(14.36)

fr'om (14.26). Then the Hardy inequality

j

flu(x) I,P w(x) dx ;;; ~'

(14.39)

P

~-1

hoZds for all

fl~~. (x) I

Denote

~

v(x) dx

Q

~

Without loss of generality we can suppose that

satisfying (14.33), (14.34).

u

J(Q ) n

Proof.

Again suppose that the right-hand side in (14.24) or (14.36) is

finite (otherwise the corresponding Hardy inequality holds trivially).

0

J

lul

I'i=1I g.Vnl .)

P

~

a~

n

N fl P Iu I P-

if 1

~

a~

lujP

Z gi v .) [i=1 n~

dS

sufficiently large. Moreover, by virtue of the assumption

for

u

and letting

n

00

n

[J(Qn)Jl P

satisfying (14.33). Dividing (14.38) by -+

00

a~, I

la

~=1

au

k

sgn u

~

[gil

~

gi dx ;;;

,I

~=1

~

f

I

+

i=1 ~

P

I u I p-l

J

~ I ax,

N

!gi l dx .

I

~

n

f

P lujP-l

i=1 Q

Holder's inequality we have

1~~.llgil ~

~

rI

;;;.

~

lu)P divg dx

Q n

f

Ig.1 dx = ~

f

p 1 lu/ -

Q n

/P '

J

(div g)l/p'

~

v~/p ~

f !u[p-l (diVg)l!P'[J ~=1

IP i

Vi dx

) lip

,

~

+

(14.38)

~

aQ

n

+

[f Q

216

lul P

lul

P

~

dx

1~~,IP)I/P vl/ p

dx .

~

lul P divg n

P

[JlgiVni) dS +

n

dx ]

I/P'[N I i=1

f Q

P

(au I --a-Xi

v

dx

)I/ P

n

'~=1

From this inequality we derive the inequality (14.36) by the same arguments l/p'

N

I

i=1 n

[f Q

rJgiVni) dS +

n

div g dX)

f lul a~

Q

which together with (14.37) yields

f

]

Now we estimate the last integral by Holder's inequality and obtain from

f lul P divg dx

n

lul P divg dx;;;

gil

n

dx;;;

n

f

I

(14.37) the following analogue of (14.38):

rr Jau aX ~

I~I x, I

II

p 1 r au , dx = f P lul l i =1 ax i Q n

Q

I

=

Using also the formula (14.35), we have

Using the formula (14.26) and estimating the last integral by

I~I ax,

IP]I/P [N,I Igil P ,]I/P' l=1

a~.

[N fa

N

-1 . l!p'r I v.I/(P-O]l!P' = [ N1auJP)l/P 2. - P (d~v g) i~11 ax i 'i=1 ~

n

n

r lulp-1 J P , Q n

in view of (14.34) we obtain the desired

,

Holder's inequality and the formula (14.26) yield

n

(i)

and, consequently,

~

N

f

;;;

<

.IN

dS ­

n

(14.37)

n

I'

(ii) dx

for J(Q)

0

>

satisfying (14.31),

gi ' u

04.33) that

~

>

(14.31),

J(Q)

inequality (14.24).

Using Green's formula, we obtain for

J lul P divg

r J lu(x) IP div g(x) dx .

J(Q)

If, II~IP ax.

/P Vi dXr

as we have derived (14.24) at the end of part (i).

.

o

~

'l

n

14.10. Remarks.

(i)

Let

Q be a domain in RN and denote by 217

(14.40)

c 1 (Q)

(14.44) u = u(x)

the set of functions on

which are bounded and uniformly continuous

Q together with their first derivatives

ou/ox

'

i

i

=

1,2, ... ,N .

1

uE C (ri)

for every

n

J(~

n

)

(d.

(14.39»

Ix

y(x)

(14.45)

neE;

this last assumption together with (14.31) again guarantees that Green's formula can be used and that

It can be shown that the solution

- X Ia o

is finite. o

~

f

lim sup n

By the same arguments we can show that Theorem 14.9 holds if the

g.

~

1 ­ C (~ ) ,

E

14.11. Example.

E W1 ,p (~ )

u

n

for every

n

->-

00

oQ

~~i (x)

I

lu(x) IP sgn a

1

(14.42)

for

n

Ix - xolE-P la[p-l sgn a •

n (x, - x ~

,) v . O~ n~

N

I

i=1

fl

o

o~i (x)

N

I

i=1

(x, - x .) v ,(x) ~ 0 o~ ~ n~

Q

1 [N lOG I P /(P-l»)P-l (lIG(x») -p j~1 oX (x) dx . j

N lOG [ i~1 oX

p = 2 , then

on

oQ

n

N

~

IP

dS .

will certainly hold if

h(x,xO,Q) = n

I

i=1

(x. - x o .) v ,(x) ~ ~ n~

then obviously the sign of this function for

~ pP

(X)]

sufficiently large. If we denote

(14.47)

~

If we set

P

/(p-l)r-

and instead of (14.30) we obtain the inequality (14.36), i.e. Ilu(x) IP lIG(x) dx

lu(x)I

This rather complicated condition

(14.46) 1

of the differential equation (14.5)

p

i=1

14.8. Using the formula (14.35) we have vex) = pP (lIG(x») I-p [Jl

,N •

a=I_~N

with

.[ I

nEE.

Let us consider the weight functions (14.29) from Example

P

1,2, ...

i

and the condition (iv) reads

pair of assumptions (14.31) and (14.33) is replaced by (14.41)

y

2-p

has the form

Obviously, the assumption (14.33) can be weakened to

(ii)

Ix. - K O ' j ] vi(x) = Ix - X IE [ ~ ~ o Ix - xol

IP /(P-l»)P-l

= I~GI

j (14.42) is exactly the inequality (14.1) with

2

and the inequality

wand

v

x

E oQ n will be important

since, for instance, if

sgn a

for

h(x,xO'~n) < 0

x

Ere o~

n n

then the condition (14.46) will be satisfied provided

u(x) = 0

given by (14.3).

for

x E

r n

Therefore, let us introduce some special sets which will be exploited in 14.12. Some applications of Theorems 14.4, 14.6, 14.9.

Let us check the

important condition (iv) of Theorem 14.4 for some special weight functions. For

1 < P <

(14.43)

218

00,

w(x)

X o

[ IE

N

(x l' x ' ... , x ) E Rand O 02 ON

=

-

~

E

E:

R,

Ec;/.

the following examples: For

N

I)

Ix

_

X

o

I E-p

,

R

N

GEe O,1

P - N , put

P

+

GC

(14.48)

x

o

ERN

denote

oG+(x ) O

{x

oG-(x ) O

{x E oG; h(x,xO,G) < O}

E oG; h(X'XO,G) > O} ,

219

[Of course,

h(x,xO,G)

is defined by (14.47) where

the i-th component of the outer normal to

v nl

. is replaced by

[aB(X O'

G.J

*

i'lJ C

)(1

aQ:(x O)

we obtain that 14.13. Example. (14.49)

Let

1 < P <

QECO,l

00

fIU(x) IP Ix - xolC-P dx

X

o

E Q . Then the inequality

a\1 (x ) O

a~:(xo)

(14.56)

$

+

a~~(xo)

·(14.55)

=

­Q

n

n

[aQ-(x ) I} rinJ U [aB(x O' n ) Ii riJ O

Q

<

[

=

P

Ic _ p + NI

'~1

JPN

flau ax.(x) ~

1

IP

1

.1

C [Ix.1 -x Ix - xol

1X

-

X

0, J

o

(for the notation see (14.48».

2 p -

(14.50) (ii)

supp u Ii {xO} E >

u

=

is such that

°

aQ - (x ) O

O

= f/J •

Now, let us go back to Example 14.13. In the case (i) we have (cf. (14.45», i.e.

sgn a.

~

1. If we put

=

n

=

Q for every

the condition (iv) of Theorem 14.4 will be satisfied if on

u E W1,p(~ )

a~-(x)

o

n ER , is such that

n

aQ+(x ) . o

=

Here

~

\ B (x ' lin) O

(14.52)

wl,p(Q)

+ (x ) = aQ,

is strictly convex then

on

B(xO,r)

=

(14.58)

°

on X

o

aQ-(x )' O 1= supp u.

°

u =

a. <

° ,i.e.

sgn a.

= - 1,

the condition (iii) of Theorem 14.4 will be satisfied if

aQ~(xo)

on

a~~(xO)~ aQ+(x O) , and (14.51)

However, according to (14.55) we have

{x ERN; Ix - xol < r}

=

°

So, we obtain the conditions (14.50).

and consequently

where

u

a. >

n E R , then

while the condition (iii) of Theorem 14.4 will be satisfied if

In the case (ii) of Example 14.13 we have

°

u

n

E f/J,

P - Nand

(14.51) Q

u

C < P - Nand

a~

(14.57)

holds provided one of the following two conditions is satisfied: (i)

~

Moreover, if the domain

dx

implies (14.58). The condition (iii) of Theorem 14.4 is satisfied 14.14. Remarks.

(i)

The conditions 14.13 (i) and/or 14.13 (ii) guarantee

that the assumptions of Theorem 14.4 are fulfilled. Before we show it let us

~

~ \ B(x O'

n

C < P - N,

n1 )

E >

B(x ,l/n) the ball from (14.52) for r = l/n , we can easily verify O (draw a picture !) that for X E Q and n sufficiently large, o

u

Eo:

WI ,p (~) ,

a~ n

=

[aQ Ii rin ]

u

[aB(x ' 1 O n

)

n riJ

.

(ii)

X

u E W1 ,p(Q ) ,

P - N

n

with

(14.54)

Q is strictly convex, then the

conditions (i), (ii) from Example 14.13 are simpler:

Taking

\1

nn =

X

If we suppose in addition that

insert some geometrical considerations.

(14.53)

~ Q . On the other hand, if we took n

o as in the case (i), the condition (iii) could be violated.

automatically due to the fact that

o

~ supp u ,

u

=

°

on

aQ

For p = 2 , the inequality (14.49) assumes the form

J U(X)!2 1

Ix ­

X

E-2

oI

dx

$

Q

Suppose

Q E CO,l • Since obviously

4 $

220

IE -

2 + NI

i~l JI~~i (x)

2 1

C Ix - xol

dx .

~

221

-...

-----_._-----

"""""""'~

~

........::-.~-

This inequality is proved e.g. in R. T. LEWIS [lJ under a little stronger assumption

on

~

[flu(x) IP Ix - xOI E- P dX)lIP

(14.61)

u.

Q

(iii)

~ 2 we have

- xOil/ix - xOIJ2-P i the inequality (14.49) implies

For

consequently

1

<

P

Jlu(X) IP Ix - XOIE-P dx

(14.59)

(Ix

~ 1 and

~

~

-

[

p

!E-p+NI

)P iL N

IE -

[f I-~(x) a IP

Ix - x

Q aX i

0

I I p lip [ x.1 - x o 1· ) dx Ix - xol J

l

IE

and

Q <

N L P + NI i=l

p

(14.62)

II ~~/x)llx-xolcdX'

~

f1u(X) jP !x - xOIE-P dx Q

p

~

Q

Compare this inequality with the classical Hardy inequality (0.2) (N

1)

P

r, _ P ,C 0 +

NIJ

fl~~ i (x) /P

I

i=l

Ix - XOI

E

-

P •

~

and with the inequalities (13.30), (13.31) (N = 2).

. r

I

'j=l

Ix. - x J

OJ

I P/ (P-O)P-1

dx

The foregoing inequalities have been derived by using Theorem 14.4. again hold if one of the conditions (i), (ii) from Example 14.13 is

Now we will use Theorems 14.6 and 14.9.

satisfied. 14.15. Example.

Let

1 < p < 00,

J1U(X) IP Ix - xolC-P dx

(14.60)

DEC

O

,l

X

o

E

~

. Then the inequality

(14.36) from Theorem 14.9 (see also Remark 14.10) where we set g.(x) = sgn (E - P + N) Ix - X IE-P x., 1. = 1,2, ... ,N

$

o

1

D $

The inequalities (14.61), (14.62) are the inequalities (14.24) and

NP- 1 r

,Ic -

p 0

N )P i~1 + NI

f/axi(x) au ,p

Since C

Ix - xol

dx

(Ix i - xoil/ix - xolJP

~

04.63)

[flu(x)

IP

Ix _

X

oIC-P

N

[Il

P $

of the differential equation (14.19) is again given by the formula

-

- IE -

I

P + Nj i=l

~

(14.45).

(14.59). In (14.60)

1

< p ~

functions

vi

)l/P

~

au /P ~(x) Ix - xOI

E

dx

)l/ P ,

1

which is an inequality of the type (12.9). Using the estimate

N

the constant is worse, but in (14.49) we have a little

more complicated weight

dx

Q

This result can be derived by using Theorem 14.6 where the solution

Compare the inequality (14.60) with the inequalities (14.49) and

1 , we obtain from (14.61) the in­

$

equality

holds if one of the conditions (i), (ii) from Example 14.13 is satisfied.

y

1

P

[ Ia.) i=l 1

while (14.59) was derived only for

N

~NP-1 Ia~ i=l

1

a

i

~

0 ,

2 . we obtain the inequality (14.60) directly from (14.63). The same inequality

14.16. Example. equalities

Let

1

<

P

<

00,

[2ECO,1,

xOEQ. Then the in­

- x I ~ Ix - xOI j Oj we can derive the inequality (14.59) again from

can be derived from (14.62) [using the fact that while for

1

<

P

$

2

IX

J,

223

.._.- ---

Ij Ix.J - xOJ Ipl :;; ( Ij Ix.J - xOJ . 12 ) P

(14.62) [using the fact, that

I

/2

-"-"

1~.19. N = 1

Example.

inequalities reduce to the classical Hardy inequality (0.2). Consequently,

(i)

one can expect also that the inequalities mentioned remain true for more general domains (unbounded, not belonging to

will say that

if there exists a sequence

o

E ~ . We

n

<

°, such that

rn +

Q be unbounded,

X

=

(14.64)

Q

(iii)

n

Q

n

°

R

Y

<

. Let

Q

n

be

°

u =

n

(lQ+(x ) , o

on

N

UEWl,P(Q)

P - N- 1 ,

h -

C(S,y,p,N)

n

p + N + 1I

°

Ix-x o I Ix - x ly - p +1

u

'

=

°

on

(lQ-(x ) O

II

Y

Ip-1

dx :;;

Q

o

E

R

N

•

We will say that

Q

belongs to

N

I fl;~. (x) \P

:;; C

i=1 Q n

N

Xo E R

•

uEW 1 ,P(Q) •

P - N - 1 ,

>

flu (x) \p /

v

{ R

C~~~(XO,oo)

Q£

following inequalities hold:

r 0,1

lim (x ,00) O if there exist two sequences

~-::_"-~,='=:;;-~~'--';::."-;;:';;"--;;;.,;;';:.; ... ..;;:.;'.:;;,",;' .. ;;;.;;-.~;;._~,:_,;'__~;;;;;;;;..:;._._;'..;:;;;;:_,_-:5

•

Further, denote

C

{r n} ,

••__

1=: supp u

o

Q \ B(xO,r ) ECO,1 n

Let

(ii)

X

X

Q belongs to

"O,1( ) v lim X o

Q

RN ,

1 < P < 00,

Y

0,

>

S

(ii)

Q be a bounded domain in

Let

is compact in

nd (i) Let

S

CO,1 ). This is really the

case for certain classes of domains.

14.17. Definition.

-:~~.~,." _

c_," -

(14.64) and let one of the following two conditions be satisfied:

Q = (0,00) , then all the foregoing

and

.: : .~ __~"..,._ _.. _:---._

for

12].

If we take formally

..

':~: --,=:=~=:"~::-;-:

} ,

{r n } ,

R

n

too,

rn +

[B(xO,R ) \ B(xO,r ) ] ECO,1 n n

slx-xol

Ix -

X

o

IY Ix . i

- x . 01

12- p

dx ,

1

° , such that lu(x) \P e

f

e

s!x-xOI

Ix - xolY-P

+1

dx;£

Q

;£ NP- 1 C

For both types of domains we define

N

()u

I

I Ia;z:-(x) i=1 f Q I

p

I

e

S Ix-x o I

Ix

- X

o

Iy-p+2

dx .

1

()Q±(x )

O

14.18. Remark.

lim nc>oo

[(lQ~(xO) n

C~~~(xO)

Tllese inequalities follow from Theorems 14.4, 14.6 where the solution y alx-xOI of the corresponding differential equation is the function y(x) = e

The inequalities derived in Examples 14.13, 14.15, 14.16

remain true if the assumption QE

()Q] •

. If we consider

Q E CO,1 QE

C~~~(xo,oo)

, we only have to add in

(14.51) the assumption supp u and

Q

n

a = 6/(1 - p) .

with

is replaced by the assumption 14.20. Example.

1 < P < 00,

QE

C~~~(xO·oo)

one of the following two conditions be satisfied:

. . RN 1S compact 1n

has to be given according to (14.64).

Let

(i)

in

R

N

a

<

°,

uEWl,P(Q) n

u

=

° on

,

(ii)

a > 0 ,

uEWl,P(Q) , n

u

=

° on

,

x ERN

°

+

(lQ (x O) ,

(lQ-(x ) , O

'

supp u

X

o

¢.

N

>

2 . Let

is compact

supp u .

Finally, let us present two examples with a little different weight Then the following inequality holds: 224

225

-":'-"0

• -

._-:._ ..... _.....

•• •

('

lJ

==::_:==:o_..:::.-=-----==_ _ ~==__::_='_______'_'__'_

_ ~ ._•.",~..

_.-

c..:'--_'-----'--'-------------'-----=--_-=-~-=----=------c--=-- ..::::__ '--_.----'__

P a I x-x O1 !u(x) I e

2-N Ix - xoI2(1-N) dx

J1/ P

.• _..-:

.o:~,

I

1

;;

~(Q)

rl

w(x) dx ]

m...(Q)

N

P

I I

2)

I

i=l

rJ ,

2 N

'I~

IP ax. (x)

rl

e

alx-x 1 -

IX

0

l

IX i

- xoil

1

-

P dx

X O'

p (N-2)+2 (1-N)

every cube

,

Y ERN

with

Q = Q(y,R)

11/ r'

v 1- r ' (x) dXJ

;; k

Q"rl

N

Q" rl

<-~ a (N -

J

l/r [_1_.

R > 0 .

In this section we will show that the (Hardy) inequality

l/p

r

q (J !u(x) I w(x) dx

J

This inequality follows from Theorem 14.9 (and Remark 14.10) where we have 2-N a I x-x O1 -N . set g.(x) = - a e Ix - xol (x.l - x l .) , l = 1,2, ... ,N . l O

]l/q

Iiaa~i (x) I P vex)

N ;; C [ i~l

P

Q

Q

for all r

dx ]l/

u

1 E CO(Q)

such that

1

<

where r

p

<

<

Q is a fixed cube, provided (w,v) E Ar (Q) Nr , l/q = l/p - l/(Nr) . The proof uses

estimates for Riesz potentials and maximal operators. Therefore, 15. THE HARDY INEQUALITY AND THE CLASS 15.1. The Muckenhoupt classes.

Let

some definitions and auxiliary assertions.

A r

1

< r

<

w E W(RN). B. MUCKENHOUPT

co

[2] introduced a class of weights denoted by A

=

r

(1)

rl

For p;;:; 1 ,

a domain in

set of all measurable functions

UIf

is now commonly called the Muckenhoupt class. It is defined as the set of such that

sup [m)Q) Q

f w(x)

Il/

(

dXJ

Q

Jwl-r' (x)

1

~(Q) Q

1 r

dx J / '

<

00

,

N Q = Q(y,R) = {xE R ; Ix. - y.1 l

the open cube with centre at r

<

R,

Since to

f = f(x)

11 / P

dXJ

for

on

rl

~

such that the

P

<

00

,

y, we will say that

If (x) I

for

p =

00

we obtain the classical (non-weighted) Lebesgue

introduced in Subsection 14.3 (i). f

belongs to

LP(rl)

if and only if the product

LP(rl;w)

fw 1 / p

and l P

i = 1,2, ... ,N}

II f p,,,,w n = Ilfw / II p,"n 1,1

it follows immediately that

LP(Q;w)

is a Banach space with respect to the

norm (15.6). (Recall Convention 5.1!)

(rl)

if there exists a number

226

l

rl

w ~ 1

is finite. For LP(rl)

A

(x) Ip w(x)

ess sup xErl

Q~ R N with sides parallel to the

This definition can be extended to pairs of weights w, v E W(rl) , N being a domain in R . Denoting for y E RN , R > 0 by

(w,v) E

W(rl) ,

I f Ii p , rl , w

coordinate axes.

(15.2)

w E

rl

r

the supremum being taken over all cubes

(15.1)

,

LP(Q;w)

equalities (cf., e.g., J. GARCfA-CUERVA, J. L. RUBIO DE FRANCIA [1]) and w

N

the weighted Lebesgue space

N A (R ) r

This class plays a very important role in the theory of weighted norm in­

all weights

R

k,

0 < k <

co

,

such that

(ii)

For

f

measurable on

Q

and

a;;:; 0 , denote

227

(15.8)

E(f,o)

E

{x

~.4. Theorem

I f (x) I > a}

t: Q;

(the Marcinkiewicz interpolation theorem; cf., e.g.,

A. ZYGMUND [lJ, Theorem 4.6; J. BERGH, J. LOFSTROM [lJ, Theorem 1.3.1).

and

(15.9)

f

W(E(f,o))

weE)

P ,P ,QO,ql O 1

w(y) dy .

Po ~ PI . Let

E [1,00) ,

be a sublinear operator

T

that

E(f,a)

I T f II ~:

L P ,

(15.10)

every

* (n;w) f = f(x)

as the set of all measurable functions

I f I ; , Q, w = sup

(15.11)

[w (E (f ,

(J

11 P

(J) ) ]

<

S c.

'qi,Q,w

Define the weak Lebesgue space (Marcinkiewicz space)

on

Pi f E L (,,;v) , 1 P

Q such that

1-

1 - 8 Po

II f II Pi' Q , v

i

that

(0,1)

put

1 1 - 0 0 -=--+­

8 PI

-=--+-,

00

oE

0,1 . FoY'

=

q

ql

qo

P Sq. Then

a>O

(iii)

For

f

measurable on

the maximal function

Mf r

(15.12)

j

(If) (x) =

RN

(Mf) (x)

(15.13)

sup

R

N

define the Riesz potential

If

by 1

f (y)

I

N 1 dy ,

x E R

N

For

f

1-8 8 c c S c 8 Co 1

,

mN~Q)

flf(y) I dy ,

x

N

ER

defined on and

Proof.

Q = Q(z,R) ,

z

E R

N

Holder's inequality with exponents 1)

l)/(r -

,

R > 0 ,

~~Q)

J I

Mf = Mf

J

for

x E Q ,

0

for

x E: R

N

(p - l)/(p - r)

and

vI-p' (x) dx

Qr:Q

N

nCR we put

{ f (x)

P E (r,oo)

for every

yields

](p-r) I (p-l)

_1

LmN (Q)

[

QnQ =

P

r

,

where _ f(x)

(w,v) E A (Q)

(w,v) E A (Q) • Then

Let

15.5. Lemma.

x

If = If

and

I x - yl ­

the supremum being taken over all cubes

(iv)

I f I p''''V n

S c

f E LP(Q;v)

Q

containing the point

111£ I q, ..n ,w

and

S [m)Q)

\ Q

J

(mN~Q))

(r-l) I (p-l)

-11 (p-l) (x)] dx

S

y

yl-r ' (x) dxJ (r-l)/(p-l)

QnQ 15.3. Theorem

(B. MUCKENHOUPT [2J, Theorem 8; J. GARCfA-CUERVA, J. L. RUBIO

DE FRANCIA [lJ, Chap.IV, Theorem 1.12). Let

there exists a constant

K > 0

such that

1 < P < 00,

w, v E Wen) • Then

cube

[~~Q) J w(x)

f

E LP(Q;v) (w,v) E

228

if and only if

A (n) p

~

<

~Q) J

dx J [m

Q.'1Q

IMfl*p, .. n ,w S K If I p, ..n ,v

for all

Q. Using this estimate in (15.3), we obtain

s

Qf)Q

N

[1 f w(x) ' l' mN(Q) Qf)" mN(Q) dxJ

VI-p' (x) dX]P-l

_1

J

v

l-r'

(x) dx

iJr - 1

S k

r

,

Qn"

229

.,

(w,v) E A (ll)

Le.

[the last estimate carr be derived using spherical coordinates;

o

P

measure of the unit sphere The proof of our main result is based on the following assertion about the continuity of the Riesz potential.

c

is the

S(O,l) = {x ERN; Ixl = I} ]. From (15.17),

and (15.18) we have (ICf)(x) ;;;

Let

15.6. Lemma. N

zER,

R>O

(15.14)

w(Q)

1 < r < p <

Let

J

1=

Nr ,

q

(w,v) E A (Q)

1 _ ~ . p Nr

Let

Q

with

Q(z,R)

;;; (~ p _ r ] (p-r) Nr - p

and denote

r

(I f)(x)

order to estimate

Then there exists a finite constant

C > 0

such that the estimate

Proof.

For

C > 0

and

x

1/

(Nr) Ilfll

p,Q,v

C

(k

B(x,c)

n+1

;;; (Mf) (x)

is the ball from (14.52), and for

(I f) (x) =

J Q

C

Ix

(lCf) (x)

f E LP(Q;v)

-

y

I

(15.21)

QC

J

J

n

N

n

I

(k

n+1

1

[c1-Nr/p

l'

[J v -r

(y) dy

p/(p - r)

~f~p,Q,v + c(Mf)(x)

]

p, The estimate (15.21) holds for every

implies

(ICf)(x) ;;; ~f~p,Q,v

1-r'

[J

v

(y) dy

J (r-1) Ip

(J c (x))

(p-r)

JC(x)

J QC

I

x -

I (l-N)pl (p-r) d <:; - ~ y Y - c Nr _ p

C > 0 . Evaluating the infimum of C > 0 , we obtain

Ip (15.22)

(If) (x)

;;; k [(Mf) (x)] 1-p/(Nr) IlfliP/(Nr) 'p,Q,v 2

[J

v

1-r'

(y) dy

) l/(Nr')

Q

where

230

)(r-1)/P

_ [ - p - r (p-r) I p Nl

k 1 - max (c Nr _ p) , 4 J

If = Icf +. I f ,

Q

(15.18)

eCMf) (x)

with

C

the right-hand side in (15.21) over all (15.17)

4

) N-1

Q

and Holder's inequality for three functions with the exponents

p/(r - 1) ,

Then

(If) (x) ;;; ;;; k

Then (15.16)

n = 0,1,2, ...

(15.19) and (15.20), we have from (15.16) that

If(Y)1 N-1 dy . Ix - y I

J

c2- n

n

(2k )N

put

If(y) I dy , N-1

C

~f~p,Q,v

(2k )N If(y) I dy ;;; n Q(x,k)

) N-1

n=O where

k

1

n

n=O

QC=Q\Q

C

(r-1)/p

J

(2k )N

;; I

E RN denote

Q =B(x,c)(JQ,

1 r v - '(y) dy )

[I

If(y)1 <:; If (y) I dy;;; I N 1 ­ N-1 n=O B(x,k )(k + ) n=O Ix - yl n 1 n Qk n \ Qk n +1

I

(I f) (x)

f E LP(Q;v) .

holds for every

1-Nr/p

put

C

C

IIIfllq,Q,w;;; CR[w(Q)r

C

Q

w(y) dy .

Q

(15.15)

Ip

where

C

(p-Nr) I (p-r)

k

= k

Z

Nr (Nr

1 p

)

pi (Nr)-l

P - 1

231

___

==-_7~~=~=~~~~~~~~_·_--O==~~=='"=~_·_='=---==---=-'=-'---~~ ._=_~~_

Since

:=--=--,-==::__

--_.._..

:_----=.:::.====_~~~

_._._--,-_._._---_._~~===.~=-

.::__-=='===:::-==-=-'_=-_

. . _. __ ._=".-=="'==

==.=-_-==~====-~=.---"=-"'=""'~=""'_==c:-=-="==-=-"'__~:_=_=~_,,__==___"'=:=~___,"'''C

£ Ar(Q) , we obtain from (15.3)

(w,v)

-

- -_._._-_ .. '

---.. --_. -

and consequently

Jl/(Nr')

r 1 r, ;;: kl/ N [m(Q)] l/N [w(Q)rl/(Nr) [ J v - (y) dy

Lemma.

Q =

- -_._- .-

w(Q)

with

k

3

from (15.14), and the last estimate together with (15.22)

Let

=

Let

every

(If) (x) ;;: k R [w(Q)rl/(Nr) [(Mf) (x)] 1-p/(Nr) IlfIIP/(Nr)

3 p,Q,v

1 N 2k / k 2

r 1 E (p,oo)

be fixed. Since

1;:; p <

;;:

for every

(15.24)

K >

°

p

=

r1

R

N

1~IP]l/P

E C~(r2)

and

x

E

r2 . Then there exists a ball

supp u C B(x,R) . Consequently, u(y) = 0 N \" {ZER ; Iz - xl = R) and from the formula

° [.I f

1=1

(Q;v)

we conclude that there exists a

~~

(x +

t

Y -

x )

Iy - xl

i

for

B(x,R)

y E S(x,R)

such

=

Yi-Xi]dt Iy - xl

R

I u (x) I for

f

E

N [au

'" If, i=I 1 -a-(x xi

°

LP(Q;v) ,

R

pi ;:; Nl/

which implies that

(II (Mf) (x) I(l-p/(Nr»)q w(x)

l/q dx

Q

J

I [1.1

' )l/q [ JI(Mf)(X)/P w(x) dx

Iu(x) I

l/p'

IYi -xii P']l/P'

I ( /Y i= 1

x

I )

dt '"

R

1 p

to) PJl/P dt = N / ' f1vu(x + to)1 P dt (x + \

°

Q

substitution dx

]l/ q

•

5(0,1) , we obtain R

c

[JIVu(x + to) Ip f 5(0,1) o

Z

x

;£ N__

IIIfliq,Q,w = [f1(If)(x)lq w(x) dxf/q '"

[J I (Mf)Cx) I (l-p/(Nr»)q w(x)

1

over the unit sphere

l/q = l/p - l/(Nr) ). Moreover, from (15.23) we have

.

~~.

IPJl/P[ N

+ to) I

o = (y - x)/Iy - xl . Integrating the last inequality with respect

Q

;:; k R [w(Q)rl/(Nr) ilfIIP/(Nr) 3 p,Q,v

11

0-

= ~MfIP/q ;;: KP / q ~f~P/q = KP/ q ~fr1-p/(Nr) p,Q,w p,Q,v 'P,Q,v

(note that

being the measure of the unit sphere in

1=1 dxi

f E Lr(Q;v) , f E L

C~ (r2)

obtain by Holder's inequality (for sums) that

G = -- - - ­ - r p r 1 such that

~Mf~ p, Q,w ;;: K~f~ p, Q,v

[.I

u(y) - u(x) for every

be a domain in R Nand u E

r2

hat

r

rIP - r

Using Theorem 15.4 for

u

c

R

KO~f~r,Q,v

IIMf I *r , Q,w ;;: KIll f II r1' Q,v 1

constant

Let

> r , Lemma 15.5 implies that 1 which together with the assumption (w,v) E A (Q) and

~Mf~;'Q,w

,let

00

o

1 p N / 'r(/vul )(x) P

c- 1

x E r2,

Ivu 1

r

(w,v) E Ar (Q) 1 with Theorem 15.3 leads to the estimates

q C = k KP / 3

inequality

kl/ N 2R[wCQ)rl/(Nr)

yields (15.23)

~~~~,_._~~~~~~~~~~

(15.15) holds with

lu(x) I '" with

=..

-_..._,_..._... _... _...._._._ .. _. .__ .....

0=-:'-".=__

+ tG

dt] dO

(with the Jacobian

t

N-1

N 1

Iz - x1 -

)

yields

Q

232

233

ju(x)j

;;; Nl/ P I

J

c

I

IVu(z)

B (x, R)

P

Nl~pl

dz

Iz -

X

I N-l

c

(iii)

I(lvu l ) (x) p

The condition

Let

N

zE R

,

R

>

1

0 . Let

r

<

<

1 =1 -

Nr,

<

(w,v) E Ar(Q)

Then thepe exists a constant (15.27)

p

c > 0

q

P

l , --N r

holds. If we compare this c ri terion, e. g., wi th the cri terion 'via solvability of differential equations' (cf. Theorems 14.4, 14.6), then

Q = Q(z,R)

with

former is relatively easier to verify in a general situation. On the

and let w(Q) be defined by (15.14). such that the inequality

rather l'estrictive, which can be illustrated by the example (see P. GURKA, A. KUFNER [IJ): If we consider the special [dist (x,aQ)r',

)l/q r [J1u(x)lqw(x)dx

represents a criterion for the

.choice of admissible weights, i.e. of such weights that the corresponding

o 15.8. Theorem.

(w,v) E Ar(Q)

:;;

shown that

vex) = [dist (x,aQ)]B,

(w,v) E Ar(Q)

a, BE R , then i t

if and only if

Q

:;; cR [W(Q)rl/(Nr)

[I

i=1

holds fop every Proof·

u

JI~(x) I Q

a > -

lip

P

vex) dx )

hi

N

E Co (Q) .

r

1,

~

a

B

Chapter 3 (see Theorem 21.5) allow a substantially bigger set of admissible

1

j Ia a~i (x) Ip

r -

instance, the results based on imbedding theorems derived in a , B described by the conditions

According to (15.26),

i~1

1. B <

Nr

a ~

III Vu Ipll ~,Q, v

vex) dx

B Nr _

Np(r - 1)

Nr - p

p

(draw a picture in the

B"p-l,

(a,B)-plane !).

.

Q

1 Let u E CO(Q)

and assume that

and Lemma 15.6 (with

f = IVul

p

IIIVul p I p, Q,v <

00

•

Lemma 15.7 (with Q

Q)

) imply

l p' IIr(jvul ) II Ilu1Iq,Q,w :;; ;:-1 N / p

q,

Q

The conditions derived in the foregoing sections which guarantee the

:;;

,101

validity of the N-dimensional Hardy inequality

p' :;; ;::-1 Nl/ CR [w(Q)rl/(Nr) !/lvulpllp,Q,v which is (15.27) with

c

=

p/ ;:-1 Nl / C .

16. SOME SPECIAL RESULTS

[J I

u (x) !q

101

Q

0

mostly sufficient. 15.9. Remarks.

(i)

l

(x) dX, l/q :;; C

The foregoing results are due to P. GURKA, A. KUFNER

[1]. This paper generalizes some results of E. FABES, C. KENIG, R. SERAPIONI

[1] and F. CHIARENZA, M. FRASCA [1], who have considered the inequality (15.4) for 101 = V •

V. G. MAZ'JA

I

P [N i~1 Iiaa~i (x) v i (x)

1/ P dx )

Q

[l]

has derived necessary and

sufficient conditions on

w,v , ... ,v under which (16.1) holds for every l N

u E C~(Q) . His conditions are expressed in terms of capacities and are

difficult to verify. In our opinion, the advantage of MAZ'JA's results lies

in the possibility of obtaining some information about the capacity of a

set, once we have derived some information about the validity of the

(ii) Using some covering lemmas, one can extend the foregoing result from cUbes to more general domains P. GURKA, B. OPIC [1].

Q

(including unbounded ones)

_

corresponding Hardy inequality by another method.

cf. On the other hand, MAZ'JA considered also inequalities of a more

complicated form, for example with right-hand sides of the type

234

235

[J [ (x;

B ~ C ~ pp(p - l)l-p B .

Vu(x))]P dXf/p

S"l (for

<1>,

.-­

16. 4 . Remark. The inequality (16.5) is a so-called isoperimet~ic inequality

see Definition 16.2), and admitted also (weighted) Orlicz norms

on the left-hand side. Here we will mention some of MAZ'JA's results for a

particular function

(16.2)

( N

l.I

[f

w(y) dy J l/q ;;;

K

~

Let us start with the definition of the capacity.

16.2. Definition.

Let

.cont~nuous . negat~ve

be a domain in

1

~

p <

(p,
N R . Let

(x;~)

be a non-

N ( ,. )xR • homogeneous of degree 1

f ' unct~on on

Let

~.

S"l

For a compact set

00

capacity of K with respect to (16.3)

KC S"l

form

IPJ lip i

without proof.

respect to

of an arbitrary compact

ith the (p,¢)-capacity. Using (16.3) and (16.2), we can rewrite (16.5)

I~

v. (x)

~:1 ~

J w(y) dy

=

K

, namely

(x;O

w(K)

S"l

f

u

E

fl~uXi (x) I

v.(x) dx ~

S"l

J

.

w,v , ... ,v under which the Hardy inequality (16.1) N 1 ~olds means to investigate (16.6) for any compact set KC S"l , which

K C S"l , the (p,
[(x; llu(x))]P dx;

L

u E M(K,S"l)}

conditions on

with

represents a rather complicated task.

is defined as the number

inf {

1 {Ii=l

B inf

lip

P

V. G. MAZ'JA

M(K,S"l)}

considered also the case

p > q . Let uS formulate a

result which illustrates the complexity of the problem.

S"l where (16.4)

M(K,S"l) = {u E C~(S"l); u

=1

(V. G. MAZ'JA [lJ

on

K} .

E W(S"l) ,

v.

E W(S"l)

continuous on

S"l,

viE W(S"l)

v.

v.

~

S"l,

constant C if and only if A <

i = 1,2, ... ,N

e

[lJ

Corollary 2.3.3.1).

continuous on

W(S"l)

a finite constant that

'J w(x)

l

Let

1 ~ p ~ q <

,

where

A

i1/q

dXJ

~ B [(p,
1/

B

for every compact set

00

•

u

EC~(S"l)

with a finite

is defined by l(P-q)/q

Q.

1 iE-l L I

~ q/ (p-q) [(p,¢)-cap (Qi,Qi+l)]

J

with

such

the supremum is taken over all sequences boundaries are

{Q.} ~

em-manifolds and which satisfy

of open sets

Qi C Qi+1

'

QiC S"l i

E

Z •

p Less complicated necessary and sufficient conditions have been derived

K

the special case

KC S"l .

Moreover, the best possible constants estimate 236

00

,

A = sup {Qi}

u E C~(S"l)

if and only if there exists a finite number

C

00

given by (16.2).

S"l,

Then the inequality (16.1) holds for every function

(16.5)

the inequality (16.1) holds for every function

from (16.2) with

P q [f w(x) dx J / (V. G. MAZ'JA

16.3. Theorem

wE W(S"l) ,

continuous on

~



1;;; q < P <

be given by (16.2). Then

¢

~

Here we will consider the particular function

Let

Theorem 2.3.5).

c

and

B are related by the

S"l = EN ,

vi ­

for

i

1, ... ,N

237

__ _

•

_zn-

"'_~

16.6. Theorem

__

n

...

='-

-

':.

w£i£W>£bW­

(v. G. MAZ'JA [IJ, Corollary of Theorem 1.4.1.2, Theorem N 1 < P < q < 00 P < N ,01" 1 = P ;;; q < co • Let w € W (R ).

1.4.2.2). Let

[J R

holds for

lu(x)

I q w(x) dxJ 1/q ;;;

[I J I~

C

IP dX. (x)

i= 1

N

RN

evel~

u E C~(RN)

function

Q

sup xER N

B

sup R1- N/ p R>O

In the case

C

>

0

Let if and

<

co.J

Q

1 < P < q <

<

co

co

,

p < N .

N

Let

w, v E W (R ) ,

. Then the inequality [Jlu(x)l

•

B(x,R)

< p < q <

1

In p- (y) d Y ]

similar way as we have derived Theorem 15.8.

R

q

l/q w(x) dx ]

;;; C

N

for every 16.7. Remark.

~

~---

---

~

1/ q

[J

----toT J

"~7-b::T.4::E

Using the inequality (16.10) and Lemma 15.7, we can prove the following

J l/p dx

with a finite constant

w(y) dy ]

(ex p

d Y ]

Q

only if (16.8)

_ _. ~-b:z .. '~:"

-

[~~Q) J p(y)

sup

Then the Hardy inequality (16.7)

..:_~

[J1 JNI~~. R

u

E C~(RN)

~

P

I

(x)

l/p

vex) dx ]

if the condition (16.11) is satisfied.

co

p < N , the proof of Theorem 16.6 is based on the estimate for the Riesz potentials Remarks. (16.9)

Ilull

N

;;; c

q,R ,w

Ilf I

(i)

Let us present a result due to K. A. DZHALILOV [IJ who

N

p,R

investigated the inequality

This analogue of (15.15) (cf. Lemma 15.6) is due to D. R. ADAMS [IJ and [Jlu(x)l

states that the inequality (16.9) holds if and only if the condition (16.8) is satisfied. Moreover, D. R. ADAMS [2J extended the result just mentioned to the case of two weights. Assuming that v E W(R N) is such that vI-p' E A (RN), co he showed that the inequality (16.10)

I IfjJ

N ;;; c q,R ,w

holds for every (16.11)

B

I fll

N

sup [Jx(R)J xER N R>O

1/

p

I

[J w(y)

Il/q < dyJ

B(x,R)

[Note that

J t(I-Np)/(p-l) [ J

R B(x,t)

PEW(RN)

1/2 vex) dx ]

C~ (16)

with

[l

R

a bounded domain in

N

, i.e. the special

(16.1) with p

=

2 ,

q > 2 ,

w

=

1,

VI = v 2 = ...

v

N

v

•

co

-1/2

[ J

v(y) dy ]

<

00

B(x,R)

v E A (RN) and the (2,~)-capacity of a single point is zero 2 N 2
provided

condition (16.14) with the condition (16.8).

co

=

class

2 1

[l

N q sup Rl+ / sup xE[l O
where Jx(R)

dX]l/q;;; C [J I \lu (x)

[l

i f and only i f

= sup

q

has shown that the inequality (16.13) holds if

p ,R ,v

f E LP(RN; v)

There are several other results in the direction

belongs to

A"" (RN)

V

1-p I (y) ] dt.

(ii)

In the one-dimensional case we substantially exploited the of the operators

if

H and H from (1.6), having shown that R L the Hardy inequ4lity (1.12) [(1.13)J holds i f and only i f HL [H R J maps the LP(a,b;v)

continuously into

Lq(a,b;w) . E. T. SAWYER [IJ extended

this approach to the two-dimensional case: he has given necessary and 238

239

sufficient conditions on the weight functions

under which the

w , v

defined as the closure of the set

operator x

y

f

r

.

is bounded from q <

wiZl always assume that the weight functions f(s,t) ds dt

J

o

~

LP((O,oo) x (0,00); v)

into

Lq((O,oo) x (0,00); w) ,

,Mostly we will consider special collections

00 . ~ p

<

00 , let

11

be a domain in

R N,

u E LP (I1;v ) such that their distributional O derivatives dU/aX.l belong to LP (I1;v.), i - 1,2, ... ,N • On this linear l 1 set we define the norm of u E W ,P(I1;S) by the formula

(16.18)

space

-

1,p,I1,S

[

Iluli

)l/ P

N Iia~ liP

P + I p,l1,vO i-I

dX i p,l1,v

i

.

I

-l/p E LP (11) vi loc

Wl,P(I1;S)

- ... - v - v . In this case the corresponding spaces (16.16) N

2 and (16.20) will be denoted by

-l/p

v.l

i

p'

E L

(K)

) , W1,p( l1;v ' v O

0,1, ... ,N for every compact set

I •III ,p , 11 , v 0 ' v

W1,p("><;vO,v )

o

K

C

11 J then the

is a Banach space.

•

Further, let us introduce the seminopm N

(16.23)

Illu

which will be

" , see (15.6) J. If, in addition, p''''V

[which means that

of weights, namely

vI - v

(16.22)

as the set of all functions

HI

S

and the norm (16.17) will be denoted by

Wl,P(I1;S)

Ilull

i.e.

) (16.21)

and define the weighted Sobolev space

(16.17)

satisfy the conditions

S - {vO,v,v, ... ,v} ,

S - {v O'v 1 , ... ,v N}

(16.16)

l

In Chapter 3 we will deal with weighted Sobolev spaces in detail.

v ,v , ... ,v E:: W(I1) . Denote N O 1 (16.15)

v.

(16.18), (16.19).

0

16.10. Weighted Sobolev spaces. Let

[for

a

is also a Banach space. Therefore, let us point out that in the sequel we

(Hf) (x,y)

1 < p

with respect to the norm (16.17)

COO (11)

111

1 ,p,l1,s

11

vi(x) dx

Illu I11 1 ,p,l1,v

[in fact,

Illu lll l,p,l1,v

JlIP

l

for the spaces from (16.21)

(16.24)

16.11. Remark.

dX. (x) IP -- [i-I I fl du

III Vu Ip I p, 11, v

denoted by

see (15.6) and (15.26)J.

The Hardy inequality (16.1) can be rewritten in terms of

the seminorm (16.23) as

The inclusion

C~(I1) C Wl,P(I1;S)

(16.19)

240

u ~ C~(I1)

[or for every

u E W1 ,p (11; S) J

is continuously imbedded in

LP (I1;w)

then

, i.e.

i - O,l, ... ,N •

Consequently, under the assumptions (16.18), (16.19), the space (16.20)

IluI1q,l1,w <: CIllu 111 1 ,p,I1,S

If (16.25) holds for every W~'P(I1;S) [or W1 ,P(I1;S) J

holds if and only if 1 v.l E Ll OC (11),

(16.25)

(16.26)

W~'P(I1;S)

c;

L

q

(l1;w)

W~,p (S2; S) 241

[or (16.27)

Chapter 3. Imbedding theorems for weighted Sobolev spaces

W1 ,p(Q;S) ~ Lq(Q;w) J.

Moreover, the following assertion holds

16.12. Lemma.

Let the expY'essions

equivalent nOY'ms on

W~,p(Q;S)

II-Ill

COY' on

n Sand 111-111 n be 1 ,p,o"S 'P"" 1 W ,p(rl;S) J. Then the imbedding

(16.26) COY' (16.27)J holds if and only if the HaY'dy inequality (16.25) is valid foY' eveY'y u € C~(rl) COY' foY' every u E W1 ,p(rl;S) J.

17. SOME GENERAL NECESSARY AND SUFFICIENT CONDITIONS

117.1. Introduction.

Let

1 W ,p(rl;S)

and

W~,p(rl;S) be the weighted SObolev

'i~';'5paces introduced in Subsection 16.10. Our aim is to establish conditions on (,1'

the collection weight function

S

of weight functions

vi'

i

= O,l, ... ,N , and on the

w which guarantee the continuity or the compactness of

the imbedding of the weighted Sobolev space into the weighted Lebesgue LP(rl;W) . More precisely, we will deal with the continuous imbeddings

space (17.1)

W1 ,p(rl'S) , (,.~ Lq(Q'w) '

(17.2)

W~,p(rl;S)

C;

Lq(rl;w)

the compact imbeddings (17.3)

W1,p(rl;S)~~Lq(rl;W) ,

(17.4)

W~,p(rl;S) 4.~ Lq(Q;w)

(for the notation, see Subsection 7.1). These problems have been investigated by many authors. Here, we will mention only some of them, mainly by giving references to results contained in books and survey articles. One of the first authors who studied imbedding theorems for weighted Sobolev spaces was L. D. KUDRYAVTSEV [lJ; he considered unbounded domains and weight functions of the type

(1 + Ixl)a,

investigated weights of the type

v(x)

bounded domains

aE R. J. NE~AS [lJ a , aE R, on

= [dist (x,aQ)J

rl. Spaces with these weights have been considered by

S. M. NIKOL'SKII [lJ (who is mentioned here as a further representative of the numerous Soviet school) and by H. TRIEBEL [lJ who, moreover, has investigated also weighted Besov spaces. The case of weights of the type v(x) = ;

242

(dist(x,M))

with

;€

W(O,oo)

and

Me arl

is analysed in detail

243

- - - - - - - - - - - - - - - - - - - ---

P =

in A. KUFNER [2J . All these authors consider the case

q ;

a survey

of results till 1977 can be found in A. KUFNER, O. JOHN, S. FuclK [lJ

,',:, "", "i,'iyor a given :'

A rather general approach for the case

1

~ p

~

q <

00,

~

£

>

0

£1 EO (0, £q/(2C Q +

choose

"'/iii,;'

'~' tl, umber "{;·!'t

­

lul:,~,w ~ £llu~~,p,n,s + ~ul:,Q;'w

a bounded

uE: W1 ,p(Q;S) . The imbedding (17.7) implies the existence of a {un}

J.

k

C {u} n

which is a Cauchy sequence in

'onsequently, there exists a number ~un

P. GURKA. The exact references will be given at the respective places.

~

be a domain in

R

N

and let

{Qn}

denote

Qn C Qn + 1 c:;, ~,

n E :N ,

U Qn

Il

.

~

q - un I Q_ < £1 1 q, n'w

k, 1 ~ k

for

0

,

un

k

- u n Il qq,~,w l

q

s £ 1 Ilu n - u n llq1,p,Q,S + Ilu nk - u n l Il q,Qii.'w s­ l k q - £ 1 2C + £ 1 S­ £q

n=l

S

Further, denote Qn

(17.6)

{u

=~ \ Q

n

.

We will suppose that there are

~oca~

n

}

in a Cauchy sequence in

Q instead of Q) and look for additional conditions n which guarantee the validity of the ,globa~ imbeddings (i.e. on the whole

•.. ~. Lemma.

every

(17.7)

Let

1

W1,p(Qn;S)

Assume that for every

~

p,q

C; C; £

<

Lq(Qn;w)

> 0

for

n E:N nEN

(17.8)

~u!qq, ~ ,w ~ £Iul~ ,p, Q, S + ~ulqq, Q ii.'w

for every

u

(17.9)

Wl,p(~;S) C;~ Lq(~;w)

Proof·

Let

E W1,p(~;S)

{u} n

Ilu n III , p

n

such that

S ~ C

there exist

(17.11)

nE:N

and suppose that

there exists an

0

n e:N

such that (17.8) ho~ds for

contrary, that the statement of our lemma is false.

W1,p(~;S) , i.e.

{v} n

and a sequence

£ > 0 >

{un}

C W1,p(~;S)

Eiun~~,p,Q,S

+

~unl:'Q n ,w

I u n III ,P,ll&, n S

7'

0

cn =

II v n 1\ : , n, w > E

+ II v n II :

taking

v

n

such that

u /c n n

we obtain

, Q ,w •

n

W1,p(~;S) and (17.9) holds, there exists a {v n } and a function v E Lq(Q;w) such that v nk -+ v k

is bounded in

subsequence for every

>

IUn~:,~,w implies that

. Then

be a bounded sequence in ,~£',

E

Proof. Assume, on the

-

there exists an

00

uEW1,p(~;S) .

and suppose that

00

1 ~ p,q <

Let

wl,p(n;s) ~ ~ Lq(~;w) .

Q).

17.3. Lemma.

o

Lq(Q;w)

k

imbeddings of the type (17.1) - (17.4)

(i.e. on the subsets

domain

n

together with (17.10) yields

its special covering, i.e. suppose that (17.5)

k

Lq(Q-;W) .

k O E:N such that

The results established in the sequel are mainly due to B. OPIC and

Let

Then there exists a

n E:N such that

domain, can be found in the extensive papers by P. I. LIZORKIN, M. OTELBAEV

17.2. The basic method.

1)J .

'~

(Chapter 8, Section 8.10).

[l

.=------.===-==-------------- ­

--------------------_.,----------------- --_._--­ .,-.-----.-=.::======c-=-===-'=--=~__.::=.-_,,~;--.-.;:,~-~-=-==---=---=--==.::--=~

in

Lq(Q;w) . Now, (17.11) yields 245

244

~~~""~~---_ --~_~~=~~~~~-C=~~

q q ivi q,,,,w n ~ E + Ivi q,,,,w n which is a contradiction since

7.15)

,

E >

0

lim n-+ oo

Ilull sup :<;1 lull n 1,p,O,Sq,Q ,1.'

o 1

W ,p(O;S)

The condition (17.8) is equivalent to the condition

17.5. Remark. (17.12)

where

lim n-+ oo

sup

~Ui1,p,O,SSl

~u~

,:;c:: Lq(O;w)

.

Conversely, if (17.16) holds, then the condition (17.15) is satisfied.

= 0

q,Qn,w

I:7.

Notation.

Let

X be a closed subspace of

1 W ,p(O;S)

and for

nEll

is given by (17.6).

Qn

Indeed, according to (17.6), the inequality (17.8) can be rewritten in the form (17.13)

lul

sup

~u11 , p ,')G, n SS~

= {u; u = vl Q ' v EX} n

, we consider the norm

n

Further, we denote for

n?; n , we have (17.12) from (17.13).

Conversely, suppose that (17.12) holds. Let

-

Then there exists a number

Xn X

q q ~ S Elul 1,p,O,S ' q,Q ,1.'

q

and since II u Il q s Ilull q,Qn,w q,Qn,w

nEll

lui

(

0

>

and denote

such that S (1

E

1

q,Qn,w

lui u

€

q,Q

n

,1.'

S Ellul

W1 ,p(O;S)

Let

Theorem.

1 S p, q <

n?; n . In view of (17.6) this implies that

+ iul:,Q ,w

•

If

n~:N

forevery

x

XC:~Lq(O;W) .

Conversely, if (17.19) holds, then the condition (17.18) is satisfied.

n

Le. the inequality (17.8) holds ( E~ = E ).

00

sup Ilull = 0 ,

u E x,l u l s1 q,Qn,w

1,p,O,S

E~lul~,p,o,s

lul:,o,w s

I'~x = ~'~l,p,O,S .

Xn~~Lq(Qn;w)

lim n-+oo

and

n

= (l/q

n ~ n

for every

I'I X = 1'~l'P,Qn'S

The next theorem can be proved analogously as Theorem 17.6.

and, consequently,

for every

o,

o

Summarizing Lemmas 17.3 and 17.4 and using Remark 17.5 we have

Let us again point out that Theorem 17.6 implies that under assumption (17.14) the condition (17.15) is necessary and sufficient for the compactness of the global imbedding. An analogous result holds

17.6. Theorem. (17.14)

and

1

W

Let 1

,p(Qn;S)

S p,q <

00

also for continuous imbeddings:

.If

~c;;. Lq(Qn;w)

for every

n E:N

17.10. Theorem. 07.20)

xn~ r

Let

1 S p,q

L q(Q ;w)

n

<

00

for every

If

n E :N

246

247

and

as well as

require in both cases ( N

(17.21)

lim n .... OO

sup

uEx,llullx~l

liull

<

N

>

1 ) that a certain limit

should be zero. 00

q,Qn,w 17.12. An application.

then

C n

and subsets

(17.22)

xC; Lq(Q;w) .

constants

Q c c

Conversely, if (17.22) holds, then the condition (17.21) is satisfied.

c

Proof. As

u EX, II u I X~ 1

q , n, w ­

n

such that for a.e.

:::; w(x)

~

C

n

E Qn

x

,

:::; v. (x) :::; C , l n

i

0,1, ... , N •

=

Lq(Q ;w) n

and

1 W ,p(Q ;S)

are isometrically 1 isomorphic to the classical (non-weighted) spaces Lq(Q) and W ,p(Q), n n and the local imbeddings (17.20) and (17.17) can be derived using the

we have Ilu II

n

, C n

Then the weighted spaces

Ilull q, Q,w ~ ~u~ q,Q n ,w + ~u~ q,Qn'w '

sup

n

Let us consider weight functions W,V O,V 1 '·· .,v N Q with the following property: there exist positive

n

classical Sobolev imbedding theorems provided some additional conditions

"'­

are fulfilled. For example, if

Q E CO,l n

then these conditions read as

follows: sup ~u~ + u EX, I u I X~ 1 q, Qn , w

~

sup u E X ' n

II u I X ~ 1

Ilull q, Qn' w •

n

I:'q! _ I:'! +

1 ;;:

I:'q! _ I:'p! +

1

p

(17.23)

This inequality implies (17.22) since the expressions on its right-hand

>

° for continuity, ° for compactness

side are bounded due to (17.21) and (17.20). (for references, see the end of Subsection 12.13). We will indeed meet The converse assertion follows by a contradiction argument from the

these conditions later in some particular cases.

inequality Ilull q,Qn'w

17.11. Remarks.

(i)

~ I u II q , n, w •

If we take

o X

=

1 W ,p(Q;S)

in Theorems 17.8 and 17.10,

we obtain assertions about the imbeddings (17.3) and (17.1), while for

X

=

w~,p(n;s) (ii)

we obtain assertions about the imbeddings (17.4) and (17.2).

18.1. Introduction. The domain

Our aim now is to reformulate the

conditions (17.21) and (17.18) [which are expressed in terms of norms of the global imbedding] in terms of the weight functions

v1 = v

The limits appearing in the conditions (17.21) and (17.18)

w,v O,v 1 '···'v N · Here

we will suppose that

imbeddings mentioned.

always exist since they concern monotone sequences in

n.

spaces considered and which together with the local imbeddings guarantee the

The foregoing results can be found in B. OPIC [2J together with

further necessary and sufficient conditions for the compactness of the

(iii)

:::;P:::;--'l
18. IMBEDDINGS FOR THE CASE

and

consequently

v

2

N

we will deal with the spaces

R+. A comparison

with the one-dimensional case (cf. Theorem 7.13) indicates a certain

(18.1)

w1 ,p(n;v 'v ) O 1

and

w~,p(n;vO,v1)

relationship. For instance, for the compactness of the imbedding, we (cf. Subsection 16.10). 248

249

~~~C__ C ' __ C

,~

-

""

_-'"

We will consider domains

_ _ '._

QC R

N

------

---

-----::;

Further, we suppose t~at there exist positive measurable

(iii)

with the only restriction

functions

b

(18.10)

w(y) ;::; bO(x)

aQ ;< (/)

(18.2)

Let us denote

O

' b

defined on

1

Qn

and such that

for a.e. d(x) = dist (x, aQ)

(18.3) and let

Q C Q , n

(18.4)

(18.11)

~These

n ER, be such that

{x E Q;

1:.n

< d(x) < n}

C

Q n

C

:=:

{x

Q; _1-1 < d(x) < n + 1} n +

(draw a picture!), put

(18.5)

E

-

Qn

n

n E

last conditions connect the weight functions

~: auxiliary function (functions

w

or

r = r(x)

Q n

C

and ensure that on the ball

Let

be a bounded domain in

A

B(x,r(x») , the

R

N

and

p

a positive

A. Then there exists a sequence of points

U Q = Q . The sets n=l n

Q

n

will play the role

(i)

U

A =

B k

k

x

k

~

A ,

with Bk = B(xk,P(xk ») ;

from Section 17.

n

with the

1

such that

Q 1 c. Q, n+;<

Q

of the subsets

w, v

v

l'function defined on

n

Obviously,

.

are bounded from above or below, respectively, by 1

constant depending on x.

18.3. Lemma.

CO, 1

y E B(x,r(x»)

and

b 1 (x) ;::; v 1 (y)

Qn = int (Q \ Q )

and suppose that

(18.6)

x

(ii)

e depending only on the dimension

there exists a number

N

and such that 18.2. An important auxiliary function. The weight functions.

(i)

In view

I

X (z) ;::; Bk

k

of the conditions (17.21) and (17.18) we need some estimates of the norm in

Lq(Qn;w)

,

there exist a number r

=

r(x)

(18.7)

(18.8)

(

)

< d (x)

-

3

nn

for

n~

2 , a positive measurable function

c r ~- 1

x

a.e.

for

such that

Let

18.4. Lemma.

;::; p,q <

00

(18.12)

!u(y)l

[ J

q

p

~

°,

r

>

°,

x E R

N

•

Then

;::;

B(x,r) x E Qn ,

y

<::

B(x,r(x») ;::; K rN/q-N/P+1[r-p J lu(y) P dy + J l'i7u(y) P dy ] l B(x,r) B(x,r)

I

N

;

Iy - xl <

r}

J.

The condition (18.7)

B(x,r(x») belongs to Q for x € 3n x E n (see Lemma 18.5 below). v

o'

v

Q

and, moreover,

holds for every x , rand

u

E

1 W 'P(B(x,r»)

(18.13)

such that

(x);::;KOvO(x)

I

!vu(x) P =

N I ja~(x) I

i=l

for

a.e.

with the constant

I

l/p

K independent of

u .

there exists a constant

250

q

l/q dy ]

[Note that here and in the sequel

-p

•

the inequality

E nn

for a.e.

The weight functions

vj(x)r

-N - -N +

,

appearing in the weighted Sobolev 1 spaces (18.1) are supposed to be connected by the following condition:

(18.9)

N

. For this purpose, we suppose that

n

and a constant

K O

z E R

q,Q ,w

B(x,r) = {y E R

ensures that the ball

(ii)

-=R

n Q

c -1 ;::; r~ ;::; c r r(x) r

[recall that

belongs to

n

defined on r x

~u~

i.e. of

e for every

P . ]

Xi

n

x€ : :rl.

251

-=~~=_=_~,=c----o-

~=:-c==-

..

_--.------_.-:::=:=:::"~_:::==-_.-------------_

Lemma 18.3 is part of the famous Besicovitch coveping lemma and its

..----

d(x) ;;; rex) + dey)

13

<

proof can be found, e.g., in M. DE GUZMAN [IJ. Lemma 18.4 is in fact the Sobolev imbedding theorem for the ball the unit ball

B(O,I)

r.

~~th

n ~ 3m,

m E ~,

B(x,r(x»)Cn

be the function fPom Subsection 18.2 (i). Let

r = rex)

m ~ IDax (2,~) . If

B(x,r(x»)(J nn ~

n

3m

~

Le.

d(x) < n

m+ 1

regard to the choice of

E 0

Let

18.5. Lemma.

1n

< ~ ;;; ~ :;; __I_

d(z)

B(x,r) , we express exactly

the dependence of the imbedding constant on the radius

d(x) +

---=.::=--~-:==---=--=----=-==---.-...:::---:-:;==__----~:::-..;..-=-

together with (18.16) yields

B(x,r) . Applying this theorem to

and then dilating it to

-==--==:.:::.-.------;:=-==.:..----=-------:::-...

----------""----.--.. --=-::.-:-...::.__ ---------....- -

m

0

.

' and (18.14)

(ii)

0 , then

Let

z

E

m, n . The last inequality implies that

proved.

1S

B(x,r(x») ,

E. B(x,r(x»)

y

O~ . Then

n

-

Iz - ~I ~ Ix - ~I - Iz - xl > Ix - ~I

ffi •

rex)

for

~ E dO

for

~

consequently

Ppoof. Let us write' nn

n

n d(x) ={ x EO;

<

i11f '

n

0",

n d(x) ={ x En;

>

n} .

(18.15) (i)

B(x,r(x»)n O~ B(x, r (x») Let

n

'" 0

; ; Iz -

d(z) =

(18.16)

Ix - yl

>

Iy - ~I

dey)

>

n

*

~

y E B(x,r(x») (J n~

d(z) <

2

3

d(z)

inf Iz - ~I E dO

since

y E

0: .

-

n~

;;

<

implies

dey) - rex)

n -

>

2n

~

3m

Z-

~

3

d(x) , i.e.

-

rex)

-2 d(x) 3

E dO

n 2

> -

m+ 1 .

2r(x) + Iy - ~I

for

~E

the imbedding of weighted Sobolev spaces. dO,

we have

18.6. Theorem.

2r(x) + dey) .

(18.18)

, and by virtue of (18.7) we obtain

Let

Let

1 ;;; p ;;; q < '" ,

wl,p(nn;v 'v ) ~ Lq(On;W) O 1

r = rex)

d(x) + i

N

q

foP

~ + 1 ~ 0 . Let p

nE~

be the function fPom Subsection 18.2 (i) and suppose that V

the weight functions

1

o

'

VI

,w

satisfy the conditions (18.9), (18.10),

(18.11). Denote (18.19)

63n = sup

x E nn

Ix - ~I ~ Ix - yl + Iy - ~I < rex) + Iy - ~I

o

The following theorem gives sufficient conditions for the continuity

Then

b~/q(x)

252

~

~ d(x)

z EnID, and (18.15) is proved. '"

Further,

and thus

d(x)

together with (18.17) yields

d(z) >

yl + Iy - ~I

and from the definition of

~

d(x)

B(x,r(x») C O~ ,

o~ '" 0 ~ B(x , r (x») C

z E B(x,r(x»)

Iz - ~I

dey) <

-»

y E nn '"

thus, since

Obviously, it suffices to prove the following two implications: (18.14)

31

nn o u '" Ix - ~I ~ Iy - ~I

00

d(x) -

~

d(x) - rex)

nn

where

Now,

~

d(z)

for

~ E an

rN/q-N/P+1(x)

b~/P(x)

If (18.20)

lim n-''''

GJ n =

U3

< '"

253

--

-

-

-="­

then (18.21)

Proof·

wl,p(I"I;VO'V

l

c;

)

~ [Kb~/q(xk)

Lq(O;w) .

We will use Theorem 17.10 with

. 3n According to this theorem, it suffices to verify that the condition (17.21) is satisfied. Taking

R

Q~

{x E Qn; Ix[

Q

=

n

<

+

n

rI~ C U Bk , k

J

B k sing here the inequality (18.11), the condition (18.8), the estimate (18.9)

n~

R} ,

d3n

definition of

(cL (18.19»

we obtain

{x E nn; Ixl < R} .

=

A = n R and p(x) and of a number e

{Xk}~ n~

(18.22)

and

1

J

n

Lemma 18.3, used for sequence

o

0 , we denote

>

=

X = Wl,p(l"I;v ,v)

rN/q-N/P+l(Xk)]q [r-P(x k ) J lu(y)I P dy + B k q/p \Vu(y) \P dy ] , k E Kn,R .

r (x) , ensures the existence of a

lu(y)

~

w(y) dy

\q

[K

bl/q(x ) 0 k rN/q-N/p+l ]q b l/ P ( ) (x k ) 1

B k

such that

f

vl(Y)

!l7u(y)\P [c~ BJ lu(y)IP-p~-dY+ r (y) B

•

Bk = B(xk,r(x ») , k

xk

k

L XB

(18.23)

k=l For

n

~

n

k

e ,

(z) S

­

z ERN.

~

n 3n '" 'rl.} f-I.

;;; K

U B C nn ~ nn , and therefore we can use k kEKn,R all estimates from Subsection 18.2. Further, by virtue of (18.22), According to Lemma 18.5,

where

Ilull

q

q

Ilu "

n

q,QR'w

3n

'11

f

w

q,I"I R ' 1"1

~

lu(y)

q

I

~

f

dy .

kE Kn R B , k

Bk

lu(y) ,q w(y) dy

~

bO(x ) k

q

dy

+

B

k

~

;;;

J

vO(y) dy +

J

[Ilu(y) jP v (y) dy 1

J

::;

d3~ cL

;;; K l

63~[k€~

q p / Kl

63~

/

[f

k ... Kn, R

;;; e q dy

q/p

q/P

IP

[lu(y)!P vO(Y) + Il7u(y)\P Vl(y)] dyf P::;

Bk

J [lu(y)!P vO(y) n,R

f lu(y) I

J Illu(y) IP vl(Y) dyl B k

lu(y)

;;; K l

+ Illu(y)[P Vl(y») dyf/P;;;

B

k

\lull{,p,Q,VO,Vl

where we have used the fact that

q/p ~ 1 , the estimate (18.23) and the

00

iJ

inclusion

k=l For 254

n

R

The inequality (18.10) and Lemma 18.4 imply (18.24*)

l

[f

q,QR'w

3n

f lu(y) Iq w(y)

L

w(y) dy

63~

;;;

B B k

k q q q/p

This inequality together with (18.24) yields Kl = K max (CrK O ' 1) . Ilu[l

(18.24)

J Iu (y) \ P v 0 (y) B k

denote

Kn,R = {k E :N ' • Bk''"I

]q/P

k

V3 ~ [ c ~ KO

Kq

v1 (y)dY

R

-~

B. ~ n

K

00

we immediately obtain the estimate 25

..- . - - - - . , - - , - - - , - - . - - - - - - .---.-..- - - -..

-----------

-------~,--

q

Ilull q,Qn,w s e

(18.25)

q ~nq ~ul'I 1.p,n,v

q p / K1

w(y) ~ bO(x)

,v • O 1

o Analogously, we can formulate a sufficient condition for the

18.7. Theorem.

Let

1

~

1

(18.26)

W ,p(nn;V 'V )

O

1

p s q

<

<; c;

Lq(nn;w)

00

q

- + P

for

~

and

yEO B(x,r(x)) .

Further, introduce the numbers bl/q(x)

63'"n

0 . Let

n E:N •

Theorem.

Let r , Vo ' VI ,w satisfy the assumptions of Theorem 18.6 and let be defined by (18.19).

for a.e. x E nn

~::__--'--··-'---..,.---·_::::_-·:_c~_~=__-',__=__·--·-----

with (18.9). (18.10) and (18.11)!J

compactness of the imbedding in question.

N

•.•.,=-

b (x) ~ VI (y)

1

Finally, (18.25) and (18.20) imply that the condition (17.21) is satisfied.

N

-:=~_':.~_=::,,=.:.=_=c::.=,_:____"_":':"_-.:"

l0. ij

.J

ISubsection 18.2 n

o

sup x E nn

Let

b1lip (x)

1 S p,q

<

r N/q-N/p+l(x)

00

•

Let

r

=

r(x)

be the function from

and suppose that the weight functions

(i)

conditions (18.29), (18.30), (18.31). Let

"­

63 n

Vo ' VI ' w

be given by

18.32).

If (18.27)

lim n->-oo

13 n

If

(i)

o

wl'P(n;vO,vl)~ Lq(n;w)

18.33)

then 1 w ,p(n;v 'v ) O 1

(18.28)

<; C;

Lq(n;w)

lim n+ OCl

Proof·

Using the estimate (18.25) and the condition (18.27) we immediately (ii)

obtain that the condition (17.15) is satisfied, and (18.28) follows from Theorem 17.6.

o

18.8. Necessary conditions.

terms of the weight functions

v

o '

r

=

r(x)

(18.29)

<

kOvO(x) = VI (x) r

-p

(x)

nn

k

O

> 0

for a.e.

such that x

w1 ,p(n;v 'v )

(18.36)

lim 63n

1

<;

~ Lq(n;w)

o.

n->-oo

(i)

Suppose that (18.33) holds and that the condition (18.34) is

b' l/ q (x )

(18.37) ,

bO ' b 1

o

k

b'l/P(x ) 1

defined on

r

N/q-N/p +l (x ) > k , k

k E :N •

k

Put (18.38)

256

O

1\

-

E nn ;

there exist positive measurable functions

such that

00

fulfilled. Then there exist a sequence of natural n~mbers n ~ 3n , k nk .

k E :N, and a sequence {x } , x E n , such--etr3t

k

k

,

(ii)

(18.35)

be the

function from Subsection 18.2 (i) and suppose that there exists a constant

<

VI ' w , let us change the assumptions

from Subsection 18.2 (ii), (iii). More precisely, let

(i)

63"

n

If

In order to derive necessary conditions for

the continuity or compactness of the above-mentioned imbeddings again in

a

Uk

Rr (Xk)/8 X3Bk / 4 '

kE:N

257

_.. -.._._._._._. __

~_.-.-

where

RE

jy - xkl

~~"'"--"

_~

.. -

"~·o~c:-_

- _-

_"'~

is the mollifier with the radius

(18.40)

_.. ~

E

and

aB

ar(x k ») . Then we have

<

(18.39)

i;;;L","~:·:

uk

EC; (B k )

uk ::: 1 auk

(18.41)

aX i

on

(x)

1

=

{y €

R

N ;

~ [(k~lc~

+ Nc) r-P(x ) k

Ib

l/ P L r

(x ) dy ] 1 k

N/p-1(

xk

) h1/p(x ) 1

k

B k

o ;'; 2

k

uk ;';; 1 ,

1

L = [(k~ c~ + Nc) mN(B(O,l))J

Bk

l/p

Using the assumption (18.33), i.e.

I<_c ~ r (x )

xE ll,

Ilull q,,,,w () ;';; C Ilulll ,p,,,,v () ,v O 1

1,2, ... ,N,

i

k

for

u E:

1

w ,p(n;v O'v 1) ,

obtain from (18.43) and (18.44) that with a suitable constant ' (18.42)

c

independent of

W~'P(ll;VO'V1)

uk E

k,

h~/q(xk) .45)

.

[For details concerning mollifiers and their properties, see, e.g.,

h~/P(xk)

r N/q-N/p +1 ( x );';; C k

suitable constant

for every

independent of

C

k

k. However, (18.45)

R. A. ADAMS [lJ, Section 2.17, or A. KUFNER, O. JOHN, S. FucIK [lJ, Sections 2.5 and 5.3; the property (18.42) is a consequence of our (ii)

vo' VI E Lioc(lt) , cf. (16.19).J

assumption

Then there exist a positive number E , a sequence of natural _ 3nk n , n ~ n, k E:N, and a sequence {x ) , x E n , such that k k k k

Using (18.40) and (18.30) we obtain [fluk(y)

Iq

w(y) dyJ l/q

~ [J

It

08.43)

Suppose that (18.35) holds and that the condition (18.36) is not

1 q w(y) dyJ /

~

bAl/q(x ) o k Al p / (x ) b

B /2 k

/

~ 2- N q [~(B(o,l»)J

l/q

1

b~ q(x ) r N q(x ) , k A

/

k

;';; [f l'v O(y) dy + Nc P J r-P(xk ) VI (y) dy J1/

;';;

P

P

B k ;,;;

B k

r-p(y) dy + NcPr-P(x ) k

for

k E :N .

x E n k

B(xk,r(x ») k

=

C

3nk

implies

nnk .

from (18.38), denote Uk =

;';;

B

k

[k~l J VI (y)

E

k

to Lemma 18.5,

[Jluk(y)!P vO(y) dy + fIIJUk(y)IP v (y) dy ]l/ 1 It It

B k

N/q-N/p+1(x ) ~ k

/

while (18.39), (18.41), (18.29), (18.8) and (18.31) imply

(18.44)

r

uk/~uk~1'P,It,VO,V1

(18.39), (18.43) and (18.44), in view of (18.47) we obtain that P

f VI (y) d y J1/ ;';;

lili

k

I

=

q,1t

B k for

kE:N

with

nk L

IlIi 1

k q,n,w

,w

1

=

~ L

hl/q(x ) 0 k rN/q-N/P+1(x) 1 t;l!p(x ) k 1

k

2- N/ q L- 1 [m (B(0,1»)Jl!q, and consequently, due to N

(18.46), we have 258

259

sup

Ilull

II u III , P , n, v 0 ' v 1;;; 1 for every

;;: II;:;kll

nk q,n

,w

q,n

nk

;;;

cObO(x) ;;; w(y) ;;; CObO(x) ,

L1 £

,w

c b (x) ;;; v (y) ;;; C b (x) 1 1 1 1 1

k E R. However, then the condition (17.15) from Theorem 17.6

18.10. Remark.

x E Qn

a.e.

is not satisfied, which leads to a contradiction with (18.35).

o

and

y E B(x,r(x»)

Then w1 ,p(n;v 'v ) ~ Lq(Q;W) O 1

The sufficient as well as the necessary conditions derived

in the foregoing theorems have concerned the imbedding of the space 1 w ,p(n;v 'v ) . Obviously, the assumptions of Theorems 18.6 and 18.7 O 1

guarantee also the validity of the imbeddings

w~,p(n;vO,v1)~ Lq(n;w)

lim n+ oo

and

w~,p(n;vO,v1) C; ~ respectively, since

53

Lq(n;w) ,

W~,p(Q;vO,v1) is a closed subspace of w1 ,p(n;v 'v ) O

of the functions

uk

from (18.38), which belong to

w~,p(n;vO,v1) by

virtue of (18.42). Combining Theorem 18.6 and part (i) of Theorem 18.9 or Theorem 18.7 and part (ii) of Theorem 18.9 we immediately obtain 18.11. Theorem (the continuous imbedding). Let

Let (18.48)

Let

- Lq( nn;w ) W1,p (Qn;v O'v 1 ) ~

r = rex)

f or

;;;p;;;q
V

o '

v

1

,w

N

q

-N + 1 ;;; O. p

n E R •

00

is defined by (18.19).

(l8.48) by W1 ,p(nn;v 'v ) ~ O 1

en w1 ,p(n;v 'v ) O 1

c..:, Lq(nn;W)

c; <:

W~,p(Q;VO,vl) ~

c,.

for

n E:N •

Lq(Q;w)

Lq(Q;w) ]

lim n-~oo

o.

Gn

_____ . Remarks.

(i)

For bounded domains the foregoing results have been

derived by P. GURKA, B. OPIC [2J. For certain sp~i~l/unbounded domains,

satisfy the following conditions:

There exist positive constants k ;;; K ' Co ~ co' c ~ C and O O 1 1 positive measurable functions b ,b defined on Qll such that

o

<

-if

be the function from Subsection 18.2 (i) and suppose that

the weight functions

(J

Theorem (the compact imbedding). Replace in Theorem 18.11 the

1

On the other hand, in the assumptions (18.33) and (18.35) of Theorem 1 18.9 the space w ,p(n;v 'v ) can be replaced by w~,p(n;vO,v1) . This O 1 follows from the fact that in the proof we have used only the properties

n

Cn

1

kOvO(x) ;;; v (x) r-p(x) ;;; KOvO(x) , 1

the problem was solved (by another method) in p~ GURKA, B. OPIC

B. OPIC, P. GURKA

[2J,

[IJ. The approach used here, i.e. th: application of

Q~

the Besicovitch covering lemma to the bounded domain of Theorem 18.6) and then the limiting process

R

-+

00

,

(see the proof exploits some

ideas of R. C. BROWN, D. B. HINTON [IJ, [2J. Another possibility is the application of an extension of the covering lemma to unbounded domains (see

W. D. EVANS, J. RAKOSNtK [lJ). 260

261

-- =-

We have dealt here with the spaces w1 • P (n;v 'v ) . Results for O 1 1 p the spaces w • cn;S) with S = {v ,v , ...• v} can be easily derived if,

v (x) 1

(ii)

o

1

N

c

>

,v 1

Vo

for instance, there exist two weight functions

the form

Vo(x)

$

cvo(x)

V1 (x)

;;; cv. (x) 1

'''tor all

x En. where

18.14. The functions

we,) ; ;

cw(t) ;;;

v

o

' v

1.p r 1.p( - - W (n;S) ~ W ~;vO.v1)'

r . b~b1_'

(O,n--1 )

t €

, then the conditions (18.49) can be

Cw(t) ,

cV (t) 1

$

v1 (,)

;;; CV 1 (t)

6n

The numbers

appearing in the

wl!q(t) -N/q-N/p +1 (t) r -l/p(t) v

lim sup t->O+

(iii)

o ).

(or

< w

1

For functions

r(d(x)) , the assumptions (18.7), (18.8)

rex)

replaced by

' b . Since we 1 O have supposed that such functions exist, it would be useful to know ho~ to

r (t)

choose them. Thus, let us give some hlnts

c

are expressed in terms of the auxiliary functions

) . In this case, 'E. ( t --r(t), t + ret)

and for a.e.

conditions (18.50) and (18.52) can be expressed as follows:

'18.54)

criteria of continuity and compactness of the imbeddings mentioned above r, b

in this direction.

1:- t

<

for

- 3

-1

<

r

~

t

--1

E (O,n

reT) ~ ret) -

c

Let

;;; p ;;; q

for

r

t

) ,

--1 E (O,n )

and

,E (t - ret), t + ret)).

A trivial choice is provided by the formulas

Example.

bO(x)

ess sup w(y) •

bO(x)

ess inf w(y)

b 1 (x)

ess inf v 1 (y)

b 1 (x)

ess sup v (y) ; 1

w(x)

the suprema and infima (here and in the following point) are taken over

we can take

y E B(x,r(x)) •

x E nn

(ii)

~(d(X))

can be

1,2, ... ,N

i

satisfy the conditions of Theorems 18.6, 1 18.7. Indeed. from (18.53) we obviously have

(i)

=

rex)

r

by simpler but slightly more restrictive conditions

(18.53)

for a.e.

w(x) = w(d(X))

,

d(x) = dist (x,an) ] and if also the function

that

and a constant

such that

0

v1 (d(x))

=

Suppose that

there exist constants

w. v c, C,

1

are defined for all 0 <

C ;;;

1 ;;; C <

w

x

E

-

nn

and that

such that

•

cw(x) ;;; ess inf w(y) ;;; ess sup w(y) ;;; Cw(x) •

(i)

-

Cw(x)

b 1 (x)

cv 1 (x)

bO(x)

cw(x)

b (x) 1

CV (x)

.

1

v

1 ' w

have the special form

rex) = d(x)/3

put

S

v (x) = d (x) 1

and bO(x)

w(x) ,

b 1 (x)

v

(x)

1

13 n has the form

= c sup [d(x)]a/q-S/p+N/q-N/p+l xE nn

a

Lq(n;d )

condition

~q

-

.ti

p

+ 1 " 0

the continuity of the local imbeddings (18.48).

­

x E ~n

(i-1) (18.57)

If, in addition.

n = 2,

w1 ,P(n;d S- P ,d S) ~

(18.31) are satisfied with bO(x)

B-p (x),

vO(x) = d

x E n

The continuous imbedding

cv (x) ;;; ess inf v (y) ;;; ess sup v (y) ;;; CV (x) 1 1 1 1

for every x E ~n . Then the inequalities (18.10), (18.11) and (18.30),

a, S E R . For

da(x)

and the number

f\

18.56)

< w,

If

n

is such that

sup d(x) xEQ

< w

,

263

262

then the condition (18.50) will be fulfilled if and only if

~q - ~p + ~q (i-2)

If

(18.58)

n

~

1

V

ro

(i)

,

sup [d(X)]a/ Q-8/p+N/q-N/P+l lIn d(x) IY/ QxE nn

c

o/ p

The continuous imbedding. The condition N N ---+1?0 q P

then the condition (18.50) will be fulfilled if and only if

~ - ~ + ~ (ii)

and have

Cn =

E rl

q

1 (x)

0

is such that

d(x) =

sup X

B + p

p

B + p

q

1

=

arantees the continuity of the local imbeddings (18.48).

0 .

sup d(x) < '" , then the condition (18.50) will be fulfilled if xE rl only i f either If

The compact imbedding

l p

W 'P(n;d B- ,d S) ~ ~ Lq(n;d a )

a

8

N

P

q

~ + 1

-+~

Q

>

p

0

The condition

~q - ~p + 1

.61)

> 0

guarantees the compactness of the local imbeddings (18.51). Suppose that (18.59)

lim d ( x) Ix I...",

xErl

q

I - ~ ~ 0 . q

p

then the condition (18.50) will be fulfilled if and

The compact imbedding. The condition

0

rl

is

quasibounded, then the condition (18.52) will be fulfilled if

P

if either

18.16. Example. w(x)

Let

1 ~ P ~ q <

a,S, y, 0 E R . For

ro,

v 1 (x)

dS(x) d(x)

<

lIn d(x)

1°

21

d(x) > 2

or

rl. Again we can take

p

q

~ + 1

Let

1

p

~

(18.57) holds. For

w(x) = vo(x)

=

VI (x)

=

d(x)/3, bO(x) = w(x) ,

0

o

and

l_i
P

put

x

1

We can take

rex)

2 d (x)

q

~

E

w(x) = ea/d(x)

rex)

>

p

~+!:!_!:!+1 P q P

18.17. Example.

dS-P(x) !lnd(x)I O

such that

x ~ rl

~+!!

CJ.

q

a q

lIn d(x) I y

da(x)

vo(x)

elsewhere in

264

,

and

the compactness of the local imbeddings (18.51). If >

The same conditions concern also the imbedding of W~'P(rl;d8-p,d8) into Lq(n;dC<) (see Remark 18.10 and Theorems 18.11,

18.12) •

x

o

p

!:!q - !:!p + 1 > 0

(iii)

for

~+

Q

sup d(x) = ro xErl if (18.61) holds. (ii)

~ - ~ + ~ - ~ + 1 p

P

= 0 ).

Then the condition (18.52) will be fulfilled if and only if q

q

If

is bounded or quasi bounded (the latter term means that

n

-CJ. - -S + ­ N

rl v

,

<

00

a, S E R . Suppose that

n

is such

put O(x)

bO(x)

d- 2p (x) e

w(x) ,

B/d(x)

b 1 (x)

v 1 (x)

= e

S/d(x)

v 1 (x) , and have

265

n

(i)

The continuous imbedding. If

N/q - N/p +

w(y) :;; bO(x) ,

;;; 0 , then the

for a.e.

condition (18.50) will be fulfilled if and only if q

(ii)

(l8.63)

p

The compact imbedding.

~q - ~p + 1

If

>

0 , and

~

is bounded

(*)

q

p

'!i)n

bi/p(x) r

4

0

>

N/q-N/p( )

x.

Lq(Q;w)

[ W1,p(~;v,v) ~

In the foregoing examples, we have apriori supposed that £;

o

sup xE Qn

W1 ,p(Q;v,v)

only if (18.62) holds.

~ - !i + 1

y E B(x,r(x») . Denote

and

Then

or quasibounded, then the condition (18.52) will again be fulfilled if and

18.18. Remark.

x E ~n

b 1 (x) ;;; v(y)

bl!q(x)

~ - ~ ;;; 0 .

(l8.62)

/3 ,

r (x) :;; d (x)

sup e (a/q-S/p) /d(x) [d (x)] 2(N/q-N/p +1)

xE rP

()

G Lq(Q;w)]

if ~

lim n"'''''

0

when deriving conditions for the corresponding continuous (compact)

n

gJ < '"

[lim :1J = 0 ] • n n"' ro

imbeddings. As will be shown later (cf. Lemma 19.14) the continuity (compactness) of the imbeddings mentioned in Examples 18.15, 18.16, 18.17

The proof is a slight modification of the proofs of Theorems 18.6 and 18.7.

implies the condition (*) and, consequently, it is a necessary condition.

Instead of the inequality (18.24*) we derive the estimate

Similarly it can be shown that the condition of the quasiboundedness

of

Q

J lu(y)l q w(y)

(cf. (18.59)) is necessary for the compactness of the imbeddings

auxiliary function

r

r.

The condition (18.8) on the

V

o=

and since the boundedness of

N

q

bounded domain. Let WI ,p(Q 'v v) r n"

'7

n'

£;

o,

let

QC

R

N

q,Qn,w

be a

18.21. Theorem.

for

and positive measurable functions

Let

there exist a number

n E :N •

Lq(Qn;W)]

Let there exist a_number n E:N b defined on Qn such that 1 266

N - + 1 p

Lq(~ 'w)

[w 1 ,p(Qn;v,v) C; C;

Q together with the inequality

we finally obtain the following analogue of (18.25): II u II q

1 ;;; p ;;; q < '" ,

,

rP(x ) ;;; (diam Q/6)P , k

Theorems 18.6, 18.7 and 18.9 hold.

Let

q/P

v )

then (18.8) can be omitted. More precisely, the following analogues of

18.20. Theorem.

J IVu(y) I P dy ]

implies

r(x) ;;; d(x)/3

vI

ju(y)!P dy +

B k

is restrictive, but it was used substantially in

is bounded and

[J Bk

+ rP(x ) k

the proofs of the foregoing theorems. If we suppose that Q

rN/q-N/P(Xk)]q

B k

appearing in these examples. This follows from B. OPIC, J. RAKOSN!K [lJ. 18.19. Weakening the conditions on

[Kb~/q(xk)

dy;;;

defined on r

, b

o

'

-

Qn

;;; 0 q / P K

1

1 ;;; p,q <

nE:N

lJ q

II u II q

n

ro

1,p,Q,v,v

,

let

Q

•

be a bounded domain in R

and positive measurable functions

r,

N

Let

hO ' hI

such that

r(x) ;;; d(x)/3 ,

267

~

w(y)

for a.e.

XEQn

b1 (x) ~

bO(x) ,

v(y)

POWER TYPE WEIGHTS

and YE B(x,r(x») . Let

1 W ,p(Q;v.v)

4

1

[w ,p(Q;v,v)

q]" n

be defined by (18.32). If

Introduction.

Lq(Q;w)

~ ~

Sobolev spaces W1 ,p (Q;d S ,dS)

Lq(Q;w)]

then '\

weighted Lebesgue spaces

/I

lim (3 = f!, < '" n n+'" [lim n+'"

"n = G

In this section we will deal with imbeddings of special

a q L (0;d )

°].

that d(x) = dist (x. (l0)

The prOOf is again a modification of that of Theorem 18.9. Using the fact that

Vo

from

(18.38). but now in a little different way. We have

= vI = v • we again derive the formula (18.45) for the function

II uk III ,p • Q, v • v

~

[J

v(y) dy

+ Nc P

B k

J r-P(x k )

uk

11/p

1

For the case ~oreover,

v(y) dYj

o

we will suppose that the domain $

P

$ q < '"

is bounded.

we will use the results from Section 18;

we will also consider the case

~

1

~

~

B k

q < P < '" • The results of

~hiS section are due to P. GURKA. B. OPIC [2J.

~

\i,

Imbedding theorems of the type mentioned above have been investigated the case

~

1/ p v(y) dy

]

[J

[

1

+ NCPr-P(x ) k

B k

1

B k $ L r

for

kEN such that

since for these

k,

case of Theorem 18.9.

18.22. Remarks.

(i)

~

(x ) dy k J

1 P / [ 1 + NcPr-P(x ) JI/ P ~

k

l!(3cN 1/p )

n

°

CO,K, and

Obviously, the space

W1 ,p(Q;v,v)

w~,p(Q;V,v)

CO. K

A bounded domain

0e EN

is said to

to the class

from Theorems 18.6. 18.7 by

'lJn

from (18.63),

K~ 1 ,

m of Cartesian coordinate systems

There exist a finite number

(Y~'YiN) ,

in Theorems 18.20,

(cf. Remark 18.10).

<

the following conditions are satisfied: (i)

03

Q •

us start with the definition of a special class of domains.

bi/p(X k )

Note that in the case of the sufficient conditions we have now

replaced the numbers

q

k

Q:k = {x E Qn k ; d(x) > n } = !/J , k NcPr-P(x ) ~ 1 . Then we complete the proof as in the k n

=

P

Domains of the class

Nip -1 (x ) k

18.21 can be replaced by the space (ii)

~

A. KUFNER [2J under certain additional assumptions about the domain

[J b

~

] 1I p

y~ = (yi1· y i2····· y iN-l)

the same number of functions

a.

1

=

a.(y~)

j

= 1,2 ..... N-l}

1

1

defined on the closure of

(N-l)-dimensional cubes (19.3)

6.

1

= {y~; ly .. 1

1J

1 <

0

for

while in the case of the necessary conditions we have used the same number as in Theorem 18.9. Consequently, we cannot combine Theorems 18.20 and 18.21 and have no analogue of Theorem 18.11 and Theorem 18.12.

( i = 1,2 •... ,m ) such that for each point i E {1,2, ... ,m}

x E (lQ

there is at least one

such that

268 269

(19.4)

x

(ii)

=

(Y~'YiN) ,

{~1 '¢2'''' '¢m}

YiN,=ai(Y~) .

The functions

a

partition of unity corresponding to the covering (19.11), 1. e.

satisfy on

~i the Holder condition with the i K , i.e. there exists a constant A > such that

°

exponent

lai(Y~) - ai(z~)

(19.5)

y~,z~E/),.

for every

1

(iii) defined by (19.6)

1

I

~ Aly~ - z~IK

(i = 1,2, ...

1

I

,m) . A

<

1

such that the sets

Q.

~

•

.L"to

[(Y~'YiN); y~ E

Qi

/),i '

ai(Y~)

D

Qili n = [(Y~'YiN); y~ E /),i

(19.8)

r.1

Q.

i

distance.

- A < YiN

<

a

i

(Y~)

1

n all

[(Y~'YiN); Y~ E

=

1

+ AJ f

i

ai(Y~) - A < YiN

<

A

>

°

x E:

for

U~: = 1

nn

U~

supp ¢. ,

sufficiently small we obviously have

u'"

i

cr ..

111

the following estimate holds:

19.3. Partition of unity.

Suppose

CO,l

be a domain from

n

E CO,K

°

la i (y i < K

~ 1 . For

n E:N let

[x E n; d(x)

>

such that

1.n Jerin C

[x E n; d(x)

>

- YiN)

1/-: ~ di(x) ~ ai(Y~) - YiN

1 + A

x

' iN ) E- U*i (Yi'Y

1,2, ... ,m

i

(see e.g. A. KUFNER [2J, Lemma 4.6).

_+1 1 }, n

The following two theorems have been proved in A. KUFNER [2J using and the one-dimensional Hardy inequality

local coordinates (Y~'Y'N) 1 1

and denote

nn

1

ai(y~)J ,

°i«)}

/),i ' YiN

dist (x,f.)

di(x)

')

(19.10)

rl .

xE

Denote

from (19.8). For d(x)

(i = 1,2, ... , m)

(19.9)

= int (n \ n ) n

with respect to the variable

YiN'

Obviously rI

n

en

d. (18.4). For

len,

n+"

[Compare these sets n

the boundedness of

nn

with

270

Q.1

n

=

un

Let

Theorem.

1

< P <

II E CO,K 00

,

n=1 n

°

<

K

~~/1

./

and

.. /

£:

>

K(p - 1) .

with the analogous sets defined in Subsection 18.1,

sufficiently large the two definitions coincide due to n.J

There exists a number (19.11)

~
1

d.(x)

(19.7)

n

for

¢. (x)

°

1

satisfy

n

supp
m

i=1

There exists a positive number

N

00


n E:N

wl,P(n;d£,d£:) ~ LP(n;d n)

such that the system

n

= {

[Ql ,Q2"" ,Qm}

from (19.6) forms a covering of the closure of the set

nn. Let

Theorem.

£/K - P £: - Kp

Let

1

<

Kp ,

fOY'

£:

foY'

K(p - 1)

P

<

ro

,

>

<

II E CO,K

£: '0 Kp

°

<

K ~ 1 and

271

E:

K(p - 1) .

;<

W~'P(ll;dE ,dE:)

Then

W~'P(Q;dE,dE)~ LP(Q;d n )

c;

;;+"iJ the inequalities f01'

where

~ LP(>l;d n ) in (19.20) are strict.

n

p

Now, we extend the above imbeddings to the case ElK - p

n ={

for

K(E - p)

>

for

~

q . In the proof

will use the following result from ~xample 18.15.

Kp ,

°< E E ~ °

for

Kp

E -

E

~

Kp

E

K(p - 1)

;<

p = q) instead of the classical

If we use the inequality (6.20) (for

~ p ~ q < <x>,

Suppose

9.22)

c,.

W1 ,P(Q;d Y- P ,d Y)

N/q - Nip + 1 ~

°

N/q - Nip + 1

[or

>

0].

Lq(ll;d(l)

Hardy inequality (0.2), then we can improve the foregoing theorems. Moreover

'I

using the methods from Section 17, we obtain assertions about the compactnes,\

r

W1,P(1l;dY-P,dY)(,; ~ Lq(Q;d(l)]

of the imbeddings mentioned:

Let

19.7. Theorem. (19.16)

1

~

Wl,P(ll;dE: ,dE)

P <

c;.

°

II E CO,K

<X>

if <

K

~ 1 .

(l

Then

ElK - P

(19.17)

for

E

n ;;; E - Kp

for

K(p - 1) < E:

n

for

E

> -

K

>

~

19.25)

Kp , ~

~

!:!+1>0].

(l

p

q

p

q

K(p - 1)

Let

9.9. Theorem.

N

N

::;p~q<<x>,

-+

q

P

;:; ° ,

1 E W ,P(Q;d ,d E) \;.

Let

19.8. Theorem.

W~'P (ll;d

n~ n ;;; n~ n

Moreover,

272

E cO, K,

°

< K

~

in (19.17) are strict.

n

1::; p < 00, E

,dE)

C;

W1'P(1l;dS,dS)~ Lq(ll;d(l)

~ LP(ll;d Tl )

liE CO,K ,

°

S <

K

~ 1 •

>

Kp ,

N - - -S +­

(l

Kp

q

!:!+ p

q

;; °

Then

L P (Q;d n )

/

(19.28)

K(p - 1) < S

(19.29)

S

~

Kp ,

S

a

N ---+-q P q

!:!+~O p/

where

(19.20)

Q

, S E. R • Then

if the inequalities for

(19.19)

I+~

1

Kp ,

Moreover, (19.18)

~P +

q

LP(ll;d n)

where Tl ;;;

°

I+~ p

q

ElK - P

for

E: - Kp

for

K(E - p)

for

> -

K

for

E

°

>

Kp ,

<

E ~ Kp ,

°

E ~ E = K(p - 1)

E '" K(p - 1)

~

K(p - 1) ,

~ _ K(p - 1) + ~ _ ~ + K > q p q p

, Proof. Using Theorem 19.7 for u (19.30)

luI p,Q,dY-P ::; c

~ul

E

wl,p (Q;d S ,dS)

°.

we obtain

S S

1,p,Q,d ,d

2/

where (19.31)

for

{ BIK B - Kp + P for

y ­

In both cases we have (19.32)

o < B ;;;

B > Kp , K(p - 1) < B ;;; KP

I ~I p

I

I

K(p - 1) + ~ _ ~ +

Cl

K(p - 1) ,

B

The inequalities (19.30), (19.32) imply (19.33)

~ - ~ + !!. - !!. + K ~ 0 p q p q

B~ 0 ,

B ~ y , and consequently,

s [ diam ~ J y-S N L ~ p i=l dX i p,~,dY 2 i=l dX i p,~,dB

L

N

Cl B N N B '" K(p - 1) , -q - -p + -q - -p + K ;;;; 0

Kp ,

p

q

,1 tmbe dding (19.23) instead of (19.22) [and, of course, the condition

q

p

K ;;;;

~ >

p

W1,p(~;dS,dB)

o.

B(w) = K(p - 1) + w , we obtain from this inequality that (19.28) holds with B(w) instead of B, and consequently,

K(p - 1) < B

S

B(w) > B implies

~

w1 ,p (~;dB ,dB) ~ W1 ,p (~;dS(w) ,dS(w)) ,

the imbedding (19.26) follows from (19.35) and (19.34).

R

D (19.40)

Similarly we can prove

Let

00

N - -N+ 1 > 0 -

P

q

,- cO ,K , < 0 K= < 1,

,~'"

Lq(~;dCl)

~

Kp

- -N + , -Clq - -pS + -N q p

K

> 0

Let

p ;;; q <

~

N 00

,

q

- ­ +

Cl, B E R • Then

W~'P(~;dS ,dB)

C;

N

p

;;;; 0 ,

~ECO,K,

0 <

K

~

1,

Lq(~;dCl)

K(p - 1) + ~ _ ~ + K > 0 .

Cl

p

q

~p;;;q
q

p

~ _ ~ + 1 q

rI E CO,K,

0,

>

0 <

K

;;;

1,

p

Then

W~'P(~;dS,dB) Cj

if 19.10. Theorem.

B

>

Kp ,

o

<

S

~

4

Lq(~;dCl)

~_lL-+~ q Kp q

Kp ,

!i +

1

0

>

P

S '" K(p - 1) ,

Cl q

f + ~ p

q

-N +

K >

0

P

or

if B > Kp ,

274

c; ~

K(p - 1) ,

Theorem.

(19.36)

q <

W1,p(~;dB(w) ,dB(w))c; Lq(~;dCl)

Since the inequality (19.35)

~

S + -N - -N + 1 > 0 -Cl - -Kp q q P

S > Kp ,

Denoting

(19.34)

p

~

Then

wE (O,K]

such that p

Let

Theorem.

(19.31) in view of (19.24). Now let (19.29) be satisfied. Then there exists a number

(19.25)

of (19.24)J.

immediately yields (19.26). The conditions (19.27), (19.28) follow from

K(p - 1) + ~ _ ~ +

0 •

is again similar to that of Theorem 19.9; we only use the

If (19.24) is satisfied, then we have (19.22) which together with (19.33)

Cl

K >

p

The proof of the following two theorems concerning the compact

W1,p(~;dB,dB) ~ W1,p(~;dY-P,dY)

q

q

Cl q

B +!!.

Kp

q

!!. + 1 ;;;; 0 p

B ;;; 0 ,

a q

KB + ~ _ ~ + K > 0 P

q

P

275

N

01'

(19.41)

B

K

(p -

Cl

1) ,

N

-+

q _K_(p_-_l~)

q

+ ~

p

~ +

q

p

K >

P

Theorems 19.9 - 19.12 give only sufficient conditions for

=

+ 1

.ti P

q

the corresponding imbeddings. We will show that for K

°,

l!._.§.+~-~+ q

p

°,

Cl

>

'§'+!:!_!:!+1>0].

q

P

q

p

~

0 . N

19.13. Remark.

°

~

P

q

N GC GC Q , and denote Let G be a domain in R , (G,oQ) > 0, D = diam Q < 00 • Then

1

°

these conditions are also necessary except for the conditions (19.29), (19.37), (19.39) and (19.41) for the imbeddings (19.26), (19.36), (19.38) and (19.40), respectively. First, let us prove some auxiliary assertions.

~Let

~ -~

<

u E W~,p(G)

d(x) -~ l2 D

x E G .

for

and define

for

=0

u(x)

xE Q \ G . Then (19.48)

:',immediately implies that

Let

19.14. Lemma.

~ p,q

<

00

•

Let

G be a bounded domain in R

N

W~,p(G) c;. W~'P(Q;dB

and

suppose that (19.42)

W~,p(G)~ Lq(G)

[w~,p(G)

c;

C;

Lq(G)] .

Lq(Q;dCJ.) ~ Lq(G) .

Then

imbedding (19.44) [or (19.45)] implies that (19.42) holds and Lemma

(19.43)

Proof·

~ - ~ + q

p

~o

[!:!-~+I>OJ. q p

the first inequality in (19.46) [or in (19.47)].

In (19.42) we consider the weighted spaces with weights identically rex) = d(x)/3,

:qual to one. In Theorem 18.21 we can take b 1 (x)

=

bO(x)

=1

,

Now, we use Theorem 18.21 and Remark 18.22 (i) where we take ~ Cl ~ B d(x)/3, bO(x) = d (x) , b (x) = d (x) . and we obtain 1

~

1 , which then yields (together with Remark 18.22 (i»

B" n =

c

" G n

lim n+ oo

0

(19.43).

UJ

"

03

n

= c

necessary condition

Let

~

p,q

<

00

,

let

<

00

[lim OJ n+ oo

Q

Cl, B E: R • Suppose (19.44)

W~ ,P(Q;d B,dB) ~ Lq(Q;d cx )

n

o]

implies

(19.45)

be a bounded domain in RN ,

W~'P(Q;dB,dB) ~ ~

Lq(Q;d Cl ) J.

<

00

[or

lim noT'"

second inequality in (19.46) [or in (19.47)].

we immediately

13 n

0]

implies the

o

Lemma 19.15 with Theorems 19.9 - 19.12 yhere we take

1)

K=

see that the conditions (19.46), .0.9-:(7) are necessary and

sufficient. More precisely:

19.16. Theorem.

[ B"

[01'

lim

1\

,\

03 n = f.>

n+'" J\

Comparing 19.15. Lemma.

'd( )]Cl/q-B/P+N/q-N/P +1

sup L x

xEQn

."\

sup [d(x)]N/q-N/p+l xEG n

and the necessary condition

Then

,dB)

p -

1] .

Let

1

~

p

~

q

<

00

,

QECO,1 ,

CJ., BE R ,

B

> p -

1

Then

W1 ,p (Q;d B ,dB)

C.

Cl Lq(Q;d )

[w~'P(QjdB ,dB)

C;

Cl Lq(Q;d )]

if and only if 277 77f..

~ - ~ + q

;t

P ­

1

J.

°,

Let

19.17. Theorem. [ S

~

p

..§+~-~+

0.

q

p

q

;;;p;;;q
~

p

rlE C

°.

O,l

0.,

S E :R ,

S

> P -

1

[w~'P(rl;dS ,dB)

c;

C;;

c:; C;

LqW;do.) ,

TR(O,b)

{u E Coo([O,bJ); supp un {b}

0}

Let

;;; q <

>

.!:!.

°,

P

<

C;; ~

1;;; q

<

B+ ­ N P q

0.

q

p

<

°

rlECO,K ,

,

1 ,

< K ;;;

0.,

S ~:R

•

N

- + 1 > P

Lq(Q;do.)

°. - -S +­1

0.

B > K(p - 1) + K E q 19.18. The case

00

Lq(rI;do.) ]

Wl,P(rl;dS,d S)

p + 1

-

0}

Theorem.

if and only if

~q

{u E Coo([O,bJ); supp un {a}

TLR(O,b) = TL(O,b)(I TR(O,b) .

Then

1 W ,P(rl;d S,dS)

TL(O,b)

Kp

q

pl

q

+

°

1 >

As was mentioned before Theorem 19.5 and

00.

Theorem 19.7, an important role was played by the one-dimensional Hardy in­ equalities (0.2) and (6.20) (the latter for

p

=

q ). In the case

K(p - 1)

p > q

<

~q - ~p +

B ;;; K(p - 1) + K E q

K(lq - 1p

+

1)

>

°

we will again substantially use the inequality (6.20). To this end, let us summarize the results derived in Examples 6.8 and 8.21 (ii):

Let

°

1;;; q < p < ro

b

(19.49)

[f lu(t) Iq

holds

°

(i) (19.50) (ii) (19.51) (iii) (19.52)

for' n

t

E

1I q

dt

J

for'

p - 1 ,

<

ro

E,

n ER . Then the inequality

b

;;; C [J1ul(t)I P t n dt

(i)

riP

First we show that under the above assumptions

c; Lq (l2;do.)

Wl,P(Q;dB,d S )

lim

11 and only if

E>n.9._L_ 1 p p'

sup

or

x

take n ;;; p - 1 ,

E > - 1

v

=

!u

=

Ilull

~u~x;;;l

n+ oo

E>n.9.-L_1 p p'

Then

v

Wl,P(g;dB,d S)

Ii u I

E COO (g);

is a dense subset of

Take

u

TR(O,b) , TLR(O,b) , respectively, where

x

n >

oo} .

W1 ,P(g;d S,d S)

u(x)

u(x)

I

{¢.} l

n,

i=l

(cf. V. I. BURENKOV [lJ). (Y~'Y'N) l

II

and the

from Subsections 19.2, 19.3, we

m

m

(19.60)

<

E V . Using the local coordinates

E gn,

gn , cf. (19.10». Let us

(for the set

S B 1,p,rl,d ,d

corresponding partition of unity have for

{)
q,gn,dCl.

if and only if

19.19. Remark. Checking the proof of the necessity of the conditions (19.50) - (19.52) we can see that these conditions are necessary for the inequality (19.49) to hold on the (smaller) classes TL(O,b) ,

278

.

to Theorem 17.10, it suffices to verify that

E>n.9._3-_ 1 p' p

u E ACLR(O,b)

nE R ,

a > - K .

if and only if

u E AC (0, b) R

n > p - 1 ,

fOr'

b

° u E ACL(O,b)

<

<

B ;;; K(p - 1) ,

¢.(x) l

)" u.1 (x)

(,

i=l

279

with u.l obtain

= u~ l. .

Consequently, using the weights

d i (x)

from (19.14), we

A

-~

c

I ~ui~ q" Dn i=l

q,lln,d a S

l1

m

da

Illuill q, Qnfl supp i=l

(19.61)

~i'

a

d

n

i

iLlluillq,u:,d~

A

S [~-1 (l1

19.66)

IUi(Y~'YiN) Iq d~(Y~'YiN)

dYiN ] dy~

p

~u,(y~,a.(y~) -

(fl oY'N 11, 0

)J (p-q)/p [f i

, au.

l

l

l

t)

l

I

t n dt J dy ~

q P

J/

E (O,A)

II-l

t)

I i Yi' a 1. (y 1I. )

t a dt J dy~

Iq

we obtain P

- t)

I

t n dt }

q/P

dy~ S c2111ullq 1,p,ll,d Kn ,d KfI .

l

We have to distinguish four cases: (i-I)

Let

~

a

0,

S

~

E =

0 . Then we put

a ,

S/K

n

and by

(19.63), (19.65) and (19.67) we have

A

i

0

11, 0

If a < 0 , then we use the first inequality in (19.15) and obtain similarly as before

(1 + A) -a/ K

~

(fl~t ui(y~,ai(Y~)

f

(19.67)

l

< =

n

q

A

f [flu ( 11, 0

l

for

yield

A

-~

q

-~ c o. aYiN II p,ll,dn -~ c 1 Iluil l,p,D,d n ,d n

a ~ 0 , then the second inequality in (19.15) and the substitution

I u. II q f' d a l q,U , i

dy ~ S

l

dx

11. a,(Y~)-A l l l

(19.64)

I

l

I[ I

l

(y~,ai «)

ui

q P

t n dt J /

II

a i (yj)

(19.63)

p

- t)

11, 0

l

q Ilu.ll l q u* da , ., .

UI~t

f

~u.~q = jr lu. (x) Iq d~(x) l * a l l q,ui,d U~

t

dy~

0

i

A

i

If

Jq/P

0 , then Holder's inequality and the estimates (19.15) together

<

Let us now estimate the norm

a,(y~) l l - y'N l = t,

I

t n dt

(19.14) obviously lead to the inequality

m

(19.62)

_ t)

dt u Yi,a i Yi

m

Ilull

p

I (lI I~ (' (')

q

J [f Iu, (' y., a, (' y,) l

l

l

l

- t ) I q t a/ K dt ] dy.I l

Ilu Ilq i q,Ui,d * a S Kq\lullq IS S l,p,D,d ,d

(19.68)

11, 0 l

The inner integrals on the right-hand sides in (19.63), (19.64) can be estimated by the Hardy inequality (19.49) on the class b = A, u(t) = u,l (y~l ,a,l (y~) l - t) , yields the estimate A

(19.65)

J

l1

i

[f Iu.l 0

E

= a

or

(y l~ ,a,l (y l~) - t) Iq t E dt) dy iI ~-

E

ACR(O,b)

(with

K

K = cc~/q ) provided (see (19.51» 1

(with

= a/K ), which finally

(19.69) (i-2)

S S K(p - 1)

Let

a

<

0,

or

S > K(p - 1) ,

B~

a

0 . Then we put

> -S -q - ~-I ~-t---; - ... K p P E

= a/K,

n

= S/K = K2 =

and by

(19.64), (19.65) and (19.67) we arrive at (19.68) (with K -a/(Kq) c l/q ) provlded , = C(l + A) (see (19.51» 2 (19.70) SSK(p-1), a>-K or S>K(p-l), a>~-K(~+l) p p (i-3)

Let

a

~

0 ,

S

<

0 . Then we put

E

=

a ,

fI = S

and the

28

;

inequalities (19.63), (19.65) and (19.66) immediately imply (19.68) (with K = K = eC l/q ). 3 l (i-4)

Let

a

0,

<

6

<

0 . Then we put €

= a/K,

n = 6

inequalities (19.64), (19.65), (19.66) yield (19.68) (with = C(l + A)-a/(Kq) c~/q ) provided (see (19.51» (19.71)

K

u E W1 ' P (rl;d fS ,d B) . This estimate implies that

which holds for every

lim

and the

n-HQ

sup

lI u l

=K = 4

Ilull s1

x

q,~,da

-

°, o

(19.53) follows by Theorem 17.6.

a > - K •

;:;;q
Let

Theorem.

Q

E CO,K,

0 < K

~ 1 ,

a, 6 E R •

From (i-I) - (i-4) we conclude th~t (19.68) holds with

K = max (K 1 ,K 2 ,K 3 ,K 4 ) provided a , 6 satisfy the conditions (19.54) _ (19.56), and then we obtain from (19.61) that flull

(19.73)

() d S d S q,rln,d a ;:;; mK II u II 1, p,,,, ,

holds for every u E V • The same estimate holds for every

1 6 6

u E W ,P(rl;d ,d ) due to the density of V in W1'P(rl;d6,dS)

t3

consequently, (19.58) is fulfilled with (ii) only for

and

~ mK . Thus, (19.57) holds.

fS > K(p - 1)

+ KE

6

1

1

- - -- + - - - + 1 q Kp q P

~_f-+1: q Kp q

q

°

<

B ;:;; K(p - 1) + K E q

6

~

0 ,

a q

~ - ~ + K(1: - l + q

q

-S + p

1:p + K (1 -

q

-

1

>

-1

p

0

+ 1)

> 0

q

p

1)

>

0 .

The proof is analogous to that of Theorem 19.20. The conditions in Theorems 19.20, 19.21 have been only sufficient.

> 0 ,

p

~

q , these conditions are also necessary

1;:;; q

<

P

Similarly as in the case there exists

a

L q(Q;d )

satisfying (19.54) (the proof for the cases (19.55),

(19.56) is analogous). Since the second inequality in (19.54) is strict, a

c; 4

01'

The compactness of the imbedding in question will be proved

a, 6

W~'P(Q;dS,dS)

£ > 0

such that the numbers

6

and

a

=a

- £

satisfy

provided

K= 1

(19.54), too. Then it follows from part (i) of the proof that

Wl,p(Q;dS,dS)~ Lq(Q;d a ) ,

K such that

i.e. there exists a positive constant (19.72)

Ilull

q,Q,d a

Using the fact that estimate liull

< =

Kllull

d(x)

<

for

x

e

q

J lu(x) Iq ;:;;

(19.74)

da(x) dx

n

£

f

lu(x)

Iq

ilull

q

-

q,Qn,d a

Q

,

E CO, 1,

a, 6 E R • Then

a

SE R ,

;:;;!S.-£ Ilull n

~ _ ~ + 1: q p q

_l + p

1 > 0 .

If (19.74) holds, then we have the compact imbedding (19.73)

according to Theorem 19.21. da(x) d£(x) dx ;:;;

rl n q

00

W~'P(rl;d6,dS) ~ (,. Lq(Q;d )

QTI , we derive from (19.72) the

Proof·

q,Qn,d a

<

if and only if

rl n

?R?

(19.73)

l,p,rl,d 6 ,d 6

~ n

Let

19.22. Theorem.

Conversely, let us suppose that (19.74) is violated for some

a , S ,

Le. assume that q

1,p,n,d 6 ,d 6

(19.75)

a <

sg--L- 1. p

p'

283

~-~____:--.--____,______~,=~c_=_=_~~~______,_,___=:===="_==:==_,,~~-~~,-______=__'-=_=_--

------=-.=----~--~- ~--~--",----

---:.--:-_- -

,;;:~

..

_._--------­

••

~___=_ ;._:.~.:.=.::.;;_.=::_=:=~-=-----===~....;;;:-~---:-...==....:;

= a, n = S J is not fulfilled C) on the class TLR(O,b) = C~(O,b). Consequently,

(with a finite constant

E

there exists a sequence of functions

un

E

C~(O,b)

such that

Jlu~(t) IP

t

S

1,

dt

~1

is Lipschitzian on 1 follows from (19.79) that v

a

(cf. (19.5»

~,

.._ ,__

.·._w.._

and consequently, it

e Wl,p(G ) . nOn

Moreover, in view of (19.80),

b

(19.76)

__,

.=====:;---=-=-...;;:;..===-~~.=;;;;;;-=-;;:-==.;;;....=;;-_-=:-_--=,=~_----,-;-_==,._=.=,

The function

In view of (19.52) and Remark 19.19, the condition (19.75) implies that the Hardy inequality (19.49) [with

.v_._._w._.__

S vn E O w1 ,P(n'd ' , dS)

nE:N,

Further,

o

d (x) = d(x) for XEU~ (d. (19.14» 1

and we obtain

b

J1un(t) I

(19.77)

q

to dt

-+

for n

00

-+

00

Ilvnll

•

p,n,d

S

=

Ilv)

* S p,U ,d

o Now, we will use the first coordinate system

from (19.2).

(yi'YIN)

Let

6, A be the constants from Subsection 19.2 [cf. (19.3) and (19.6)J

and

~1

I~I ~

The estimates Ilvnll

the corresponding function from the partition of unity described

is contained in

o

<

A*

A,

<

supp ~1

<

a 1 (yi) - A*

6*,

<

YIN

P p,n,d S

for sufficiently small numbers

A*

~

0 < 6* < 6 • Further, we introduce a function

E

~ ~(z')

~ 1

for

z' E R

(19.78) ~(z') =

and for

n E:N

(19.79) where

1

and

v (x) = {

n

u

n

for x

E:

I z' I

6'~

<

2

C~(RN-l)

un

o

Iz'l >

for

if

x

o

if

x E

=

, y IN) (Yl'

li*

~

{(yi'YIN); Iyil

~ 3~*

c1

Iax:

A* P S lun(t) I t dt J dyi

f [J

*i

I

~

for instead of

x =

(yi'YIN) E

,

b ,

and, moreover, there exists a domain

G

n

C R N

~

c4

[f

o

E (O,A:':)}

such that

and

c3 [ Iun (a 1 (y i)

I ::~II 1

U* d S

p, l' 1

a

1

imply that

YIN) I

P

+

lu~(al (yi)

- YIN) IPJ

~

A'"

A*

is contained in the set a 1 (Yi) - YIN

~

U~ and we obtain similarly as in (19.81) that

11:::II:,n'd S

n

P S J lu n (t)I t dt .

o

0

P

(x)

c 2

E U

C~(O,A*) v

d~(yi'YIN) dy lN J dy~ ~

1

The properties of the functions

4

n \ U~

A*

P

lu n (a 1 (yi) - YIN)I

a (yi)-A

dV

~(yi) u n (a 1 (yi) - YIN)

E

I~(yi) I [ J

~1

(z')

1 ) yield

,

A*

n we define

Obviously, the support of

?~4

~

'

J

,

is the function from (19.76), (19.77) with

and consequently,

(19.80)

N-l

K =

,

such that

o

and (19.15) (with

=

~1

1:* ,u

U* d S

p, l' 1

al(Yl)

(19.81)

a 1 (yi)}

<

(19.78»

(d.

in Subsection 19.3. Without loss of generality, we can suppose that the set {(yi'YIN); IYil

llvnll

1

S lu (t)I P t dt + n

J o

lu~(t) IP

t

S

dtJ

hence

supp v n C Gn C Gn C U*l '

285

--_....... ..

::--------_.....

...,

~----_.-_.

,_..

--_..

>..*

or

>..*

s [J °

Since the first integral on the right-hand side

f lu~(t)IP t dt] . ° can be estimated by the

second (cf. Example 8.21 (ii) with

= q ,

P Ilvnll S S 1,p,n,d ,d

;;;

b

have from (19.76) (with

=

C

S

lu n (t)I P t S dt +

b

>..* ,

=

p

a

=

S ) we finally

(19.86)

c

6

independent of

Theorem 19.22. For

f

c7

ly~I<6f'/2 [cf. (19.81)J with

c

8

>

°

Uf ' d a

=

we have

W1 ,P(rl'd S dS) 0""

o

19.25. Remarks. >.. -i,

[J Iun °

(t) I q t a d

independent of

t]

,

dYl

n.

c8

f

(i)

The necessity of the condition (19.87) cannot be

proved in the same way as in the case of necessity of the conditions (19.85), lu (t) jq t

a

n

dt

(19.86): If we used functions

defined analogously as in (19.79), we n 1 would not be able to guarantee that they belong to W ,P(rl;d S ,d S) since for

°

S ;;; -

v

the inclusion

Cco(~)C W1 ,P(rl;d S ,d S)

{v J , n

W~'P(n;dS,dS) , is unbounded in Lq(rl;d a ). Consequently, W~'P(rl;dS,dS) into Lq(rl;d a ) cannot be continuous, and

which is bounded in

the less so, compact.

°.

~

From (19.82), (19.83) and (19.77) it follows that the sequence the imbedding of

1 >

>

>..*

~

1.P +

q

(cf. A. KUFNER [2J, Remark 11.12 (ii», and the result follows from Theorem

ly~1 < 6*/2 , we obtain

for

p, l' 1

(19.83)

p

S;;; - 1

'"

~(y~)

Ilvnll

q

W1 ,P(rl'd S dS)

q

q,rl,d a

1

In the cases (19.85) and (19.86) the proof is analogous to that of

19.22. Ilvnll

-CI. - -S + ­ 1

S ;;; - 1 ,

(19.87)

n.

Using the fact that

a > -

1 ,

or

Proof.

with

1 < S ;;; p -

>..* ) that

Ilv n liP S S c

l,p,n,d ,d S - 6

(19.82)

-

o

does not hold. (ii)

We have derived necessary and sufficient conditions only for

O nE C ,l , i.e. for

K =

1 . In the case

°<

K

< 1

it is possible to find

necessary conditions for the validity of the corresponding imbeddings (by 19.23. Remark.

In the proof of Theorem 19.22 we in fact have shown that either the imbedding of W~'P(rl;dS,dS) into Lq(n;d a ) is compact or it is not even continuous. The same is true for the imbedding of W1 ,P(rl;d S ,dS)

the same methods as in the proof of Theorem 19.22; moreover, also for the case

p;;; q ) but the conditions are different from the sufficient ones.

(see the following theorem). 19.24. (19.84)

Theorem.

Let

rl E CO, 1 ,

;;; q < P < co,

1 W ,p (rl;d S ,dS) ~

c.;.

S

>

P - 1 ,

S E R . Then

20. UNBOUNDED DOMAINS 20.1. Introduction.

Lq(rl;dCl.)

In Section 17 we derived general criteria for the

continuity and compactness of imbeddings of weighted Sobolev spaces into

if and only if either (19.85)

CI. ,

-CI. - -S + q

p

1 q

1 p-

+

1 >

°

weighted Lebesgue spaces (Theorems 17.8 and 17.10). In Section 18 we reformulated these general results in terms of the weight functions; an important role was played by the distance

d(x) = dist (x,3rl) .

286 287

unbounded domains and the

Here we will deal with a special type of role of

d(x)

will be played by the function

20.2. The domain for some

Q.

{x E

Ix I

Q;

is such a domain that

Ix I

{x E R ;

:>

n} .

by the function

(20.3)

that the ball

E:2J

Q

Q =

R

KC

if there exists a compact set N

R

=

G where

R

N

or

{x

n

K

rf>,

K

Q = R

E Q;

N

\

{OJ

or

Q=R

Q

N

Ixl

<

n

(20.7)

ensures

B(x,r(x)) (l Q3n " rf>

provided

[see Lemma 18.sJ, and we may introduce the following

Q

~

E

[instead of (18.2)J, define

[instead of (18.4)J and assume that

~

n

r = rex)

Q

by

n

satisfies (20.6) vo

'

' b ' b O ' b 1 ' about the O 1 ) remain unchanged [compare also the identical conditions

" ~

,

n

r, b

the 'old' ones, and therefore, the formulation as well as the proofs are left to the reader.

C Qn+" Ie

Q,

UQ

Q

r

r = rex)

Now, we will give some examples in which we will use the following

n=l n

Let

n:> n

the set

Qn

coincides with the

B(O,n)

Q

E

defined on

(20.8)

.

~

Qn

~ } Ixl

!iYl

a*

n, see (20.1)J a~d a constant

[for

(20.9)

-1 < ~ c c r = rex) r

x

for a.e.

E x

=

inf

put

{Ixl; x

E

Q}

~ 0,1

the set of all for a.e.

!lJ ,

x E.

and denote by

We will suppose that there exists

such that rex)

notation: For

Q

E 'J) such that

Q

N

= R

\

G

with

GEe O,l

Qn Theorems 18.11, 18.12 together with Convention 20.5 imply the following

E

nn

and

y

E

B(x,r(x))

.

results.

20.6. Example. 288

'fJ

The proofs of these 'new' theorems are literally the same as those of

20.3. The function

(20.6)

Qn

Q E

[instead of (18. 7)J. All other assumptions (about the ~eigh~ functions

numbers

int (Q \ Q )

complement of the closed ball

1

belongs to

relation

n ~ max (n,2) . This is the situation which occurred in Section 18 due

(20.4)

from Subsection 17.2 will be

n} ,

moreover, according to (20.1), for

~

which together with the

remain true if we suppose that

­

\G.

n

n

is now 'controlled' from above

20.5. Convention. All assertions formulated in Subsections 18.6 to 18.12

Again we have

cr

B(x,r(x))

r = rex)

(20.7) and (18.8)J.

Qn

Q

with those

{oJ ,

and we denote

a function

rex)

vI ' w , about the auxiliary functions Q

(20.5)

Ixl/3

to the condition (18.7)

played by (20.4)

r

convention:

E ~ . The role of the set

Q

such that

G is a bounded domain. Then

Q =

Let

and

N

\ K •

We will mainly deal with the following special cases:

K

and

Q [see (18.4) and (20.4)J, n

[see (18.7) and (20.6)J.

The important auxiliary function

:il>

in fact,

Q

- in the definition of the sets - in one property of

N

n}

:>

If we compare the assumptions about

of Section 18, we see that there are certain differences

This class of domains will be denoted by

(20.2)

20.4. Remark.

- in the classes of domains considered,

QC R N

Let us suppose that

x ERN.

'

~ ~ 2 ,

n En,

(20.1)

Ixl

Let

~ p ~ q <

00,

0;,

S E R,

Q E: ~

o, 1

,

a*:> 0 . Then

289

--------------

O~~=~~=

W1 ,p(lt; Ixl s - p , Ix) s) [WI,p(lt; Ixl s - p ,

C;

---'=--

-

----=---~-=--==-==-'-"'=""""=

~-----

Lq(lt; lx/a)

<; ~

S IxI )

-=---_._--­

-a - -S + -N - -N + q

Lq(lt; Ixl a )

J

p

[Here we set

q

rex)

-

=

p

~"'-~~:-====::-.=:===-..;;.~~~~~."~-="",-,,==,==,-,,,-~,:~==-:=~-~

--­

°,

.r_~
bO(x)

Ixll3

p

w (x)

,

b 1 (x)

v 1 (x) .

J

i f and only i f

N N

-q - - + 1 p

S N N a -q - -p + - - - + 1 ;;; ° q p

° ,

~

[N -q - -pN + I > ° ,

~ere we set

rex)

20.7. Example. For

x E It

~q - ~p + ~q - ~p + I

Ixll3,

Let

lx/a

bO(x)

;;; p ;;; q < 00,

b (x)

a, S, y,

i

°E

a

vI (x) '" Ixl

InYlxl ,

S

Ix IS-p

vO(x)

In

Let

Ix IS

.J

C;

P

R ,

ItE;j)°,1

,

,

a,~

> 1 .

~q

[Here we set

rex)

~

°

- ~ - ~ + 1 p + ~ q p

q

(ii)

p

q

<

° °,

-+ P

1

c;

~

S $

-

P

0

q

p

b (x) = e a1xl , 0

b (x) = e s1xl 1

.J

° rl E ~0,1 ) has guaranteed the validity of

1

.r q

~

p

;; °

a"

'" ° . Nevertheless,

can use the results from Section 18 since

alt '" {a}

for ~

0

It '" R

N

\

{O}

(see (18.2»

we and

Ixl : according to Example 18.15 (and Lemma 19.14) we d(x) '" dist (x,arl) N obtain for It = R \ {a} that W1 ,p(lt; !xIS-p, IxIS)~ Lq(lt; Ixl a )

~q - ~p +

1 > 0

i f and only i f

and either q

,

-

classical Sobolev (Kondrashev) imbedding theorems.

Lq(rl;w)

i f and only i f

a

J

~-~
This approach fails if N

w ,p(n;v O 'v 1 )

a Lq(Q; e / xl )

W ,p(lt ;V 'v ) and Lq(ltn;W) are for a* > 0 isometrically n O 1 isomorphic to the corresponding non-weighted spaces and we can use the

or

~ + ~

c; c.;

the 'local' imbeddings (18.18), (18.26) since the corresponding weighted spaces

Ct

=1

a q

-

(together with the assumption

- ~p + 1

e: "1)0,1. Then

Let us go back to Example 20.6. The condition

a* >

and either

~q

°,

r N N L---+l>O, q p

Lq(.Il;w)

i f and only i f

rl

00,

i f and only i f

20.9. Remark. WI,p(rl;VO'V ) I

<

[ W1 ,p(lt; e slxl , e s1xl )

~q - ~ + 1 ~

°Ix I

q

$

lnolxl .

Then

(i)

1;;; p

W1 ,p(lt; e six I , e s1xl ) CLq(rl; e a1xl )

0].

<

put

w(x) '" Ixl

or

20.8. Example.

-S +­ N P

q

~p

+ 1

<

°

N q

N

- - + p

~

°,

a S N N

- - - + - - - + 1

q

p

q

p

=

0 •

N The same result obviously holds if we take It = R . However, there is N B a certain difference: while the spaces W1 ,P(R \ {a}; /xI S- P , IxI ) and W~'P(RN \ {a}; IxI S- P , IxI S) are well-defined since the conditions (16.18)

290 291

and (16.19) are satisfied for every S E R , in the case ~ = R N, vO(x) =

S

S = !xI - P , v 1 (x) = Ixl , the conditions (16.18) are satisfied for S p - N . Therefore, when

1 N dealing with the spaces w 'PCR ; IxI S- p , IxI S) , W~'PCRN; IxI S- p , IxI S)

o¢~

I0/

(x)

p > q • Radial weights.

20.10. The case

Now we will consider imbeddings

Let

W1'P(~;vO,v1) L LQ (I1;w) 1 ~ Q < P <

00

We assume that

•

functions

v = vex)

(20.10)

vex) = v( Ix W(a*,oo)

E

~

and restrict ourselves to weight

of the type

j)

[for

see (20.8)J. Such weight functions are called

radial weights.

I~

an unbounded interval WB(r)

or

WeI)

P <

for

R

11 E ~

00,

k

0

>

VO' vI E Wc(a*,oo) . Suppose

,

and a number

for a.e.

t

oo c: (l1) = {g E C (I1); supp g s

C~

is a dense subset of W1'P(~;vO,v1) where Proof.

Moreover, we introduce two special subclasses of the class

(20.11)

1,2, ... ,N.

j

to ~ a*

such that

> to •

Then the set (20.13)

x E 11 ,

a*

~

vO(t) ~ k v (t) t- P 1

(20.12) 11

1

that there exist a constant

of the type

vE

1 ,2,

i

The proof is standard and is left to the reader. 20.12. Theorem.

with

for

K

S E (p - N, Np - N) .

we will consider

for

I ~

J

Let u

function

u

(20.14)

1 W ,p(l1;v 'v ) O 1

e

and fix

is bounded} v.(x) = v.(lxl) l

l

i

0,1

s > 0 . Then there exists a

E: Us

E c

oo),",l,p (~ I i W (l1;v ,v 1) O

v

WC(r)

denotes the class of all

vE

WeI)

such that

which are bounded from above and from

J~

below by positive constants on each bounded or each compact interval

I ,

s

(20.15)

iju - uEij1,p,I1'VO,v1 <

~

respectively. (cf. V. I. BURENKOV [IJ). We will make use of the following two auxiliary assertions:

Let

20.11. Lemma.

R

>

0 . Then there exists a partition of unity

¢R

{¢~,¢~} with the following properties: R

R

00

N

(i)

¢1' ¢2 E C CR ) ,

(ii)

supp ¢1 ~ B(O, R + 4) ,

o

(20.16)

Choose

f ( t)

~

be such that ~

f(t) = 1 R

n

>

(for

R

(20.17)

R,N --­ (iii) supp ¢2'R \ B(O,R) ,

292

f E Coo(R)

Let

on R N ,

(iv)

o ~ ¢~ ~

(v)

R R ¢1 (x) + ¢2(x) = 1

(vi)

there exists a constant

l

1

Further, for 11

i = 1,2 ,

for

Fh(x)=f [

The function

x ERN K

>

0

independent of

s

>

0

{x E

s

F h

1

for

for

t

n

t E R , ~

5/4,

see (20.1»

f(t) and for

o

t G 7/4

for

denote

h > 0

IXI-R] N h ,xER denote 11;

Ixl < s} ,

I1 s = int (11 \ 11 ) •

from (20.17) belongs to

s

N COOCR )

and satisfies

R such that 293

= 1 for x e

Fh(x) (20.18)

IkaF h

x E R N ,

for

o ;;; Fh(x) ;;; 1

(x)

I ;;;

c f h1

J

QR+Sh/4

with

= 2 1/p ' max {1,3c

U aQ 1,2, ... ,N

j

u

Ilu E

~u - u If we define

with

u

from (20.14) and u

(20.20)

F h

h E Coo(Q),

£,

supp (u

E

- u

E,

h)

h C B(O, R + 2h) ,

£,

>

~

(20.23)

1

E

EW ,P(Q;v 'v )

u

E,

oo hE C (Q) bs

according to (20.20).

o

1 ~ q < P <

(R,

00,

0\0,1 QE ~ ,

00,

R >

n

-­ w, v E WB(a*,oo) .

such that

N-1 N-1 w(t)t , v(t)t , q,

p)

<

00

max {R, to - R}

where (20.21)

[fluE - uE,h lP Vo dXJI/ Q

~ [J

P

luEI

P

V

o dx

J l/p

[fla~~ Q

;;

P

(u E - uE,h)

~

[J Ia/I au I p vI

P

~

W1 ,p(Q;v,v)

with

=

w(x)

Proof.

luEI

<;

~ Lq(Qjw)

~(Ixl)

Ik(l a

1P dXJ / +

-

Fh )

Ip vI

dx

riP

J

/uEI P h- P vI dx

{u

e

Coo(Q);

W1 ,p(Q;v,v)

!u~l,p,Q,v,v

< oo}

(see V. I. BURENKOV [lJ). Due to this density and

to Theorem 17.6, it suffices to verify that

J

cf [

v( Ix I) .

The set

is dense in

;"

vex)

riP

(20.2S)

~

lim sup liIull n ; u E V, n+ oo q,Q ,w

Ilu!1

1,p,Q,v,v

~ I}

o

QR+h\QR+2h p +3c f

J

[J I::~IP QR+h

P

QR+h

lau jP il/ a/ex) v1([xl)dxJ

[f

K

l(f

il/P dxJ +

J

QR+h

~

dXJI/

J

I::~IP vI

[f

(20.24)

v=

QR+h ~

vI

J

QR+h (20.22)

I

A is given by (1.19). Then L

QR+h

and

294

U

E

Suppose that there exists a number

These properties together with (20.18) and (20.12) imply that for

h

£,h ~ l,p,Q,v ,v < O 1

Let

20.13. Theorem.

R+h Q .

C

. Since

Now we are able to prove some imbedding theorems.

from (20.17), then obviously

supp u

= 1,2, ... ,N

j

E,h II l,p,Q,v ,v < ~2 , O 1

Thus our theorem is proved since

x E Q ,

u E, hex) = u E (x) Fh(x) , E

f

which together with (20.1S) yields

supp FhC B(O, R + 2h)

(20.19)

k- 1/p } ,

O 1 according to (20.14), the estimates (20.21), (20.22) and the properties of nR+h imply that there exists a number h > 0 such that

x ERN,

for

K

.J

r

vI dx +

..

j

oR+h

lu E /P

V

o

[J

V1 (jxj)] !uE(x)I P pdx ~+h Ix I

dx ll/P I

= Qn+S . [Note that (20.2S) is the condition (17.1S); the condition (17.14) is satisfied due to the assumption ~, v E WB(a*,oo) .J

where we put

l/p ;.;

Qn

~u~l ,P,H,V,V n ~ 1

and let

{ep~ , ep~}

unity from Lemma 20.11 with a fixed

nEl',

n >

Let

u E V,

H ,

be the partition of H

from (20.23).

Then

J 29S

u = u

+ u

1

where

2

u

U¢~

i

1 ,2 ,

i

1

supp u C B(0,n+4) , 1 supp u

2

C

R

N

\ B(O,n)

q q,Qn,w

IU 2 (x) Iq

Ilu 2 (t,8)

8

n

1

I

w(x) dx ;:;

lu (x)l 2

q

w(x) dx

Iq

A ( H, R

N 1 w(t) t - dt d8

00,

that

RN\B(O,n)

N-1 , vI - (t)t N-1 , q, P) w(t)t

00,

w1 ,p(n;v 'v ) O 1 and

u (n,8) = 0 for every fixed 2

dimensional inequality (cf. Theorem 5.10) implies

' we have 2

8, and-the one­

C;

u

wi th

<

00

,

is given by (6.7). Then A R

where

{x ERN; Ixl = I} . According to the definition of

u 2 (·,8) E Coo(n,oo)

,-:\ 0,1 , w, - v- ' vI ­ E WB(a*,oo) Let 1;:; q < P < n E 'ou o such that (20.12) holds. Suppose that there exists a number H > n such 20.14. Theorem.

RN\ B(0,n+5)

I

0

lu(x)1 q w(x) dx

R N\ B(0,n+5)

i A~

lim An = 0 , and consequently, the n->-oo

condition (20.25) is satisfied.

I

I

(20.26)

=

v (x) dx riP ;:; c

J

Nq[~_l(Sl)](P-q)/q max {N q K q , I}

q c 1 = --

where

n

From (20.23) it follows that Ilull

81

P

(x)

J =1

and we have

with

II ~~. I

+.LN

~ Lq(n;w)

v 1 (x) = vI ( Ix I)

v0 (x) = V0 ( Ix I) ,

~ (Ix I)

w(x)

The proof is similar to that of Theorem 20.13, only instead of Lemma 20.11 Ilu 2 (t,8) I

q

N 1 wet) t - dt ;:; A~

OOI (

n

I~(t,8) dU 2 Ip

N 1

v(t) t -

dt

] q /p

n

we use Theorem 20.12. Then we can work with functions vanish for

g(t) = u(t,8) = 0

with

_

An-q

1/ q (p) I 1/ q I ,\n, (

00

-

,w(t)t

N-1

N-1 ) ,v(t)t ,q,p.

in

20.11 (iv), (vi) that

Ilull

q ;:; q,Qn,w

;:; [rn...

N-1

;:; [~

A~

I 8

1

(fl:: I 2

P

(t,8)

r

(8 )](p-q)/q Aq -lIn

20.15. Theorem.

Let

w, VO' vI E WB(a*,oo) mIl dU 2 (t,8)

8

n

q N

at

IP

vet) t N-1 dt d8 ] q/p "

P

KP Ilu(x) IP vCx) dx +

n

" ~

(20.27) ~

where (N

t

near infinity. Thus we can use the one-dimensional

g E ACR(n,oo) .

The following two theorems will deal with general unbounded domains R N . We again define a* by (20.8) and nn by (20.5) and consider

N 1 vet) t - dt]q/P d8 ;:;

I 1

for

radial weight functions.

n

(8)] (p-q)/p Aq 1 n ,

which

sufficiently large (see (20.13)), and consequently,

Hardy inequality for

Using this estimate in (20.26) we obtain by Holder's inequality and Lemma

(20.28)

with 296

Ixl

u E c~s(n)

voCx)

(a*,

00,

1;:; q

<

p

<

00

•

Let

nCRN be unbounded,

and N-1 , v- (t)t N-1 , q, p ) w(t)t 1

<

00

,

is given by (8.98). Then

w~,p(n;vO,v1) C, Lq(n;w) voclxl)

v 1 (x)

v (lx!), 1

w(x)

w(lxI) . 297

·~.c

~.

- _..

~ E WB(a*,oo)

Moreover, let and let

(20.29)

_~.

._ _

_:_!!!!!!!!~~~

be decreasing in

(H,oo)

for some

H

>

in view of the monotonicity of

a*

lim ~(t)

o . 20.16. Theorem.

W, v

Then (20.30)

w~,p(n;vO'V1) C; ~ Lq(n;wA)

i

R

N

u E c~(n) . Extending

and introducing the spherical coordinates

get) = u(t,G) E C~(a*,,,,)

we have that

for every fixed

(t,G),

f J lu(t,G)

q r

N 1

wet) t -

vex) = v(/x!)

with

Moreover, let

dt dG

w~,p(n;v,v)

with

=

A(X)

to verify that

1

2

lim

where

n+'"

Take

u ex, lul

o,

where

x

w~,p(n;vO'V1)

n

>

q

n = q,n ,WA

Lq(n;wA)

II ull x;;; 1

n

q,Q ,w

< 00 ,

l,p

X= W

nEE,

o

(n;v,v) .

H be the number from our assumptions and let uE

c~(n) ,

{¢~'¢~} lull x ;;; 1,

be the nEE,

H . Then we have q Ilul q,Q n,w ; ; J

H . Then

>

satisfy the assumptions of Theorem 20.15.

partition of unity from Lemma 20.11. Take

x

n

c;. G.

II ull

sup

Qn = nn+5 , Let

similarly as in the

that

sup lui lul ;;;l q,nn,wA

lim n+ CO

O 1

In order to obtain (20.30) it suffices to show -

WB(a*,oo)

First we will prove (20.34). According to Theorem 17.10 it suffices

(20.36)

II 1,p,n,v ,v

Consequently, we have proved (20.28).

proof of Theorem 20.13

w(lx!)

=

i(lxl) .

arrive finally at the estimate

; ; c Ilu

iE

(20.35)

Proof.

liull q,n,w ; ; c1 11~~ll otlp,n,v

w(x)

Then

by the one-dimensional Hardy inequality according to Theorem 8.17 and

(20.32)

<",

w~,p(n;v,v)C; Lq(n;w)

(20.34)

G. Due to

51 a*

(20.31)

11 ( H, 00, w(t)t N-1 , v(t)t N-1 , q, p ) Tt

u

(20.27), we can estimate the inner integral in

Ilull~,n,w =

1"

~=

N be unbounded, Let 1 ;;; q < P < 00 • Let nC R . Suppose that there exists a number H > a* such that

Then

Proof· Using the density argument we can consider by zero to the whole

E" WB (a* ,00)

(20.33)

A(X) = ~(Ixl) .

with

0

now follows by (20.29).

t+",

A and of (20.31). The condition (20.32)

f

1u 2(t,G)

r

q w(t)t ­ N-1 dt dG

51 H

f lu(x) Iq w(x)

I(lxl) dx ;;;

with

u

H

2

u¢2

(cL (20.26», and since

nn ;;; A(n)

f

lu(x)

I

q

w(x) dx ;;; i(n)

c~ lul~

J u 2 (t,G) 1

I

q

N

w(t)t -

1

dt ;;;

H

nn

298

299

~

c

It q

[Ji~~2(t,e) IP vet)

JIg~ (t) IP

t N- 1 dtf/P

H

from (20.33), we obtain analogously as in Jlgn(t)

the proof of Theorem 20.13 the estimate < =

cIA-­

<

c1

C~(~)

independent of

in

eR

n

o

foregoing theorems we have derived sufficient conditions for

Let 1 ~ q < p < 00, ~Ef1J, ;OEW(a*,oo), ~';1EWc(a*,00). Suppose that there exists a number R > n such that 20.17. Theorem.

(20.40)

J1un(X)

<

q

w(x) dx

I

-+

00

•

e RN \

for

x

B(O,R) ,

for

xE ~nB(O,R) .

JrIgn(t)

J

Iq

wet) t N-1 dt de

for

-+

00

n-+

oo

VI (x) dx

~

~

J[g~(t) IP v1 (t)

J

N 1 t - dt de

~-1 (S1)

Sl R

00 nER.

On the other hand, it follows from (20.37) that 00

W~,p(~;VO'V1) C;

f Ign (t) Ip v0 (t)

Lq(rl;w)

N 1 t - dt

vo(!xl) LI ( vv R,

00,

- - (,x I I) v (x)-v 1 1

I I) ,

w(x)

N-1 ,v - (t)t N-1 ,q, P) w(t)t 1

<

w( x

then

00

(20.42)

Proof·

Suppose that (20.39) is not satisfied. Due to Theorem 8.17 and the condition ~, ;lEWC(a*,oo). the corresponding one-dimensional Hardy inequality 00 co /q q N r ~ C Igl(t) jP v (t) t N- 1 dt J lip (J Ig(t) I wet) t -1 dt 1

)l

R

[J

C independent of

with

-

[Iu n liPp, ~ ,v

C) on the class

C~(R,co)

(cf.

C

f Ig~ ( t) [P v1(t)

t N-1 d t

o

n, and consequently,

N \1 dU liP ~ Co i=l L I~I n ~ Co Xi p,,,,v 1

Nm.,_ 1 (Sl) N

due to (20.41). By virtue of the estimates (20.42) and (20.41), the sequence {un}

c=

C~(~)

c=

W~,p(~;vO,v1)

unbounded in

Lq(rl;w)

cannot hold.

0

R

does not hold (with a finite constant

~

R

R

vO(x)

(20.39)

n

Sl R

JI::~(X) \P

(20.41)

If

with

for

while

for every

(20.38)

00

and

~

~

N-1 vtr2 ( R, 00, vO(t)t ,vI (t)t N-1 , p, p )

o

unEC~[RN \ B(O,R»)

Then

J.3 is given by (8.69).

where

-+

put { gn (I x I)

the corresponding imbeddings. Now, let us give a necessary condition.

(20.37)

N 1 wet) t - dt

un(x)

The step from (20.34) to (20.35) is the same as in the proof of

In the

Iq

n. This estimate together with the density of

X implies (20.36) and thus, (20.34) is proved.

Theorem 20.15.

nER,

1 ,

dt

R

00 For

with

1

R

due to Theorem 8.17 with ~

fluil q,Qn,w

N

v 1 (t) t -

is bounded in

W~,p(~;VO'V1)

, but it is

due to (20.40). Consequently, the imbedding (20.38)

Remark 19.19), and consequently, there exists a sequence of functions gn E C~(R,oo) 300

such that

20.18. Remark.

The reader can easily see that Theorem 20.17 remains true 301

for

1

~

p

(20.43)

~

q

j)

(R,

<

00

,

provided we replace (20.39) by

- N-1 "', w(t)t , vI (t)t N-1 , q, p)

(i)

BI

If

p - N , then the following three conditions are equivalent:

Wl,p( Q;v 'v ) o O 1

< '"

W1 ' P (Q;v 'v ) r: '7 Lq (Q;w) O 1 O

On the other hand, in Section 18 we have derived necessary conditions for (20.38) to hold without the (restrictive) assumption (20.37)

(. q( ~ L Q;w) ,

C;

,

(see, e.g.,

Theorem 18.9).

B

a

N

N

-q - -p + -q - -p +

°

1 <

Now we will apply the foregoing theorems to some special weight functions.

a - -B+ -N - -N+ q P q P

20.19. Example.

Let

;;; q <

P

a, S E R .

< '"

(ii) (i)

Let

E

Q

~,

B#

a* > 0 ,

p - N . Then the following three

S p Ixi - ,

B IxI )

~ ~

w~'P(Q;

B p Ixl - ,

B IxI )

<;

a q

a Ixl )

Lq(Q;

B

N

N

a

p

q

p

q

Q

E~O,l,

a* > 0,

B

P - N . Then the following three

>

IxI B)

<;

1 w ,P(Q;

B p Ixl - ,

IxI B)

~

~ + ~ - ~ + 1

q

(iii)

W~'P (Q;

p

Let

Ix I B-p,

q

Q = R

Ix IB)

p

n

c:

Lq(Q;

Ix[a)

Ixl ) ,

or

Q = R

N

,

Let

B 1- p - N . Then the space

is cont inuously imbedded into

;;; q < p < "',

Q

E ,'j),

Lq (Q;

a* > 1,

and put w(x)

Ix Ia In Y Ix I '

8

1xl

In <5 I x

I .

B p

v 0 (x)

Ixl

vI (x)

Ix I B

-

< 0

;;; q <

Let

P

q

q

P

.!.

p

< '" ,

< 0 •

QEfllo,l ,

I

Ix a)

In

(i)

w~'P(Q;

eB\x l , e B1xl )

c:; (.

Lq(Q; e alxl )

(ii)

W1,P(Q; e B1xl , eB\x l )

c: G

Lq(Q; e a1xl )

(iii)

w~'P(Q;

G Lq(Q;

(iv)

w1 ,P(Q; e B1xl , e B1xl )

(v)

-

for no

e B1xl , e B1xl )

a, BE R ,

B

t-

°.

c;.

e

alxl

) ,

Lq(Q; e Cl1xl ) ,

a, B, y, 8 E R a q

-

B

-

p

<

0 .

If we weaken the conditions on

302

-Y - -8 + -1

°,

p

the following five conditions are equivalent:

< 0 •

" { o}

q

20.21. Example.

a

Lq(Q;

B > P - N , then the following

,

aER.

20.20. Example.

p

q

IxjB-p,

a

-B + -N - -N + 1 P q P

~-~+~-~+

conditions are equivalent· 1 W ,P(Q;

and

0 .

,

- + - - - + 1 < 0 .

(ii) Let

~ 0,1

<

Wl,p(Q;V 'v ) ~ G Lq(Q;W) ,

O 1

1,p( w Q;v 'v ) c '7 Lq( Q;w ) ,

O 1

a Ixl ) ,

Lq(Q;

Q E

.!.

p

conditions are equivalent:

conditions are equivalent:

w~'P(Q;

If, moreover,

Y 8 --+­ 1 q P q

o,

conditions (i), (iii),

(v)

Q

and suppose only

Q E ~ , then the

are equivalent.

303

20.22. Remarks.

(i)

The results of this section are due to B. OPIC,

P. GURKA [ 1] .

(A)

n bounded (in most cases

v(x) = d S (x)

(ii) Let us go back to the estimate (20.42). Its first part is in fact the N-dimensional Hardy inequality for our special functions

u

implies that for these functions, the norms

and

'"u n III 1,p,lO,vl ()

(see (16.24»

Ilu n 1 ,p,rI'VO'Vl I1

n

and

(B)

w

and

w(x)

dCl.(x) ,

d(x) = dist (x,arl)

unbounded ( "

n

v

with

°

nEe ' 1 ) and

or

E fiJ

Q from some special subclasses of ;j) ),

radial weights.

In both cases we will make use of Lemma 16.12, and therefore, we will

are equivalent. As was mentioned in Section

start with some assertions about equivalent norms. First, let us consider

16 (see Lemma 16.12), the problem of equivalent norms is closely connected with the validity of the N-dimensional Hardy inequality. We will deal with

the case (A). Here the question of equivalent norms is solved by the

following lemma.

this question in the next section. 21.2. Lemma (A. KUFNER [lJ, Proposition 9.2).

;;; p <

Let

00

Le t

Q

E CO ,K

,

S E R . Then the norms

;;; 1,

21. THE N-DIMENSIONAL HARDY INEQUALITY 11·11

21.1. Introduction.

At the very beginning of this book we have stated our

(21.1)

[N Iiaa~. (x) I P v i (x)

[JrI Iu (x) Iq w(x) dxJl/q ;;;

C .~1 1-

Q

dx

(21.4)

Illulll

(21.5)

Ilull

JlI P .

1

1,p,Q,d S

S = [f lV'u(x) IP d S (x) dx f /p , 1,p,Q,d

n

P S S = [flu(x)I P dS(x) dx + IIIulli sf/P 1,p,Q,d ,d 1,p,Q,d Q

Some partial answers have been given in Chapter 2; moreover, Lemma 16.12 enables us to use also the results derived in Chapter 3. Since we

III • III

where

intention to describe conditions which guarantee the validity of the N­ dimensional Hardy inequality

and

1,p,n,d S ,d S

W~'P(Q;dS,dS) provided

are equivalent on the space

investigated in Sections 18 - 20 mainly the case

~ < S < K(p - 1) 1 - K

vI

= v 2 = ...

v

N

v

,

[For

we can write (21.1) in the form -

(21.2)

[J I

I

u (x) q w (x)

dX] 1/ q

;;;

C[Jll7u(x)I P

Q

lip v(x) dx

Q

J

00

<

S

< P -

1

so that the last condition reads

K)

Kp/ (1

we put

K=

.J

Lemma 21.2 together with Lemma 16.12 and the results of Section 19 (see Theorems 19.10, 19.21) imply

where (21. 3)

Il7u(x) I

P

=

a Ip L I~(x)

N

i=l

21. 3. Theorem. Let .

0.,

xi

validity of (21.2) on some classes following cases:

304

K

containing

C~(")

mainly in the

N

00

,

N P

-q - - + 1 ;;;

°,

Q E CO,K ,

°

< K ;;; 1 ,

S E R , and either

°

We will establish simple necessary and sufficient conditions for the

1 ;;; p ;;; q <

< S < K(p -

1) ,

~<S;;; 1 - K

°,

~ - f + ~q - ~p + K ;;; q p

°

or 0.

q

-KS +­ N P

q

-NP +

K ~

-

°. 305

Then

the~e

exists a finite constant

'J

)

~

llu(x) ,q da(x) dxJ 1/ q

(21.6)

I

Let

21.4. Theorem. eithe~

°

<

S

<

K

hold s

I)

°

E CO,K

p

q

lp + 1)

q

p

K

::;;

1,

a, S E R

a q

~<S::;;o, K

°

>

the~e

(21.6) holds If

fo~ eve~y

rlECO,l

~ - ~ + !'! - !i + 1 ;;; q

c such that the

Ha~dy

21.7. Theorem.

inequality

domain,

exists a finite constant

fo~ eve~y function 1 :;; p

~

q <

S

< P -

1 . Then

00

~q - !ip +

,

if and only if

1 ;;;

°,

< P <

00

a

S

N

N

-q - -p + - - -p + 1 ;;; q

B

a

1

1

-q - -p + -q - -p + 1

,

::;; q < P <

00

•

from Subsection 21.1, i.e. un­

(B)

Let

1::;; p <

00

>

•

Let

rl

C

R

N

\

be a non-empty unbounded

{O}

'" ( a*, "', vO(t)t _ N-1 , v- (t)t N-1 , p, p ) < :}) 1 C > 0

00

•

such that

(21.8)

II u II

fo~ eve~y

u E

rl S C Ill1u 11 n p, 'Vo P,H'V1

eithe~

°

o~

::;; q

°.

the~e

C such that the Ha~dy inequality (21.6) holds

u E W~'P(rl;dS,dS)

p

The following two theorems form a counterpart of Lemma

Then the~e exists a constant rlEC O, l ,

l::;;p,q
q

va' VI E W(a*,oo) . Suppose that

(21. 7)

then we can give necessary and sufficient conditions:

Let

21.5. Theorem.

p

21. 2.

u E W~'P(rl;dS,dS)

function

if and only if

P

q

exists a finite constant

°,

Now we will consider the case

KS+ K(1 - - - 1+ 1 ) >0. P

KC r.

derive analogous results for the case

bounded domains. - 1 -

B [(p,
Here we have used Theorems 16.3 and 21.5. If we use Theorem 16.5, we can

<

o~

Then

!'! + 1 ;;;

qN _

~ + K(l

a

1) ,

(p -

[l

00

~

for every compact set

u E W~'P(rl;dS,dS)

:;; q < P <

dx J 1/q

K

rI

holds fo~ eve~y function

[J da(x)

inequality

C [Jlllu(x) P dS(x) dxf/P

rl

and

Ha~dy

C such that the

°

I

W~,p(rl;VO,v1)

consequently, the (21.9)

with

vO(x)

vo(lxl) ,

v 1 (x)

V1 ( Ix

I)

[and

no~s

Illu 11l 1 ,p,rl,V1

l/ P

(J [l1u (x) Ip v 1(x)

dx }

rl

21.6. Remark.

As was mentioned in Subsection 16.1, the Hardy inequality

provides a useful tool for deriving estimates for capacities. Using the

and (21.10)

foregoing results, we can specify the isoperimetric inequality (16.5) and obtain that for

l::;;p::;;q<"',

lI

, N

(x,O =

i=l

Ie IP

rlECO,l,

S
and

lip

i(x»)

there exists a finite constant

B >

°

such that the inequality

[f

iu(x)/P vO(x) dx +

a~e equivalent on the space W~,p(r.;vO,v1)

J.

Using spherical coordinates, the condition (21.7) and the one­

dimensional Hardy inequality, we can derive the inequality (21.8) by the same method as we have derived the first inequality in (20.42) in the proof of Theorem 20.17 from the condition (20.37).

306

IIi u lll P1 ,p, rl ,VI Jl!P

rl

P~oof.

1

Ilu 11 1,p,rl'VO'V1

o 307

In the next theorem, we will deal with domains

rI E

:;u

[r1E~O,IJ

which satisfy the condition

[J !u(x) Iq w(x)

(21.14)

dXJ l/q ::; C [Ill7u(x)I P v l (x) dxf/P rI

rI (21.11)

x

E.

rI,

t

~ tx E rI .

1

>

("I'

(21.12)

jJ*

[Ci\ J,.J

0, lJ

*

Let

1

. $

that there exist numbers -

vO(t)

~

p <

00

,

k > 0

-

k vI (t) t

-p

rI E

and

vo' vI G WC(a,,,,oo)

V* ' to ~ a*

for a.e.

t

>

such that

(i)

1

p,q

$

0:,

00

SE R

!xI S- P

vo(x)

[n E .f) ~' 1] ,

n E: 'iJ

Let

<

a*

and put Ix IS

vI (x) >

0,

S-I-

p -

[S

N

>

P - NJ .

Then the Hardy inequality (21.14) holds with a finite constant C on the l class K(n) = w~,p(n;vO,vl) [K(n) = w ,p(n;v O'v l )] if and only if either

to .

N-l ,vI - (t)t N-l ,p, P) BR (a*, 00, vO(t)t

Let

w(x) = [x\O:

and Suppose

Assume that (21.13)

K (n) , which will be specified in the

following examples. 21.10. Example.

21.8. Theorem.

K

to be valid on the class

This class of domains will be denoted by

$

P

$ q <

N 00

,

-N +

q

<

;;; 0 ,

P

0:

q

-B + --N P

q

~ + P

1 ;'; 0

or

Then there exists a constant C > 0 such that the inequality (21.8) holds for every u e Wl ,P(D;V 'v ) with vO(x) = "o(lxl) , v (x) = "1(lxl) [and O l l consequently, the norms (21.9) and (21.10) are equivalent on the space l

W ,p(rI;v 'v )] O l

Proof·

Due to our assumptions, it suffices to prove the inequality (21.S)

only for functions

u

from the dense subset

C;s (rI)

(cf. Theorem 20.12). For such

u , we proceed analogously as in the proof of Theorem 21.7.

1 ;'; q < p <

Let

(11)

(21.15)

$

(21.7) and (21.13) are sufficient for the equivalence of the norms (21.9),

l

(21.10) on W~,p(rI;VO'Vl) and W ,p(rI;V 'v ) , respectively. Obviously, O l these conditions are also necessary if we suppose that the domain has the special form

with some Q

=R

N

•

r

{xE R ; ~

Ixl

>

0 . In the case of the space

Let

S, y,

Using Theorems 21.7, 21.8, Lemma 16.12 and the results from Section 20

S 1- p - N

N

N

- + 1

00

q

= RN ,

Let

°E R, w(x)

(see Examples 20.6 - 20.8, Remark 20.9, Examples 20.19 - 20.21) we immedia­

n

.Y~~ Examp~.

r} l W ,p(rI;V 'V ) , it can be even O l

\ {O},

[B > P - N ]. Then the Hardy in­

K(n) =

P

P - N

~

o,

a q

-B +­N P q

N

o .

- + P

B < Np - N . Then the Hardy inequality

<

(21.14) holds with a finite constant C on the class K(rI) = w~,p(n;vO,vl) l or K(rI) = W ,p(O;v 'v ) if and only if the condiiton (21.15) is satisfied. O l

0:,

rI

N

$ q <

P

o

In Theorems 21.7, 21.8 we have shown that the conditions

N

= R

1 < 0

equality (21.14) holds with a finite constant C on the class W~,p(O;vO,vl) [K(n) = wl ,p(n;v O'v l ) ] if and only if

(iii) 21.9. Remark.

n

~ - f + ~ - ~ + q p q P

00

$p,q
S f- P - N =

[S

Ix [a InY11xl,

>

n~j)

[ n E 1) ~ ,1

J,

a,,,

>

1 ,

P - N ] and put

vO(x) = IlxlB-p lnolxl,

VI (x)

=

Ixl

B

InO[xl

Then the Hardy inequality (21.14) holds with a finite constant C on the K(n) = w~,p(n;vO,vl) [K(n) = wl ,p(n;v 'v l ) J if and only if one of class O the following two conditions is satisfied:

(i)

1 $ P :;: q <

N 00

,

q

t-J

p

+ 1 ;;; 0

tely obtain necessary and sufficient conditions for the Hardy inequality 30S

309

and either

another method, using the theory if fractional integral operators in a.

q

§. + !J

p

q

!J+

l
S

N

P

!J +

1

q

p

weighted Hardy spaces.

or a.

- +­

q

p

;;; q < p <

(ii)

y

o,

~ ;;; 0

q

21.14. Concerning equivalent norms.

p

will summarize the conditions which guarantee the equivalence of the norms (21.9) and (21.10) for the particular weights appearing in Examples 21.10,

00

21.11 and 21.12. Some of these conditions can be derived directly from the

and either ~_'§'+!J q

p

!J + p

q

Hardy inequality, since (21.8) is nothing else than a special case of (21.14)

1 < 0

-a - -S + ­ N q P q

!J + p

Let

21.12. Example.

-a + -1

1

1 = 0 ,

1;;; p , q <

w(x) = ea.lxl ,

P

q

r2 E

00,

1

q

p

'lJ,

a. ,

S < 0

.

a",

S > 0

;;; p <

1

(i)

Wb,P(n;v 'v )

=

a*

=

0,

S

0,

<

C on the

q

a.

§.

~ - .§. ;;; 0

q

p

<

P

<

00

q

p

<

0 .

on

Wb,P(n;v 'v )

if

l

O

n E :JJ,

a*

> 0,

n E q; ,

a*

>

0 ,

S

RN \ {O},

S

# p -

n

S # p - N,

a.;;;

S - p

p - N ,

a. < -

N

n

E J)~,1

, replace (21.16) by

S

> 0

[/ = R

and consider (21.14) on the

1, p (0' ) . W n;v 'v

O l

21.13. Remark. vI (x) = Ixl

S

The inequality (21.14) for the special case

(i-2) w(x)

=

Ixl

=

N,

S-

p

a.

S-

a. =

or

The same conditions are necessary and sufficient also if we suppose

on

N

,

S E (p

N, Np - N)

W1, p ( n;v ,v ) O l

r2 E ;I)*,

Cl

a*

> 0,

,

=

p

if

S > p - N,

a.;;; S -

p

,

appears very frequently in the literature, mostly in

or

n = R N \ {O},

connection with estimates used in the theory of partial differential equa­ tions. From the numerous results let us mention at least the paper by A. E. GATTO, C. E. GUTIERREZ, R. L. WHEEDEN [IJ who derived exactly the necessary N and sufficient conditions (21.15) for r2 = R \ {a} and K(r2) = C~(r2) by 310

a., S E-R ,

or

1 ;;; q

class

,

p > N •

or

that

IxI S

vI (x)

or

~p + 1 ~ 0,

N 00

For the weight functions

if and only if either

O l

;;; p ;;; q <

Cl,

the norms (21.9) and (21.10) are equivalent

or

Then the Hardy inequality (21.14) holds with a finite constant

K(n)

However, the set of admissible values

00

vo(x) = Ixlo.,

(i-I) class

o.

Suppose

S E R • Pu t

S "I 0

> 0

V

assumptions of Examples 21.10 - 21.12.

v (x) = v (x) = e s1xl

o

w =

and

is in fact bigger, including also values excluded apriori in the

(y , a )

Suppose that one of the following two conditions is satisfied:

a;, = 0

q = p

where we put

or

(21.16)

For the convenience of the reader, we

13 > p - N,

a.

= S - p

or

n

= R

N ,

S E (p - N, Np - N),

a. =

S - p .

311

(ii)

Q

For the weight functions

vI (x) = Ixl B lnoixl '

lx/a lnYlxl ,

vO(x)

a, B, Y, 0 E R ,

the norms (21.9) and (21.10) are equivalent

W~,p(rl;VO,v1)

on

(ii-l)

B,

p -

N,

a <

B-

B

ex :;:;

a* > 0 ,

1) ,

(a; B) i

(0;0)

or

n = R N \ {O} ,

u. < 0 ,

a :;:; B ,

\ {a} ,

B > 0 ,

a :;:; B

\ {O},

B

p > N

or

n E fj),

if

e

a* > 1

and either

rl = R

N

or

p

rl

= R

(iii-2)

on

N

0,

~

p - N,

ex =

B - p,

y:;:; 0

or p - N ,

B

ex < -

W1 ,p(Q;v 'v ) O 1

rl E: ~* '

N

p-N,

B

a=-N,

o1p-1,

y:;:;o-p

or p -

N,

a

- N,

0

p -

1,

1 ,

N rl = R \ {O}

or

rl = R

N

and either

or 0,

B B

>

a :;:; B

B > 0 , or

N

if

or

a* > 0

1

1 < P < N if

or B

=

p

a < 0 ,

p

a
=

1

1 < P < N if

N

>

1 .

Y < - 1 21.15. Some extensions.

(i)

In this section we have been in fact concerned

with two special types of weights depending on W1 ,p(rl;v 'v ) O 1

on

(ii-2)

B

>

P - N,

d(x) = dist (x,3Q) if

rl E: 'i{) *

'

a* > 1

and either or on

Ixl = dist (x,{O})

a < B - p

It is possible to extend many of the foregoing results to the more general

or B

>

P - N,

ex =

B - p,

case of weights of the type

y:;:; 0

(21.17)

or p-N,

a<-N,

o>p-1

B=p-N,

a=-N,

o>p-1,

B

where

vex) v E W(O,oo)

or

;(dM(x»)

and

dM(x) = dist (x,M) , y$O-p. Me (iii)

For the weight functions vO(x) = e a1xl ,

vI (x) = e BIxI ,

and a, BE R ,

MC"0,

mN(M) = 0 . (See also Example 12.10 where

M was its edge, i.e.

Me 3rl

used with an auxiliary function rex) :;:; on

Hz),P(n;v 'v ) O 1

M

~

was a polyhedron

3Q .)

One can expect that some of the general theorems from Section 18 can be

the norms (21.9) and (21.10) are equivalent (iii-i)

but

rl

r = rex)

of the type

1

"3 dM(x)

if

313 312

!,__ !!!ll~__

or

__ ~

~~_:J

• _':'.-

more precisely,

""'""

rex)

~}

(l8.7), (20.6)J. The dimension

~_

-,.

~~-

•

_~~._~ ._:=..:::- .=:~

min {d(x), dM(x)}

_~ __ ~

~~_'-:"'7'-=_: __ -:__"-_: __-?~?~!:5:1~.;;:~~~~~;:'~!ff,~;z2'~~~~"]fii;-~3~~~~~::'~:::~

[compare with formulas

m of the manifold

M will play some role.

j~'f::~~,~~~;ii;:~'~~;;:c;-;Z:~,,:,"~~:~=1;.;::~,","'x"~~~':::-,,"-=.E''2c~~j1~~~~ ~~:::o.'-'ii5""-»'~"{;-"="'"

_~c~'" ""•• _"" .... ':"''"''-'',;.;.-,-.'.~"_. ~".="'=

Appendix

Some results concerning the continuity and compactness of the imbedding

w1 ,p(n;vO'v 1 ) c=

Lq(~;w)

with weight functions of the type (21.17) are mentioned in A. KUFNER,

B. OPIC, I. V. SKRYPNIK, V. P. STECYUK [lJ; the case

p = q,

Me aQ

is

22. LEVEL INTERVALS AND LEVEL FUNCTIONS

dealt with in A. KUFNER [2J, J. RAKOSNIK [1J and E. D. EDMUNDS, A. KUFNER,

J. RAKOSNIK [lJ. (ii)

In this additional section, we will give the proof of HALPERIN's Theorem 9.2 which is a fundamental tool for the proof of the Hardy in­

In Section 10 we have investigated the Hardy inequality for higher

order derivatives in the one-dimensional case. Obviously, imbedding theorems

equality with

0 < q <

. The proof will be divided into several auxiliary

assertions. Let us start with some notation.

and Hardy-type inequalities involving derivatives of higher orders (and even

fractional derivatives) can again be derived for N-dimensional domains N > 1 . Some results concerning the case

p = q

can be found in

n,

A. KUFNER

22.1. Level intervals. and for

(0.,

B)

C

(a, b)

[2J; as concerns the approach described in Sections 17, 18, cf. B. OPIC,

J. RAKOSNIK [lJ, where also further references can be found.

-,

dt,

f

p(o.,B)

to

(o.,B)

1

(a,b~

pet) dt,

f(o.,B) p(o.,B)

R(o.,B)

0.

0.

The interval

f EO M (a,b)'1L

B

f f(t)

f(o.,B)

1)

+

and

let us denote

B

(22.

1

p E W(a,b) r : L (a,b)

For

C (a,b)

is called a level interval (of

f

with respecl

p) if

(22.2)

R(o.,x)

~

R(o.,B)

If the level interval then it

(o.,B)

x E (o.,B) .

for every

is not contained in any larger level interval

is called a maximal level interval.

By (22.3)

L (a,b,f,p)

L

L

M

=

LM(a,b,f,p)

we denote the system of all level intervals and of all maximal level intervals

(o.,B)

22.2. Remark.

C

(a,b) , respectively.

A natural question arises whether the systems

Land

LM

can be empty or not. The answer is given by the following example.

22.3. Example.

Let us take

(22.4)

=: 1

f (t)

pet)

(a,b) t

= (0,1)

for

t

and

E (0,1)

According to Subsection 22.1, the interval

(o.,B)

with

0 ~

0.

< B ~ 1

is

314 315

~

sign

a level interval if and only if

by the sign

<

, =

~

,

or

>

simultaneously in all three

conditions (i), (ii) and (iii).

S

x

J pet) dt

I s I

a

a

r

J f(t) dt a

(22.5)

~

x

f(t) dt

22.6. Theorem.

a

r

for every

x 6

(a,S) .

(ii)

pet) dt a

Using (22.4) we obtain after a simple calculation that (22.5) is equivalent to the inequality ~

S

x

for every

~

a

l

f ,p

L

as well as the

Let

22.4. Lemma.

(a,S)

C

The system

= LM(a,b,f,p) is either empty or it is a M denumerable system of non-overlapping intervals.

f/p

is decreasing on

S

Let

(i)

level interval

from (22.4).

L

I

be the system of all level intervals containing the (a

o' b O)

. Int roduce in

S

Land

L

(a,b) .

--<

Ii

M

1

2

~

We have to show that

'1 S

C

Ii

x

c

(a,S) . Then the following three

~

R(a,S)

(ii)

R(a,x)

~

R(x,S)

(iii)

R(a,S)

~

R(x,S)

definition of

R(a,S)

CS

is bounded

{I

S

. I

y'

(a ,b ) E S y y

Y

y

E

r}

be ordered and

define 1

After some elementary calculations we succesively obtain from the

Proof·

S

from above. Therefore, let

R(a,x)

by

contains a maximal element. By virtue of the Zorn

lemma, it suffices to verify that every ordered subset (a,b) ,

-<

12 .

conditions are equivalent: (i)

a partial ordering

the rule

Moreover, the reader can easily verify that the systems are empty if the function

<

2

(iii)

Proof· L is empty for M

a

<

(a ,b ) , (a ,b ) are level intervals with l l 2 2 then (a ,b ) is a level interval. b < b ~ b l 2 2 l

If

x E (a,S) ,

which obviously cannot hold. Consequently, the system system

Every level interval is contained in a maximal level

(i)

interval.

that the following inequalities are equivalent: R(a,x)

~

R(a,S)

pea,S) f(a,x)

~

f(a,S) p(a,x)

Obviously

M

(aM,b ) M

=

aM = inf

h n }, {Y n } C

r

aM Since

S

I

r

b

a

yEr

U YE

y

Y =

M

sup yE r

b

y

. Then there are two sequences

such that =

lim a n-+oo Yn

b

M

lim b­ n-+oo Yn

is ordered, we have also

[p (a, S) - p (a, x)] f(a,x) ~ [f(a, S) - f (a,x)] p (a,x) , b

p (x, S) f (a,x)

~

f (x, S) p (a,x)

R(a,x)

~

R(x,S)

Thus we have obtained that (i) (ii)

B

(iii).

4-p (ii) . Similarly we can prove that

o

M

= lim b n-+oo Yn

If we show that

1 M

=

(aM,b ) M

is a level interval, i.e. that the impli­

cation (22.6)

x E 1

M

~

R(aM,x) ~ R(aM,b M)

holds, then the proof of the assertion (i) will be finished. But for 22.5. Remark. 316

Obviously, Lemma 22.4 remains true if we replace the

xe 1M 317

E r such that n interval, we conclude that

there exists a

R(a

Y

Yn

,x) '" R(a

Yn

,b

Yn

x

E Iy

,

and since I

n

n

-+

00

•

to

The inclusion yields

(a 1 ,b 1 ) E L

f';

(a ,b ) 2 2

R(a ,b ) 2 2

E

of

,: E (a 1 ' b 1)

if

L , Lemma 22.4 and the first in­

f';

R(a ,b ) 1 2

if

x E (a ,b ) , 2 2

x E (a , b )

2 Z

if

follows from point (ii) above.

L

318

n

M

I.

J

{I n =

o

if

(0.,6)

If (a

n

L

x

E

for

x

E (a,b) \ I .

f(x) (a,b)

R (0.,6) O

1

L (a,b)

for

I

with respect

= (a ,b ) ,

n

n

n

denote (a,6) O p(a,6)

f

and

Let

L = LM(a,b,f,p) t 0 . Let

M I = (a ,b ) n n n

M

.

*0

f

O

be the level function

a maximal level interval. Then for

x E In '

:;; fo(a,x)

for

0.

= fo(a,x)

for

a, x

(i)

f (a , x) n

(ii)

f(a n ,b n ) = fO(a n ,b n )

(iii)

f(a,x)

(iv)

f(a,x)

:;; fo(an,x)

(a ,b ) E L

1 2

R(a ,x) :;; R(a ,b) n

n

n

E (a, b) \ I ,

x E I

for

x

e

E (a,b) \ I ,

x

(a,b) , >

a

n

and consequently, by (22.12) we have x

o

,b n );

i I j

C

p (x)

n

Proof. It follows from Subsection 22.1 that

LM is nonempty (cf. Example 22.3) then the assertion

22.7. Level functions.

O

22.8. Lemma.

The inequalities (22.8) and (22.9) immediately imply that

1

f

together with the second inequality in (22.7)

R(a 1 ,x) :;; R(a , b ) 1 2

I.

= 0

M

The following lemma is an easy consequence of the definitions.

and consequently, again by Lemma 22.4 we have

If

= { R(a n , bJ

f

equality in (22.7) yield R(x,b 2 )

(x)

f

(22.12)

R(a ,b ):;; R(a ,b ) 1 1 1 2

R(a 1 ' x) :;; R(a 1 ' b 1) :;; R(a 1' b 2)

Analogously, the inclusion

where

L

[of f E M+(a,b) O 1 p E W(a,b)(\ L (a,b) ] by the formula

(22.13)

R(a 1 ,b 2 ) :;; R(a ,b ) , 2 2

(22.10)

if

In

R(a ,b ) :;; R(b ,b ) 1 1 1 2

and Lemma 22.4 implies

(iii)

LM~0,

and define the level function

For

(22.9)

if

The definition of level intervals and Lemma 22,4 imply that

R(a 1 ,a 2 ) '" R(a ,b ) 2 2

(22.8)

I

)

R(a 1 ,a 2 ) '" R(a 1 ,b 1 ) '" R(a 2 ,b ) :;; R(a ,b ) :;; R(b ,b ) . 1 2 2 1 2

Consequently,

(22.7)

I:

is a level (22.11)

Now, (22.6) follows by passing to the limit (ii)

Yn

f(a ,x) :;; R(a ,b ) p(a ,x) n n n n

J a

, then

R(a ,b ) pet) dt n

n

n

x n

= 1,2, ... }

Denote

J fO(t) dt a

fO(an,x)

n

Thus (i) is proved. The proofs of assertions (ii) - (iv) are similar.

o 319

Let

22.9. Theorem. (i)

be the level function of

fa

every level interval of

(22.14)

f . Then

is a level interval of

f

This implies fa ' ~.e.

every maximal level interval of

is a cevel interval of

fa

f

and since

>

i.e. L ( a , b , fa, p) M

C

have the same maximal level intervals

fa

J

for each level interval

J

of

fa ' there exists a constant

(22.17) (v)

for

kp(x)

fO(x)

x

E

Let

In

J

E

=

fO(t) dt

a x

R(a n ,b n )

J

pet) dt

a

=

Let

f pet) dt

a

a

R(a ,b ) n n

E L(a,b,fO'p)

~

n

such that

n

JC I n

x E I

n

Let

C

(a,S)

(a,b)

o x

E

(a,S) . Then

RO(X'S) ~ RO(a,S)

and RO(a,S) ~ RO(a,x)

Proof.

(i)

Suppose that

a, S

are finite, i.e.

a, S E R . In order to

(22.20)

RO(a,S)

According to Theorem 22.6 (ii) a

nor

S

exists a point X

If for every

x E (a,S) .

RO(y,S)

can be interior

f . Consequently, from Lemrr;a 22.8 (iii),

fO(a,S)

fO(a,x)

RO(x,S) .

<

.

= (a,S) E LM(a,b,fO'p)

points of some level intervals of (iv) we obtain

f(a,x)

(a ,b ) E LM(a,b,f,p)

n

prove (22.18), suppose on the contrary that

J

=

I

x E (a,S) ,

for every

and the above proof of point (i), neither

f(a,S)

,

According to Theorem 22.6 (i) and to

E L(a,b,fO'p)

which immediately yields the assertion.

(22.19)

n

J

J

R(a ,b ) = RO(a ,b ) ,

n n n n

The function Let

J

(22.16) there is an interval

(22.18) =

x

f pet) dt

RO(a,x) = RO(a,S)

(ii)

E

(iv)

22.10. Theorem.

x

(a,b ] , and consequently,

which implies that

x

(v) Lemma 22.8 (ii) implies that

x

x

for

The assertion follows from (22.14), (22.15).

= R(a n ,b n ) p(x) for every This implies (22.17) with k = R(a n ,b n )

= (a n ,b n ) E LM(a,b,f,p) such that J C I n . By the definition of the level function fa [d. (22.12)J we have

there is an interval

for every

R(a,S)

=

( iii)

= (a,S) E L(a,b,f,p) . According to Theorem 22.6 (i)

J

RO(a,x)

RO(a,S)

fO(x)

J

(fo)o = fa (i)

~

RO(a,x)

and by (22.12)

such that

k(J)

Proof·

= LM(a,b,fo'p)

LM(a,b,f,p)

(iv)

=

~

i.e. J E L(a,b,f,p) .

and

f

i.e. (22.16)

x E (a,S) ,

for every

L ( a , b , f , p) ;

the functions

(iii)

RO(a,x)

E LM(a,b,fO'p) , we obtain

J

R(a,x)

(22.15)

k

~

R(a,x)

L(a,b,f,p)C L(a,b,fO'p) ;

(ii)

RO(a,S)

R(a, S)

o

X

o

is continuous on

[a,S) , and consequently, there

such that

= max {y €

[a,x]; RO(y,S)

=

X = x , then (22.20) would imply that

o

contradicts the definition of

min

RO(S'S)}.

s E: [a,x]

x

o

RO(a,S)

. Consequently,

<

Xo

RO(XO'S) , which

<

x

which imme­

diately yields 320

321

(22.21)

RO(xO'S)

RO(s,S)

<

for every

and (22.18) follows by passing to the limit

s E (xO,x] .

(iii)

Using Lemma 22.4 (and Remark 22.5) we obtain from (22.21) that

n -+

RO(xO's)

The function

RO(xO'S)

<

RO(xO'S)

there exists a point

R (x 'X 1) O O

(22.23)

s = x

Putting (22.24) Now

x

Xl

Xl

<

[x,S]

Proof.

RO(xO'x) ~ RO(xO'S)

RO(xO'S)

R (CJ. ,S2) O 2

leads to a contradiction:

while (22.22) (for

(x ,x )

O

1

containing the point

x

while for

E

s

(xO,x]

1

'

a

<

and

2 6 = S2

and

<

2 ,

6

1

~

S2 . Then

S2 . Using first the inequality

x = a

and

S = S2 ~

SI

2

' and then the inequality

x = Sl ' we obtain

R (a ,6 )

O 1 2

CJ.

CJ.

1

2

or

62

Sl

s

E

(x,x ) 1

"

we have from (22.23)

is similar and is left

Proof.

For

(22.28)

x

Now, let

k

p(xO's)

for every

RO(xO'x) = R (x 'x ) O O 1

(a,S)

C

(a,b)

s E (x ,x ] . O 1

f

D = D(x)

with respect to on

(a,b)

p •

such that

(a,b)

x

t > 0

and

E (a,b) define

H(x,t) = R (x,t ) O 1

t

1

=

+ a , Sn t S and that n (i) of the proof we have a

(22.29)

be a general interval (i.e.

x E (a ,S) n n ,

nE N,

min {x + t t

[0 <

t

1

2(x + b)} • 1

< 2(b

- x) ] we have

which contradicts (22.24).

possibly infinite). Then there exist two sequences

RO(x,Sn) ~ RO(an,Sn)

for a.e.

= D(x)

E

Thus for small

(ii)

be the level function of

O

where

fO(xO's)

In particular, we have

f

fO(x)

pw

(22.27)

yields RO(xO's)

Let

Then there exists a non-increasing function

RO(xO'S) ~ R (x 'x 1 ) , O O

<

According to (22.25), the formula (22.17) together with the definition

322

CJ.

D

22.12. Theorem.

then (22.22) and (22.23) yield

RO(xO's) ~ R (x 'x 1 ) . O O

that

a

The proof for the case to the reader.

RO(xO's)

S

;';;

(x ,x ) E L(a,b,fO'p) O 1

Indeed, if

R O

1

which implies (22.26).

and such that

of

a

R (CJ. ,S2) ~ R (a 1 'Sl)

O 1 O

s = x ) implies

.

Thus, we constructed an interval

(22.25)

CJ.

1 (22.19) with a = a ' 1

RO(xO'S) ~ RO (x O'x 1 )

x = Xl

a =

(a,b)

R (CJ. ,6 ) . O 1 1

~

Suppose that

(22.18) with

C

(a ,Sl)' (a ,S2) 1 2

R (a ,6 ) O 2 2

(22.26)

max RO(xO's). s E [x , S]

<

Let

22.11. Lemma.

[x,S], and consequently,

such that

since the assumption

(22.23) implies RO(xO'x)

is continuous on

in (22.22) we obtain in view of (22.23) that

RO(xO'x) <

E

D

s E (xO,x] .

for every

•

The inequality (22.19) follows from (22.18) by Lemma 22.4 and

Remark 22.5. (22.22)

00

{a}, n

for every

a

{S } n

and/or

CR

H(x,t) = RO(x, x + t)

Assume that

such

nE N . By part

(i)

for every fixed

decreasing for (ii)

t

x E (a,b)

the function

H(x,t)

is non­

+0 ;

for every fixed

t > 0

the function

H(x,t)

is non-increasing 323

in

x .

(22.32)

According to (i), the limit of (22.30)

for

t + 0

exists. If we define

a < xl < x 2 < b

(a,b)

due to (ii). Indeed,

i

we have

=

G: lim H(x ,t)

D(x ) .

2

t+O

i

+ b) ,

If

= RO(x i , ~(Xi + b))

x x+t

= liml. uo t

pes) d'S

x

J

~(b

J

- x ) 2

~

t

<

~(b

RO(x l' xl + t) ,

1

B. = x. + t ,

~

~

~

i

=

1,2 ,

a.

~

=

x., ~

S.

~

- xl) , then

H(x ,t) 2

= RO(x 2 , ~(x2 + b))

1 1 xl + t < xl + 2(b - xl) = 2(x j + b) < ~(x2 + b) , the inequality (22.32) follows again from Lemma 22.11 where we put a 1 = xl ' B1 = xl + t , 1 a 2 = x 2 ' B2 = 2(x 2 + b) 0 Since

fOes) ds fO(x)

x x+t

= x. ,

1,2

H(x , t) x+t

lim l.

t+o t

a.~

1

~ =

(ii-3)

fOes) ds

i = 1,2

for

~

.

D(x) = lim H(x,t) = lim RO(x,x+t)

UO UO

J

, then

t G: 2(b - xl) , then

If

1

On the other hand, we have

x+t

x2

<

and (22.32) follows again from Lemma 22.11 where we put

2

2(x

lim t+O

i(b - x 2 ) ~

H(xi,t)

f

xl

1, 2

(ii-2)

= lim H(x 1 ,t) t+O

for

and (22.32) follows from Lemma 22.11 where we put

and consequently,

for a.e.

0 < t <

~

H(x , t) G: H(x ,t)

1 2

D(x ) 1

H(x ,t) 1

H(x.,t) = RO(X., x. + t)

is non-increasing on

D(x)

If

(ii-I)

D(x) = lim H(x,t) ,

UO

then the function for

H(x,t)

~

H(x ,t) 2

= p(x) pes) ds

x 22.13. Proof of Theorem 9.2.

x E (a,b)

be the level function of f with O

p. Then the property (9.4) is a consequence of the definition

respect to Thus, we have arrive at the formula (22.27). In order to complete the

of

proof it remains to show that the assumptions (i), (ii) are fulfilled.

f

O

Let

f

and of Lemma 22.8.

The property (9.5) is a consequence of Theorem 22.12.

(i) It suffices to verify that

(22.31)

H(x,t ) G: H(x,t ) 2 1

for

Thus, it remains to prove that (9.6) holds, i.e. that

0

<

t

1

< t < -(b - x) . 212

bf[fO(X)]P

-p(;)

From Theorem 22.10, formula (22.19), we have RO(a,y) G: RO(a,B) Putting here

a

=x

,

for

B = x -;- t 1

RO(x, x + t ) 2

~

y

E (a,B)

and

y

= x + t 2 ' we obtain

RO(x, x + t ) , 1

which implies (22.31) according to (22.29). (ii) 324

We have to verify that

p(x) dx ~

a

If

bJ[f(X)]P p(x) p(x) dx a

xc (a,b) \ I

with

I

from (22.11), then

fO(x)

f(x) . Consequently,

it suffices to show that

b

b n

(22.33)

r l,fO(X)JP

J

a

where

n

(an,b ) n

p(x)

p(x) dx

n

~ f [~ ~:n

p

p(x) dx

a

n

are the intervals from (22.10). 325

If

p = 1 • then (22.33) follows from Lemma 22.8 (ii) (even with the

References

equality sign). Thus. suppose that

p > 1 . From the definition of

[cf. (22.12)J

x E (a n .b n )

we have that for

b (22.34)

fO

c

with

fO(x) = cnP(x)

n

b

n

n

= f f(t) dt/ f pet) dt . a

an

n

If

ADAMS, D. R.: [lJ

A trace inequality for generalized potentials. Studia Math. 48 (1973). 99 - 105. MR 49 # 1091

[2J

Weighted nonlinear potential theory. Trans. Amer. Math. Soc. 297 (1986). no. 1, 73 - 94. MR 88m:31011

b J

n

= ([fO(X)( ---- p(x) dx = 0 a

[lJ Sobolev spaces. Academic Press, New York-San Francisco-London, 1975. MR 56 II 9247

n ~

I

then (22.33) holds trivially. If

n 22.8 (ii) and Holder's inequality yield b I

n

ADAMS, R. A.:

p(x)

=

b

n

f c~-l a

a

n

ANDERSEN, K. F.; HEINIG, H. G.:

that

[lJ Weighted norm inequalities for certain integral operators. SIAM J. Math. Anal. 14 (1983), no. 4. 833 - 844. MR 85f:26012

n

f c~-l

fO(x) dx =

f(x) dx

f [fO(X)J ~

[lJ Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators. Trans. Amer. Math. Soc. 308 (1988), no. 2,547 - 558.

n

p

[lJ Hardy's inequality and its extensions. Pacific J. Math. 11 (1961). 39 - 61. MR 22 # 12187

n

,; Jl/ p ' - n

[f a

[2J Integral inequalities involving a function and its derivatives. Amer. Math. Monthly 78 (1971). 705 - 741. MR 48 # 4235

lip n

(f(X») p(x)

p

p(x) dx

J

BENSON, D. C.

n

However, this inequality implies (22.33) since b n

BEESACK, P. R.:

l/p' (x) f(x) p1/p(x) dx ;;;

p(x)

b

J

ANDERSEN. K. F.; SAWYER, E. T.:

p-1

bn a

0 , then the formula (22.34), Lemma

J

n

~

0

and

[lJ Inequalities involving integrals of functions and their derivatives. J. Math. Anal. Appl. 17 (1967). 293 - 308. MR 34 # 2809

n

= c P f p(x) dx < n

a

n

BERGH. J.; LOFSTROM, J.: 00

•

o

[lJ Interpolation spaces. An introduction. Springer-Verlag. Berlin-Heidel­ berg-New York, 1976. MR 58 # 2349 BLISS, G. A.:

[lJ

An integral inequality. J. Math. Soc. 5 (1930). 40 - 46.

BRADLEY. J. S.:

[lJ Hardy inequalities with mixed norms. Canad. Math. Bull. 21 (1978). no. 4, 405 - 408. MR 80a:26005

326

327

;=-----=

-=~======--===~-"-=::=-==~~--=====~'~~:""-=~"'''-=-=''''~'''''':~~-~-=t""~~~~~?~~=~.~~~;~~:~~~~~=~~.1~~~~~~.;-'':~:,'.'~~::~c~~~~=~~~~~=b~~

---=-o;=-=':...=:=.:::'.:~'=:~:::~";"~

GRISVARD, P.:

BROWN, R. C.; HINTON, D. B.: [lJ Weighted interpolation inequalities of sum and product form in Proc. London Math. Soc. (3) 56 (1988), 261 - 280. [2J Weighted interpolation inequalities and embeddings in in Canad. J. Math.).

~~~~,=:::=~",~~:~,~:~~::~~~_===="~=~-","""",===-=:c,,,-,--

n

R

n

R

.

(to appear

BURENKOV, V. I.: [lJ Mollifying operators with variable step and their application to approximation by infinitely differentiable functions. Nonlinear analysis function spaces and applications, Vol. 2 (Pisek 1982), 5 - 37, Teubner- ' Texte zur Math., 49, Teubner, Leipzig, 1982. MR 84a:46068 CHIARENZA, F.; FRASCA, M.: [lJ A note on a weighted Sobolev inequality. Proc. Amer. Math. Soc. 93 (1985), no. 4, 703 - 704. MR 86h:46052 DUNFORD, N.; SCHWARTZ, J. T.: [lJ Linear operators. I. General theory. Interscience Publishers, Inc., New York-London, 1958. MR 22 # 8302 DZHALILOV, K. A.:

[lJ

Espaces intermediaires entre espaces de Sobolev avec poids. Ann. Scuola Norm. Sup. Pisa 17 (1963), 255 - 296.

GURKA, P.: [lJ

Generalized Hardy's inequality. Casopis Pest. Mat. 109 (1984), no. 2, 194 - 203. MR 85m:26019

[2J

Generalized Hardy's inequality for functions vanishing on both ends of the interval (to appear in Analysis).

GURKA, P.; KUFNER, A.: [lJ

A note on a two-weighted Sobolev inequality (to appear in Banach Center Publ.).

GURKA, P.; OPIC, B.: [lJ

Ar-condition for two weight functions and compact imbeddings of weighted Sobolev spaces. Czechoslovak Math. J.38 (133) (1988), 611 - 617.

[2J Continuous and compact imbeddings of weighted Sobolev spaces I, II, III. Czechoslovak Math. J. 38 (133) (1988), no. 4, 730 - 744; 39 (134) (1989) 78 - 94; to appear. MR 89j:46034

[lJ On an imbedding theorem with weight (Russian). Izv. Akad. Nauk Azer­ baidzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 7 (1986), no. 4, 25 - 31. MR 88f:46071

DE GUZMAN, M.:

EDMUNDS, D. E.; KUFNER, A.; RAKOSNfK, J.:

HALPERIN, 1.:

[lJ Embeddings of Sobolev spaces with weights of power type. Z. Anal. An­ wendungen 4 (1985), no. 1, 25 - 34. MR 86m:46032a

[lJ Function spaces. Canad. J. Math. 5 (1953), 273 - 288.

EVANS, W. D.; RAKOSNfK, J.:

n [lJ Differentiation of integrals in R . Lecture Notes in Math., Vol. 481, Springer-Verlag, Berlin-New York, 1975. MR 56 # 15866

HARDY, G. H.: [lJ Note on a theorem of Hilbert. Math. Z. 6 (1920), 314 - 317.

[lJ Anisotropic Sobolev spaces and quasidistance functions. Preprint no. 55, Math. Inst. Czech. Acad. Sci., Prague, 1989.

HARDY, G. H.; LITTLEWOOD, J. E.; POLYA, G.:

FABES, E. B.; KENIG, C. E.; SERAPIONI, R. P.:

[lJ Inequalities. University Press, Cambridge, 1952. MR 13 # 727

[lJ The local regularity of solutions of degenerate elliptic equations. Corom. Partial Differential Equations 7 (1982), no. 1, 77 - 116. MR 84g:35067

HEINIG, H. P.: [lJ Weighted norm inequalities for certain integral operators II. Proc.Amer. Math. Soc. 95 (1985), no. 3, 387 - 395. MR 87h:26027

GARCfA-CUERVA, J.; RUBIO DE FRANCIA, J. L.: [lJ Weighted norm inequalities and related topics. North-Holland Math. Studies, 116, North-Holland Publishing Co., Amsterdam-New York 1985. MR 87d:42023

HORMANDER, L.: [lJ Linear partial differential operators. Springer-Verlag, Heidelberg­ Berlin-New York, 1963.

GATTO, A. E.; GUTIERREZ, C. E.; WHEEDEN, R. L.: [lJ Fractional integrals on weighted HP spaces. Trans. Amer. Math. Soc.

289 (1985), no. 2, 575 - 589. MR 86k:42037

KADLEC, J.; KUFNER, A.: [lJ Characterization of functions with zero traces by integrals with weight functions I, II. Casopis Pest. Mat. 91 (1966), 463 - 471; 92 (1967), 16 - 28. MR 35 # 3430, 5924

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KOKILASHVILI, V. M.:

LEVINSON, N.:

[IJ On Hardy's inequalities in weighted spaces (Russian). Soobse. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 37 - 40. MR 81i:26014

[IJ Generalizations of an inequality of Hardy. Duke Math. J. 31 (1964), 389 - 394. MR 30 # 2111

KUDRYAVTSEV, L. D.:

LEWIS, R. T.:

[IJ Direct and inverse imbedding theorems. ApplicatiQns to the solution of elliptic equations by variational methods. Translations of Mathematical Monographs, Vol. 42, Amer. Math. Soc., Providence, R. I., 1974. MR 49 # 9618, 33 # 7838

[IJ Singular elliptic operators of second order with purely discrete spectra. Trans. Amer. Math. Soc. 271 (1982), no. 2, 653 - 666. MR 84a:35215

KUFNER, A.: [IJ Imbedding theorems for general Sobolev weight spaces. Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 373 - 386. MR 40 # 6253 [2J Weighted Sobolev spaces. Teubner-Texte zur Math., 31, Teubner, Leipzig, 1980 (first edition); J. Wiley and Sons, Chichester-New York-Brisbane­ Toronto-Singapore, 1985 (second edition). MR 84e:46029, 86m:46033 KUFNER, A.; HEINIG, H. P.: [IJ Hardy's inequality for higher order derivatives (Russian) (to appear in Trudy Mat. Inst. Steklov). KUFNER, A.; JOHN, 0.; FuciK, S.: [IJ Function spaces. Academia, Prague, Noordhoff International Publishing, Leyden, 1977. MR 58 # 2189 KUFNER, A.; OPIC, B.: [IJ Some imbeddings for weighted Sobolev spaces. Constructive func.tion theory '81 (Varna, 1981), 400 - 407, Bulgar. Acad. Sci., Sofia, 1983. MR 85d:46042

[2J A Friedrichs inequality and an application. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 185 - 191. MR 86i:35111 LIZORKIN, P. I.; OTELBAEV, M.:

[IJ Imbedding and compactness theorems for Sobolev type spaces with weight: I, II (Russian). Mat. Sb. (N. S.) 108 (150) (1979), no. 3,358 - 377; 112 (154) (1980), no. 1 (5), 56 - 85. MR 80j :46054, 82i:46051 MAZ'JA, V. G.: [IJ Sobolev spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1985. MR 87g:46056. (An extended and improved version of the books Einbettungs­ satze fur Sobole~sche Raume 1, 2, Zur Theorie Sobole~scher Raume, Teubner-Texte zur Math., Vols. 21, 28, 38, Teubner, Leipzig, 1979, 1980, 1981) MUCKENHOUPT, B.: [IJ Hardy's inequality with weights. Studia Math. 44 (1972), 31 - 38. MR 47 If 418 [2J Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207 - 226. MR 45 # 2461 MYNBAEV, K. T.; OTELBAEV, M. 0.:

[2J How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25 (1984), no. 3, 537 - 554. MR 86i:46036

[IJ

KUFNER, A.; OPIC, B.; SKRYPNIK, I. V.; STECYUK, V. P.:

NECAS, J.:

[IJ Sharp embedding theorems for a class of weighted Sobolev spaces (Russian). Dokl. Akad. Nauk Ukrain. SSR Ser. A (1988), no. 1, 22 - 26. MR 8ge:46038

[IJ Les methodes directes en theorie des equations elliptiques. Academia, Prague and Masson, Paris, 1967. MR 37 # 3168

Weighted function spaces and the spectrum of differential operators (Russian). "Nauka", Moscow, 1988. MR 89h:46036

NIKOL'SKII, S. M.: KUFNER, A.; TRIEBEL, H.: [IJ

Generalizations of Hardy's inequality. Confer. Sem. Mat. Univ. Bari 156 (1978), 1 - 21. MR 81a:26014

LANDAU, E.: [IJ A note on a theorem concerning series of positive terms. J. London Math. Soc. 1 (1926), 38 - 39. LEE, K. C.; YANG, G. S.: [IJ On generalization of Hardy's inequality. Tamkang J. Math. 17 (1986), no. 4, l09 - 119. MR 88i:26047 330

[IJ

Approximation of functions of several variables and embedding theorems (Russian). Second edition, "Nauka", Moscow, 1977. MR 8lf:46046

OINAROV, R.: [IJ

Density of smooth functions in weighted spaces and weighted inequaliti (Russian). Dokl. Akad. Nauk SSSR 303 (1988), no. 3, 559 - 563.

OPIC, B.: [IJ N-dimensional Hardy inequality. Theory of approximation of functions, Proceedings (Kiev 1983), 321 - 324, "Nauka", Moscow, 1987. 331

[2J Necessary and sufficient conditions for imbeddings in weighted Sobolev

spaces. Casopis Pest. Mat. 114 (1989), no. 4, 165 - 175.

OPIC, B.; GURKA, P.:

SHUM, D. T.: [lJ On a class of new inequalities. Trans. Amer. Math. Soc. 204 (1975), 299 - 341. MR 50 # 10183

[lJ N-dimensional Hardy inequality and imbedding theorems for weighted

Sobolev spaces on unbounded domains. Function spaces, differential

operators and nonlinear analysis, 108 - 124. Pitman research notes

in mathematics series, 211, Longman Scientific and Technical, 1989.

SINNAMON, G.:

OPIC, B.; KUFNER, A.:

[2J Operators on Lebesgue spaces with general measures. Thesis, McMaster University, 1987, 149 pp.

[lJ Weighted Sobolev spaces and the N-dimensional Hardy inequality (Russian). Imbedding theorems and their application to problems of mathematical physics, 108 - 117, Trudy Sem. S. L. Soboleva, No. I, 1983, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1983. MR 85j:46053 [2J Some inequalities in weighted Sobolev spaces. Constructive theory of functions '84 (Varna, 1984), 644 - 648, Bulgar. Acad. Sci., Sofia, 1984. [3J Remark on compactness of imbeddings in weighted spaces. Math. Nachr. 133 (1987), 63 - 70. MR 88m:46041 OPIC, B.; RAKOSN!K, J.: [lJ Estimates for mixed derivatives of functions from anisotropic Sobolev­ SlobodeckiI spaces with weights. Preprint no. 54, Math. Inst. Czech. Acad. Sci., Prague, 1989. PACHPATTE, B. G.: [lJ On some extensions of Levinson's generalizations of Hardy's inequality. Soochow J. Math. 13 (1987), no. 2, 203 - 210. MR 89i:26018 PORTNOV, V. R.:

L~l~(Q x R+) and their applications (Russian). Dokl. Akad. Nauk SSSR ' 155 (1964), 761 - 764. MR 28 # 4243

[lJ

A weighted gradient inequality. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 329 - 335. (See also: Weighted Hardy and Opial type inequalities to appear.)

STECYUK, V. P.: [lJ Thesis, Inst. of Appl. Math. and Mech., Akad. Nauk. Ukrain. SSR, Donetsk, 1986. STEPANOV, V. D.: [lJ Two-weighted estimates for Riemann-Liouville integrals. Preprint no. 39, Math. Inst. Czech. Acad. Sci., Prague, 1988. [2J Two-weighted estimates for Riemann-Liouville integrals 1, 2 (Russian). Preprints, Akad. Nauk SSSR, Far East Branch, Comput. Center, Vladivos­ tok, 1988. SYSOEVA, F. A.: [lJ Generalizations of a certain Hardy inequality. Izv. Vyss. Uceb. Zaved. Matematika 6 (49) (1965), 140 - 143. MR 33 # 240 TALENTI, G.: [lJ Una diseguaglianza integrale. Boll. Un. Mat. Ital. (3) 21 (1966),

25 - 34. MR 33 # 4220

[lJ Two imbeddings for the spaces

[2J Sopra una diseguaglianza integrale. Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 167 - 188. MR 36 # 1600

RAKOSNiK, J.:

TOMASELLI, G.:

[lJ On imbeddings of Sobolev spaces with power-type weights. Theory of approximation of functions, Proceedings (Kiev, 1983), 505 - 507, "Nauka", Moscow, 1987

[lJ A class of inequalities. Boll. Un. Mat. Ital. (4) 2 (1969), 622 - 631.

MR41#411

SAWYER, E. T.:

TREVES, F.: [lJ Relations de domination entre operateurs differentiels. Acta Math. (10C (1959), 1 - 139.

[lJ Weighted inequalities for the two-dimensional Hardy operator. Studia Math. 82 (1985), no. I, 1 - 16. MR 87f:42052

TRIEBEL, H.:

[2J A weighted inequality and eigenvalue estimates for Schrodinger operators. Indiana Univ. Math. J. 35 (1986), no. 1, 1 - 28. MR 87m:35164

[lJ Interpolation theory, function spaces, differential operators. VEB

Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 80i:46023

SEDOV, V. N.:

ZYGMUND, A.:

[lJ Weighted spaces. The imbedding theorem (Russian). Differentsial'nye Uravneniya 8 (1972), 1452 - 1462. MR 46 # 9715

[lJ Trigonometric series. Vols. I, II. Second edition, Cambridge University Press, London-New York, 1968. MR 38 # 4882

332

333