Hardy-type inequalities

B Opic and A Kufner Czechoslovak Academy of Sciences

Hardy-type inequalities

Longman NNW

Scientific &

- Technical

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

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

© Longman Group UK Limited 1990 All rights reserved: no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1990 AMS Subject Classification: 26D10, 46E35 ISSN 0269-3674

British Library Cataloguing in Publication Data Kufner, Alois, 1934Hardy-type inequalities 1. Mathematics. differential inequalities 1. Title II. Opic, B. 515.3'6

ISBN 0-582-05198-3 Library of Congress Cataloging-in-Publication Data Kufner, Alois. Hardy-type inequalities / A. Kufner and B. Opic. p. cm.-- (Pitman research notes in mathematics series, ISSN 0269-3674 ; 219) ISBN 0-470-21584-4 (Wiley) 1. Inequalities (Mathematics) I. Opic, B. II. Title. III. Series. QA295.K87 1990 512.9'--dc2O

89-14502

CIP

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn

Contents

Introduction Chapter 1.

1

The one-dimensional Hardy inequality

5

1. Formulation of the problem

5

2. Historical remarks

14

3. Proofs of Theorems 1.14 and 1.15

21

4. The method of differential equations

35

5. The limit values of the exponents

p

45

q

,

6. Functions vanishing at the right endpoint. Examples 7. Compactness of the operators

HL

HR

and

8. The Hardy inequality for functions from 9. The Hardy inequality for

0 < q
0 ) and w,vl,...,vN are bers (in fact, we will consider p ? 1 ,

,

weight functions, i.e.

measurable and positive a.e. in

S2

.

We are concerned with the question what conditions on the data of our problem

-

on the domain

i.e.

w,v1,...,vN

weight functions

(0.1) for all functions

u

12

-

,

on the parameters

p

,

q

and on the

guarantee validity of the inequality K ,

from a certain class

K D CO(Q) C > 0

with a constant

independent of the function

estimates for the best possible constant

C

u

.

In some cases,

in (0.1) will be given.

The inequality (0.1) will be called here

Terminological remarks. (i)

the (N-dimensional) Hardy inequality, the reason being the following: In 1920, G. H. HARDY proved an inequality (see Section 1) which can be easily rewritten in the form 1l1/P

l11/P

u'(t)lp to dtJ

lu(t)1p to-p dt]

(0.2)

p + 11

0 J' J

0

wher e

u' = du/dt

.

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

it holds, e.g., for all functions

p>

1

and

u E CO(0,-)

provided

e x p - 1.

The inequality (0.2) is a special case of the general (one-dimensional) Hardy inequality 1

bll1/q IIbl/p

[JIu(t)j

(0.3)

CIJu'(tp v(t) dt1

w(t) dtl J

lllla

1J

a

- - = a < b < -

where

and

w(t)

,

are weight functions.

v(t)

So, we obtain (0.2) from (0.3) taking p = q

>

1

a=0

,

b=0,

,

w(t) =

tE-p

v(t) = tE

,

.

On the other hand, the inequality (0.3) is again a special case of the inequality (0.1) for the case

N=

1

S2 = (a,b)

,

It can be said that the conditions of validity of the inequality (0.3) are investigated (almost) completely; we will deal with them in detail in Chapter 1.

In the literature, the following inequality has been intensively

(ii)

investigated:

CIfIVU(x)Ip dx]

_

S2

S2

where

l1 /p

((

ll1/q

{Jju(x)I q dx]

(0.4)

(Du

x = (x1,x2,...,xN)

Vu =

,

N

,

1

au

p

x

.

ax

and

8x

IDulp =

2

This inequality is known, e.g., as the SoboZev inequality and

i=1 axi

holds (e.g.) for 1

provided

u E C0(0)

0

independent of

f

.

For a proof see, e.g., G. H. HARDY [1] or G. H. HARDY, J. E. LITTLEWOOD, G. POLYA [1]. The exact value of the constant

was given in 1926

C

by E. LANDAU [1]: it is P

1e-p+1I lp 1.2. Definition.

Let

-

I = (a,b)

a < b < +

,

and denote by

AC(I)

the set of all functions absolutely continuous on every compact subinterval [c,d] C I Further, denote by .

AC

(1)

L

and

AC R(I)

the sets of all functions

u E AC(I)

for which

5

lim

(1.4)

u(x) - 0

x+a+ and

u(x) = 0

lim

(1.5)

x+brespectively. (So, the indices tion

and

L

express the fact that the func-

R

vanishes on the Left and right end of the interval

u

I

,

respect-

ively.) Finally, denote by ACLR(I)

ACL(I) flACR(I)

the intersection

.

If it is necessary to point out the concrete form of the interval I - (a,b)

, we will use the notation AC(a,b)

ACL(a,b)

,

,

ACR(a,b)

,

ACLR(a,b)

Further, let us introduce the notation xr

(HLf)(x) =

f(t)

dt

f(t)

dt

J

a

b (HRf)(W )

= J

x

Using the operator

(1.7)

I

HL

(HLf)p(x)

we can rewrite (1.2) in the form

,

xe-p dx s C

0

for

1

fp(x) xe dx

0

e < p - 1

e > p -

J

,

and similarly with the help of the operator

HR

for

.

From the inequality (1.2) we obtain the Hardy inequality (0.2) as an easy corollary:

1.3. Lemma. (1.8)

Or 6

Let

1

c< p- 1

< p
p -

(1.9)

and

1

u

AC R(0,00)

.

Then

JIu(x)P X'-p dx

(1.10)

0

C I1u'(x)jp xE dx

0

.

0

Further x(

u(x) = J u'(t) dt + u(c)

for

c > 0

C

since

u c- AC(0,-)

.

Moreover,

u E ACL(0,-)

,

and therefore, we obtain for

e -` 0+ that

7

u(x) = J

u'(t) dt

0

Finally,

J

Jiu'ti dt = (HLlu'I)(x)

u'(t) dt

0

0

and (1.10) follows from (1.7) for (ii)

f = lu'I.

The case (1.9) can be handled analogously. 11

1.4. Definition.

denote by

W(a,b)

or

W(I)

I = (a,b)

For

the set of all weight functions on

I

,

i.e.

the set of all functions

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

I

.

Further, denote by M+(I)

M+(a,b)

or

the set of all measurable functionsnon-negative a.e. on

1.5. Problem.

Let

1

< p,q < -

is there a (finite) constant

C > 0

v, w C- W(a,b)

Under what conditions

.

such that the inequality

111/q

b((

(1.11)

Let

.

I

l/p

((b

CIIIu'(x)lp v(x) dx

IJIu(x)Iq w(x) dxl

a

a

holds (i)

for every

u E ACL(a,b)

(ii)

for every

u E ACR(a,b) ?

1.6. Example and remark.

the inequality (1.11) for to

.

(i)

,

or

The inequality (1.10) is a special case of

p = q

,

a = 0

,

b = -

,

w(t) = tE P

,

v(t)

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

1.3. (ii)

8

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

HL

problem concerning inequalities involving the operators

and

HR

from

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

Let

7. Problem.

1

< p,q


1

,

but also with

p < 0

and even with

0 < p
0

and

lim

tib-

a(t) = m . Here

,1(t)

and

such that ,1(a) _ - W

(2.7) holds for every

v

,

u ; ACL(a,b)

2.6. Formulas; the case

p

O

*1

1/pr
0

0< Define 50

.

IM

.

and if we denote

Mn = {x E (a,E);

IMnI

x E (a,b)

v-1(x)

> S(E) - n

l

,

Therefore, there exists a subset

I



0

.

B E R consider the inequality

Jlu'(x)I peBx 2

p = q = 2

for

,

dx

a = B > 0

) 1/P

-

in F. TREVES [1]

(see L. HORMANDER [11, p. 182). The investigation of the validity of this

72

inequality on the class ACLor ACR(-°°,m)

leads to the calculation

of the integrals

x

x eat2

eat2

dt

J

,

dt

,

_m BL

which appear in the definition of the numbers a > 0

a < 0

(6.40)

or

8 5 0

dt ,

f

x

x

bers are infinite if

e-Bt2/(P-1)

(e-St2/(P-1) dt ,

,

BR , AL

,

AR. These num-

Consequently, the condition

.

g > 0

,

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

mentioned above seems to contradict our necessary condition

a = 8

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

special class C0defined in Subsection 7.11. We will resume the study of this inequality in Section 8.

7. COMPACTNESS OF THE OPERATORS

HL

AND

HR

For two Banach spaces

7.1. Notation and some auxiliary results.

X

,

Y

we denote by [X,Y]

(7.1)

K[X,Y]

or

the set of all linear mappings from

into

X

which are continuous or

Y

compact, respectively. If

X(_ Y

(7.2)

,

then the symbols

X(,, Y

and

X c (, Y

denote that the identity mapping [X,Y]

and

K[X,Y]

,

I

lu = u

,

X

u E X

,

belongs to

respectively. We will say that the imbedding

continuous (compact) or that the space bedded into

for

I

is

is continuously (compactly) im-

X

.

The symbol

(7.3)

un - u

will denote the weak convergence of

u

n

to

u

(in

X

The symbol 73

X = Y

(7.4)

and

T = [X,Y]

Finally, if

and

X

will denote that the spaces

Y

X

are isometrically isomorphic.

,

Y

T

,

are the dual spaces to

X

,

Y

then T*

will denote the adjoint operator to

acting from

Y*

X

into

We will use the following assertions whose proofs can be found, e.g., in N. DUNFORD, J. T. SCHWARTZ [1]. (i)

If

T E [X,Y]

(iii)

T E [X,Y]

Let Let

Tu

then

and

T* E [Y*,X*]

= II T*

T II

(ii)

,

.

T E K[X,Y]

Then ,

if and only if

T E K[X,Y]

{un} C X

,

un - u

.

T* C- K[Y*,X*].

Then

n -+ Tu

We will work mainly with the weighted Lebesgue spaces introduced in Subsection 5.2. For

1

and

< p < -

LP(a,b;v)

v E W(a,b)

,

the mapping

defined by

(P

@(u) = uv1/P

(7.5)

is obviously an isometric isomorphism of

LP(a,b;v)

into

r

simultaneously an isometric isomorphism of

LP(a,b)

,

r

LP (a,b)

into

LP (a,b;v

and 1_

r

P ).

This fact together with Riesz' representation theorem leads to the following assertion: [LP(a,b;v)]*

(iv)

Let

g E LP

< p < W ,

1

G E

.

Then there exists an element

such that

(a,b;v1-P

)

b

G(u) = Jg() u(x) dx for every u e LP(a,b;v) a Moreover, G II

=

,

Ilg

.

p,,(a,b),v1-P Consequently, (7.6) 74

[LP(a,h;v)]*

--

Lp'(a,b;v 1-p?

Further, the following two assertions will be used: (v)

(R. A. ADAMS [1] , Theorem 2.21) is precompact in

S C LP(a,b)

there exists a number such that for every

LP(a,b)

1
0

G = [c,d] C (a,b) Ihl
a

dtj p v1-p' (x)

ge (t)

J[ 1

dx

4 p,

Ig

l-P

II

q',(a,b),wq r

=

IUJ

vl-p (x) dx

t p'

(br

I

-p'/q'

w(x) dx]

Fp,() L

J

a (II)

Let the conditions (i), (ii) of the theorem be fulfilled. Let LP(a,b;v)

be the unit ball in

B

H

,

L

B

its image in

Lq(a,b;w)

and

,

denote by S = (HLB)w1/q

(7.19)

the set of all functions of the form

with

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

u C S

(7.8) (with

instead of

q

p

)

c

implies the existence of numbers 0 < FL(x)
0

(7.20)

f C B

l/q

E

.

The condition (ii) of our theorem

,

d E (a,b)

c < d

,

,

such that

x c-: (a,c) U (d,b)

for every

.

4 k(q,p) 3 If

u E S

,

then

u =

u(x)lq dx

(7.21) J

(a, c) U (d,b)

b

x

c

I

a

d

I1

w(x) dx = 11 + 12

a

using the inequality (1.12) for the interval

and (7.20): c

I1 = (7.22)

I

J

a

We estimate the integral (a,c)

[JJf(t) dtj

w(x) dx +

dt I

J

xll

( J

a

x

IJ If(t) I dtj

w(x) dx

a

[k(q,p) BL(a,c;w,v,q,P)-1A

P,(a,c),v

]q
0

and

wl/g E Lg(c - H/4, d + H/4)

such that for

h E R ,

IhI

< H0

,

there exists a

we have

r

lwl/q(y) -

w1/g(y - h)Ig dy < eg / 13.28

c

a

which together with (7.28) implies (7.29)

eg

J31
0

we have

x+h sup c'<x

v-1/PII

P',(Yk'yk-1)

then we proceed analogously using (8.42) and obtain

102

,

yk r

(8.46)

skq

w(x) dx < (s

f

s

1

lu'vl/pljq

Bq

p,(yk'yk-1)

xk

The formulas (8.44) and (8.46) yield yk

skq I w(x) dx < xk

s


1 1<s

arbitrary. Consequently, for the best possible constant we obtain the estimate 2

C < inf

ss- 1 B = 4B

.

s>1 Thus we have proved the implication (8.38).

8.5. Remarks. (8.50) 104

(i)

11

Note that in the proof of the implication

C

2-1/p B< C

1

we have not used the assumption

p

q

-
p -

Let

a= 8 P- P - 1.

,



1

ponding Hardy inequalities hold respectively on the subintervals

(0,1)

,

107

and for the classes

(1,m)

ACL(0,1)

,

ACR(1,-)

. According to Example

6.10 and Remark 6.11, the conditions which ensure the validity of these Hardy inequalities are given by (6.31) and coincide with the conditions (8.60). However, the approach used in this remark guarantees only the

sufficiency of the conditions (8.60), while their necessity follows from Theorem 8.2.

Therefore, a natural question arises whether this coincidence is of splitting the interval

accidental or if the 'trick' subintervals ACLR(a,b) ACR(c,d)

(a,c)

,

into two

(a,b)

and investigating the Hardy inequality on

(c,b)

via the investigation of the Hardy inequality on

ACL(a,c)

and

could be used also generally. Let us describe the general situa-

tion.

We investigate the Hardy inequality (8.63)

luwl/g1Iq,(a,b)

For this purpose choose

C

llu'vl/pllp,(a,b)

c E (a,b)

jj

(8.64)

on

ACLR(a,b)

and investigate the Hardy inequalities

uwl/g q,(a,c) 5 CLllu'vl/PI p,(a,c)

on

ACL(a,c)

uwl/gllq,(c,b) < CRl u'vl/pllp,(c,b)

on

ACR(c,b)

II

and (8.65)

The inequality (8.64) or (8.65) holds for

1

< p < q < -

.

if and only if

BL(a,c) = BL(a,c,w,v,q,p)
0

c E (a,b)

,

then we will consider two different cases:

Let

1

5_

q


(9.12)

A

(9.13)

AL
- P-L

[FL(bn)

lim (An k)r ' k.w

due to (9.18) and

r

lim (An* k)r q

k1-

FL(bn) > 0

.

Letting

n---*

AL ?

(A* )r lim (lira (A*n,k )r) = f'! q

q

k-w

n+w

and the implication (9.13) is proved. 11

In the next assertions we will deal

9,6. Remark.

with the inequality

-

equality (9.1)

b

[J(Hf)(x) w(x) dx j

(9,19)

instead of the in-

-

b

1/ q

1 1/ p

(

J fp(x) v(x) dx

C

a

1

]

a

for

This is possible since the assertion of Lemma 1.10 holds

f E M (a,b)

also for

p

from (9.2). The reader can easily realize that the proof

q

,

of Lemma 1.10 remains valid if we replace the assumption assumption

q > 0

9.7. Lemma.

Let

number

AL

f E M+(a,b)

(i)

(9.21)

1

by the

The same is true as concerns Remark 3.7 and Lemma 3.4.

0 < q


Let

.

f E M+(a,b)

and assume in addition that

f E L1(a,b)

and (9.22)

P = vl-p, E L'(a,b)

Consequently, the assumptions (9.3) of Theorem 9.2 are satisfied and there exists a function

f0

satisfying (9.4) - (9.6). From (9.4) we have

((b(

(9.23)

}1/q

IJ(HLf)q(x) w(x) dx 1J

a

x(

b(

(J

a

a

(b(

xr

q

f(t)

I

q

f0(t) dt)

IJ

1/q

w(x) dx

dt)

l

1/q

w(x) dxj

(J

a

and since

p,(a,b),P

lg/pIi=

Igv1/p p,(a,b)

a

due to the definition of

p

-

133

see (9.22)

we can rewrite (9.6) in the form

-

l1/P

br

b(

fP(x) v(x) dxl

Ifa

=

IJ fp(x) v(x) dx l

JJJ

11/p

a

This together with (9.23) implies that it suffices to verify the inequality the inequality i.e. instead of f f0 (9.19) for ,

x

b

f0(t) dt)

(9.24) If

(J

a

a

b

1/q

q

w(x) dx)

q1/q AL IJ

11/p

fp(x) v(x) dx

a

Since b

x(

l f0(t) dtl w(x) dx = 4

1J

1

a

a

)))

b

x

y

[J

l)

a

a

= q

f0(t)

l

a

dtl-1 q f0(y) dyI w(x) dx

the Fubini theorem yields q

0(t) dt)

(9.25)

w(x) dx =

a

a

b lq-

f0(t) dt) a

Let

f0(y)

II w(x)

a

dx)

dy

y

be fixed. The condition (9.5) implies

y c (a,b)

f0(Y) p(t) < f0(t)

t G (a,y)

for every

P(Y)

and consequently (note that

0 < q
0

(9.33)

An > 0

a.e. in

(an,bn)

and

IJ w(t) dt

dx1

w C W(a,b)

,

we conclude

.

The Fubini theorem yields (b(

b(

1/q

(

An/q = i

J

fn(x)

=

I

lx

l

a t(

b(

( j

( J

W(t)

II

a

l 1/q l fn(x) dx] dt1

a

and since

x

x

f(t) dt =

fn(t) dt

If [

a

a

1/q

-q' [qJ[J a

we have

r/q

b(

(

(9.34)

q An

J

a

If we denote

138

fn(s) dsJ

f n (t)

a

t1/q' fI

x[In If

a

(s) ds

a

dtj

1/q

q

fn(t) dtI

w(x) dx

gn(t) =

(t

1/q.

l

fn(s) dsJ

IJ

fn(t)

a then we can rewrite (9.34) in the form b

1/q

[J(Hg)(x)

q

w(x) dx J

a

and the inequality (9.19) yields b ((

q An /q - C

II

11/p = gn(x) v(x) dxJ

a rb(

(xr

1/p

p/q,

l

C

J

a

fn(x) v(x) dxJ

U fn(s) dsJ a

bn

x

dsIpli-q1 v(1-p')q(x)

V'-p'(s)

= C {J fn (x) vp'(x) an

x

lJ

an

r

I

lan

1/p

p(q-1) v(1-p')(1-q)(x)

fn(s) dsJ-p/q

dx

V'-p' (s) ds I lJx

an Using Holder's inequality with exponents

1

and the definition

1/(1 - q)

q

of

fn

-

(9.31)

cf.

q

-

we obtain

Ar/q n

bn (

0

=Cj (9.35)

J

bn

x

fn/q(x)

l

vp,/q(x)

llJan V'-p'(s) II( ds

}q/p

1-p,

p/q' v

(x) dx

J

x

p/q

x

fn(s) ds

}(1)/P V'-p' (s) dsJ-p v1-p,(x) dx

=

I

an

an

an

139

bn

x

P/q

rq/P = C An

j

fn(s) dsJ

IJ

I

(1-q)/p

_

w(x) dx

}

flan flan

where

x I-P

(9.36)

vl-pa(s) dsv1-pI(x)

w(x) a

x E (an,bn)

,

n

Since, in view of (9.29), bn

x dsll-P

w(t) dt 5

(9.37)

p

1




1

_

t s

,

0 < t < s < x

therefore,

;

0 < x - t < x

and

and consequently s

K(x,s) < ( xm-1(s 1 1

-

t)n-1 dt

xm-1 sn n

0

147

S

m-1

(

xm+1 -

s -n K1(x,s) = s

I(1 J0

x

s t

(1 - S

1

=

s

n-1

(1 - 5

dt

m+n-2

dt

0 K2 , where we use the fact that in (10.25) we have

similarly for

0

1

-

-

Thus, the function

(J1f)(x) = J

K1(x,s) f(s) ds

0

is equivalent to the function x

x

J

m-1

s

n

f (s) ds

0

and the inequality (10.26)

f

Jlfllq,(0,-),w < C

p,(0,°°),v

will hold if and only if the inequality x (I

(10.27)

x

(f

n

m- 1

s

q

f(s) ds)

l1/q < C w(x) dxJ

ll1/p

(I

fp(x) v(x) dx] lJ

lJ

0

0

0

holds. However, the last inequality is nothing else than the inequality replaced by

(1.12) with f(s) w(x)

,

replaced by w(x) = x(m-1)gw(x)

v(x)

r

(r

m-1

x

IJ

n s

and with the weight functions

f(s) = snf(s)

f(s) ds

,

v(x) = x-npv(x)

.

Indeed,

w(x) dx = I

0f

0

. =

(

J

lJ

0

and 148

X

fr

°(D

x I

sn f(s) ds q x(m-1)q w(x) dx = 0

I

(J 1

0

U 0

l

f(s) dslq w(x) dx 111

fP(x) v(x) dx = J(xflf(x))P x-np v(x) dx =

I

fp(x) v(x) dx

J

0

0

0

But then, according to Theorem 1.14, the inequality (10.27) holds if and

only if (10.28)

B1(0,-,w,v,q,p)
q

AR

,

replacing the

from (1.19), (6.7). According

to Theorems 1.15, 6.3 and 9.3 we immediately obtain

10.8. Theorem. Let

Iq w(x) dx = ju(x)I

[Jiux)Ip w(x)

dx

1

q/p

W4/P(x) w-q/P(x) w(x) dx

(I(o,w,w,q,P))

(p-q)/p

J Q

(12.12)

I = I(0,w,w,q,p) =

f

w4/(q-P)(x) wP/(P-q)(x) dx

J

0

The estimates (12.11), (12.10) imply dil/P

(12.13)

[Jiux) q w(x) dxjl/q < C I1/r i=1

0

where

1/r = 1/q - 1/p

q < p

with

(i.e. for

fax.(x)P

vi = vi

,

.

0

i

vi(x) J

However, (12.13) is the inequality (12.7) for

and consequently

p = q ) provided the number

we have derived (12.7) from (12.10) I

from (12.12) is finite.

12.5. Some special weights. In the theory as well as in applications, weight functions appearing most frequently are of the type

(12.14) with

M L S2

(12.15)

with

and

a E R or, more generally, of the form

v(x) = v(dist (x,M))

v = v(t) E W(0,-)

notation

172

v(x) = [dist (x,M)]a

(see Definition 1.4). Sometimes we will use the

(12.16)

dM(x) = dist (x,M)

for

x (E

12

.

Investigating the inequalities from Subsection 12.2 for these special weight functions, we can use with advantage the 'one-dimensional' approach, exploiting the results derived in Chapter 1. Let us illustrate this approach by some examples.

12.6. Example.

Let

be the cube

12

N

x = (x1,x2'...,xN) E R

,

M

and let

Further, let K = CO(Q)

for

;

dM(x) = xN

(12.19)

for

w(x) = w(xN)

(i)

Q

,

RN-1 ,

i.e.

.

w,v1,...,vN E W(0,1) p = q

For

= (x11x2,...,xN-1) E

'

xN = 0}

and consider the inequality (12.7) on

with the weight functions

w(x) = w(dM(x)) Since

x

be the 'basis' of

M = {x E Q

(12.18)

.

denote

with

x = (x',xN)

(12.17)

Q = (0,1)N

,

x E Q

vi(x) = vi(dM(x)) ,

we obtain

vi (x) = vi(xN)

Under the notation from (12.17) we have 1 11

(12.20)

[Ju(x',xN)Ip w(xN) dxN] dx'

JIu(x) S2

P w(x) dx = 1 M

0

Using the one-dimensional Hardy inequality, we can estimate the inner integral on the right-hand side of (12.20) obtaining 1Iu(x',xN)l

1(

w(xN) dxN

(12.21)

Cp IIau (x',xN)IP vN(xN) dxN

0

0

N

provided (12.22)

with

43 (O,1,w,vN,P,P)
ql/q

w(xi,xi) dxiJ dxi

9q(x',x i) i n

Arq/p(xi)

11/q An(x') dxiJ

l

J

J II

p i (0)

1/g/ gl/q Er

f

P

1/q

r

An,k(xi) dx' A

i (s2)

Using now the inequality (13.39) for the function

un,k E ACi,L(Q)

,

we

obtain in view of (13.47), (13.48) that (2

ql/q

1/q

r

r

)

An, k(xi) dx1

An,k(xi) dxiJ

1/q'

< COl

J

P

i

J1/P

(0)

and since 0
0

v

IYIp-1

i

and consequently

the inequality p

r

N

LNP-1

(14.22)

N 1 gulp IQYIP

v +

lax.l

v

IYIP 1 Iulp-1

p ax.

with

yi

from (14.20) holds a.e. in

sgn u v yi] ? 0

12

Since au u p-1

p ax.

sgn u v y

ulP

=

a aX.

1

v Y1)

Iulp ax.

(v Y1)

the inequality (14.22) implies N

(14.23)

au

Np-1

ax,

i=1

v + (p -

p

1)

lulp

IVYIP

IYIP +

C

i=1

Using (14.20) and the fact that

lulp ax.(vyi)

Z:

C

i=1

i

(1u1p v yi)

ax

.

i

is a solution of the equation (14.19),

y

we have

ax (vyi)

L

i

[,Iv Ip-2 N

ax,

L

1

y

ax

i=1

Y

y ax

P-2

ax.

r

YI1-P sgny]I

=

1

-

2

NC

IvYlp-2

1=1

212

IYIp-2

y

i

+

C

i=1

(1 - P)IYI-P ('Y ) axi

v =

w+(1 -P)

ivy

P I

v

and this formula together with (14.23) yields N Np-1

p

au ax

/

v - gulp w

L

axi

pulp v yi)

-

i=1

i==1

The rest of the proof repeats the arguments used in the proof of Theorem 14.4 (integration over

Stn

Green's formula and the application of the

,

condition (iv)). C]

14.7. The approach via formulas.

The foregoing theorems have shown that

the investigation of the Hardy inequality is closely connected with the

solution of a certain boundary value problem. Now we will prove a theorem

which is due to B. OPIC [1] and represents an analogue of the approach described in Subsection 2.4. Before formulating the general result, let us illustrate it on a certain simpler case.

It can be shown that the Hardy inequality 1/p

(14.24)

i?1

J

v.(x) dx

ax.

sz

St

holds for every

1/p

p

au (x)

N

[Ju(x) P w(x) dx

if the weight functions

u E C - (St)

w,v1,...,vN

are given

by the forrm1ias (14.25)

w(x) = div g(x)

(14.26)

vi(x) = pP lgi(x)lp [div

where

g(x)]1-p

g = (g1,g21...1gN) N

(14.27)

,

i=1

i = 1,2,...,N

,

is an appropriate vector function such that ag.

div g(x) _

,

aX

i

(x) > 0

a.e. in

Q

The formula (14.26) can be rewritten in the form ,

(14.28)

div g - pp

1-PI

vi

lgilP

r

= 0

,

i

and the formulas (14.25) - (14.28) can be exploited in two ways:

213

If we suppose that

(i)

w

function

are given, then the weight

v1,v2,...,vN

for which (14.24) should hold can be determined by solving

the system of non-linear differential equations (14.28) (for the unknown functions (ii)

gi ) and using (14.25). If

is given, then we have to solve the equation

w

div g - w = 0

N = 1

If we take

by (14.26).

vi

and then determine the weights

0 = (a,b)

,

g1 = g

, write

,

v1 = v

and assume

in addition that x

1 vl p'(t) dt


0

for

the function

and consequently

x E (a,b)

w

from (14.25)

is given by

x

w(x) = g'(x) = W)

-P

vl-p, (xW

vl-p'(t) dtJ-p

J a

This formula coincides (except for a multiplicative constant) with the formula (2.6), and thus

the approach just described is a natural extension

of that in the one-dimensional case.

The formulas (14.25), (14.26) are also extensions of the formulas (2.11): the function

g

from (14.25) is connected with the function

from (2.11) by the formula

A = (1 - p') In g

.

14.8. Example.

Let a function

x E 0

g = grad G

214

and put

C = G(x)

be such that

in (14.25), (14.26). Then

4G(x) > 0

for

A

(14.29)

aG p aXi

(AG) 1-P

and the inequality (14.24) assumes the form

11/p f(Ilu(x)lp

(14.30)

4G(x) dx]


0

(14.31),

for

n

> 0

and, consequently,

sufficiently large. Moreover, by virtue of the assumption

J(O ) < -

[J(Qn)]1 /p

J(S2)

for

u

and letting

n

satisfying (14.33). Dividing (14.38) by in view of (14.34) we obtain the desired

,

inequality (14.24). (ii)

Holder's inequality and the formula (14.26) yield au Igil

ax

L

N

Du

p

pJ1/p IiNjlg1!P']1/p, L

ax.

i=1

i

i= 1

P-1

(div

I

i=l[axi Ip)1

g)1/p Ii=1

vl/(P-1)1/P 1

1

Using also the formula (14.35), we have N Iulp-1

P

J

f

lulp-1

Igil dx = J P

axiI

i=1

N (1Lllax gi

12

n

n N

lulp-1

(div g)1/P

= J R

i=1

au ax.

pl1/P v1/P dx

.

1

n

Now we estimate the last integral by Holder's inequality and obtain from (14.37) the following analogue of (14.38):

iNjgi' N

gulp divg

dx 5

J Q

asp

n

+ lJ Iulp divg

n P

i

N

dx I

au

J

p

ax.

11=1 12

n

iJ dS +

=

111

1

p

u

J

ll1/p

v dxI

n

From this inequality we derive the inequality (14.36) by the same arguments as we have derived (14.24) at the end of part (i). 11

14.10. Remarks.

(i)

Let

Q

be a domain in RN

and denote by 217

C1(Q)

(14.40)

on

which are bounded and uniformly continuous

u = u(x)

the set of functions

au/axi

together with their first derivatives

Q

i = 1,2,...,N

,

.

Obviously, the assumption (14.33) can be weakened to

for every n E N ;

u E C1(Qn)

this last assumption together with (14.31) again guarantees that Green's formula can be used and that (ii)

J(S2n)

(14.39)) is finite.

(cf.

By the same arguments we can show that Theorem 14.9 holds if the

pair of assumptions (14.31) and (14.33) is replaced by

gi E

(14.41)

C1(S2n)

uE

,

for every n E N .

W1'p(S2n)

Let us consider the weight functions (14.29) from Example

14.11. Example.

14.8. Using the formula (14.35) we have ( N

v(x) = pp (AG(x))1-P

I

Y

1=1

ac

ax.

x

P/(P-1)lp-1

)

1

and instead of (14.30) we obtain the inequality (14.36), i.e.

(14.42)

AG(x) dx

flu(x) S2

N

p

< PP

(

i=1

S2

L

i

p = 2

then

,

dx

ax (x)

.

j=1 N

If we set

P/(P-1) lp-1

N``

r (AG(x))1-P

ax (x)

p/(P-1)P-1

8G x

2

and the inequality

= IVG

1111=1a

j

J

(14.42) is exactly the inequality (14.1) with

w

and

14.12. Some applications of Theorems 14.4, 14.6, 14.9.

v

given by (14.3).

Let us check the

important condition (iv) of Theorem 14.4 for some special weight functions. For

1

< p < -

,

x0

(x01,x02""' x ON

) C RN

p -

(14.43)

218

w(x)

(IE - p + Nil l

p

E-p I

x - xO

and

E E R ,

E

p - N

,

put

(14.44)

vi(x) = Ix - x

Ix i - x Oi1

le

2-p

(

lIx --X01

G

It can be shown that the solution

of the differential equation (14.5)

y

has the form (14.45)

y(x) = Ix - x0la

with

a = 1 -

N

e

,

P and the condition (iv) reads 0 < lim-sup

aQ

alp-1 sgn a

x0IE-p

Iu(x)Ip Ix -

J

n N

(xi - x01) vni(x) I ds L

i-1

This rather complicated condition

will certainly hold if

N

(14.46)

Iu(x)lp sgn a

Y (xi i=1

for

n

(14.47)

vni(x)

- x01)

>=

0

on

BQ n

sufficiently large. If we denote

h(x,xO,Qn) _

(xi - x01) vni(x) i=1

then obviously the sign of this function for

x ` aQ

n

will be important

since, for instance, if

sgn a h(x,xo,1 n ) < 0

for

x E F n C 30n

then the condition (14.46) will be satisfied provided

u(x) = 0 for x E P n Therefore, let us introduce some special sets which will be exploited in the following examples:

For

G C RN

,

G C- C 0, 1

,

x0 E

RN

denote

8G+(x0) _ {x E 8G; h(x,x0,G) > 0} (14.48)

@G-(x

0

)

_ {x E 8G; h(x,x0,G) < 0}

219

[Of course,

h(x,x0,G)

is defined by (14.47) where the i-th component of the outer normal to G .]

Let

14.13. Example.

1

< p < W

S0EC0'1

vni

x0 E 0

,

.

is replaced by

Then the inequality

f Iu(x)IP Ix - x0IE-p dx

(14.49)

R

p iN

P
0 (cf. (14.45)), i.e.

sgn a =

1

.

If we put

0n = 0

for every

the condition (iv) of Theorem 14.4 will be satisfied if

u = 0

while the condition (iii) of Theorem 14.4 will be satisfied if

n E IN

on

,

then

aQ (x0),

x0 0 supp U.

So, we obtain the conditions (14.50).

In the case (ii) of Example 14.13 we have and consequently

u=0

(14.58)

a < 0

,

i.e.

sgn a = - 1,

the condition (iii) of Theorem 14.4 will be satisfied if

aStn(x0

on

aQn(x0)C a0+(x0)

However, according to (14.55) we have

,

and (14.51)

implies (14.58). The condition (iii) of Theorem 14.4 is satisfied automatically due to the fact that Stn = 0

x0 It 0n

.

On the other hand, if we took

as in the case (i), the condition (iii) could be violated.

If we suppose in addition that

is strictly convex, then the

0

conditions (i), (ii) from Example 14.13 are simpler:

(ii)

c 0

(i)

is compact in RN

supp u

B p - N -

,

uE

,

1

W1,P(Sl n)

u=0

,

on

852+(x 0) ,

;

y 0

c > 0

(17.11)

As

vnllqq

{v

n

, ,

p,

,

n

such that

} C W1'P(Q;S)

IunIIq,Qn,w

S x 0

;

taking

v

n

= u /c

n

n

we obtain

q

subsequence vIN } .

(u n lll

{u

a sequence

and

,2,w > E + IIv nq,Qn,w

is bounded in

n}

Lq(Q;w)

c

holds for

contrary, that the statement of our lemma is false.

Ilun q,Q,w > Ellun 1,p,0,S +

This implies that

such that (17.8)

n c N

.

Proof. Assume, on the Then there exist

and suppose that

W1'P(0;S)

and a function

and (17.9) holds, there exists a

v E Lq (S2;w)

such that

vnk

-' v

in

Now, (17.11) yields

245

1v11q,Q,w - E + li"11,,12,w

which is a contradiction since

E > 0

.

11

17.5. Remark.

The condition (17.8) is equivalent to the condition sup

lim

(17.12)

ull 1

Qn

where

= 0

n

q,Q ,w

is given by (17.6).

Indeed, according to (17.6), the inequality (17.8) can be rewritten in the form jjujj

(17.13)

q Qn,w


0,

xE

RN .

Then

the inequality 11/q < u(Y)lq dy]

(18.12) [

J

B(x,r)

K r

1/p

N/q-N/p+1(r-p

dy + J

J

B(x,r)

B(x,r)

holds for every

x

,

r and

u

Vu(Y)IdY

lu(Y)Ip

I

with the constant

u E wl'p(B(x,r))

K

independent of

.

[Note that here and in the sequel N

(18.13)

p

IaX (x)

Iou(x)Ip = i=1

.

i

251

Lemma 18.3 is part of the famous Besicovitch covering Zemma and its proof can be found, e.g., in M. DE GUZMAN [1]. Lemma 18.4 is in fact the Sobolev imbedding theorem for the ball the unit ball

B(x,r)

.

and then dilating it to

B(0,1)

Applying this theorem to B(r.,r)

the dependence of the imbedding constant on the radius

Let

18.5. Lemma.

n ? 3m

,

r = r(x)

r

.

be the function from Subsection 18.2 (1). Let

m > max (2, n)

mEN,

we express exactly

,

.

If B(x,r(x))n On x 0

,

then

2m

B(x,r(x)) C

Proof. Let us write*

n = S2 0 U Q

St

where

S20 _ {x e On; d(x) < n}

,

On =

f x E Stn; d(x) > n}

Obviously, it suffices to prove the following two implications:

(18.14)

B(x,r(x))n Stn z 0 > B(x,r(x)) C S2m

(18.15)

B(x,r(x)) n

(i)

Let

On

z E B(x,r(x)]

Iz - El

B(x,r(x)) C O

x 0

y C B(x,r(x)) n On

.

.

Then

Iz - yl + Iy - EI < 2r(x) + ly - El for


n -

3

d(x)

,

i.e.

d(x) > 2

3

which together with (18.17) yields

d(z) > 2 ? 2m > m + 1 m z E SZ

Consequently,

and (18.15) is proved.

The following theorem gives sufficient conditions for the continuity of the imbedding of weighted Sobolev spaces.

Let

18.6. Theorem.

1

p < q

n

n

xk e S2 k

,

,

is f

such -that

1/q

(18.37)

b0(xk b

/p l

rN/q-N/p +1(xk) > k

k c- N

(xk

Put (18.38)

uk - Rr(xk)/8 X3Bk/4

'

k E N

257

where

is the mollifier with the radius

RE

ly - xkI

< ar(xk)}

uk e C0(Bk)

(18.40)

U

au (18.41)

on

1

0 < uk
0

such that

A


(19.9)

ai(Y,) - A < yiN < ai(Y,)}

,

.

19.3. Partition of unity. 0

yi E Ai

}Con C {x E 0; d(x) > n + 1 }' n

and denote

SZn = int (0 \ 0n)

(19.10)

Obviously

n

C 0

C n+l x

[Compare these sets

cf. (18.4). For

n

the boundedness of

0

L)

S2

n=1S2n n

with the analogous sets defined in Subsection 18.1,

sufficiently large the two definitions coincide due to 0 .1

There exists a number {Q1,Q2,...,Qm}

(19.11)

with

from (19.6) forms a covering of the closure of the set

Q i

270

n E N such that the system

Stn

.

Let

{a1' 2,...,Om}

be the partition of unity corresponding to the covering (19.11), i.e.

Oi E C'(RN) (19.12)

supp i

,

for x E an

Oi (x) = 1

WW_

19.4. The distance.

L2th ri (19.14)

1

m

i=1

(19.13)

0`i6

Qf ,

Denote

di(x) = dist (x,r.) from (19.8). For d(x) = di(x)

A

> 0

sufficiently small we obviously have

x E U

for

Vhere Ut i

= S2 n supp of

U* C L'

,

Moreover, the following estimate holds:

(19.15)

ai(yi) - yiNl 1/K 1

1 + A

for

Kp

for

K (p - 1)



if x e 0 \ U 1

0

where

Iz'I

we define

(YP) u n (a (Y') - Y 1 1 1 1N

vn(x) _

for

supp v C Gn C Gn C U n

1

al(Y1) -

G C RN

IN

(0,A )}

such that

instead of

b

The function

(cf. (19.5)) and consequently, it

O1

is Lipschitzian on

a 1

follows from (19.79) that

vn E Wp'P(G

n

Moreover, in view of (19.80), vn E Further,

W10,P(SI;d6,ds

d1(x) = d(x) for x E Ui (cf. (19.14)) and we obtain

IIvnil,Q,d p

The estimates

IDI

IIvnIIp

s - IIvnilp,U

0

From (19.82), (19.83) and (19.77) it follows that the sequence Lq(St;da).

is unbounded in

which is bounded in the imbedding of

W0'p(S2;d8,d8)

the less so, compact.

19.23. Remark.

into

Lq(R;da)

cannot be continuous, and

In the proof of Theorem 19.22 we in fact have shown that

is not even continuous. The same is

Lq(SZ;da)

into

W0'p(S2;d8,d8)

true for

is compact or it

the imbedding of

W1'p(Q;dd

(see the following theorem).

(19.84)

Theorem.

Let

S2EG'0'1

< q < p
p - 1

+

6 + 1 -

1

q

286

-

,

p

q

p

> 0

a,

,

Consequently,

E

either the imbedding of

19.24.

{v n}

8 E R . Then

or

- 1
- 1

p

or (19.87)

S- 0

,

In the cases (19.85) and (19.86) the proof is analogous to that of

Proof.

Theorem 19.22. For

6 < -

we have

1

W1,P(Q;d6,d6) = W01,P(O;d6,d6)

(cf. A. KUFNER [2], Remark 11.12 (ii)), and the result follows from Theorem 19.22.

E

19.25. Remarks.

The necessity of the condition (19.87) cannot be

(i)

proved in the same way as in the case of necessity of the conditions (19.85), (19.86): If we used functions

vn

defined analogously as in (19.79), we W1'p(0;d6,d6)

would not be able to guarantee that they belong to 6 < -

1

since for

the inclusion

C(0) C W1'p(Q;d6,d6) does not hold.

We have derived necessary and sufficient conditions only for

(ii)

0 C

0,1

,

i.e. for

K =

1

.

In the case

0 < K
n} = {x E RN;

IxI

> n}

IxI

.

This class of domains will be denoted by

(20.2)

;b

in fact,

;

if there exists a compact set

12 E .0

(20.3)

S2 = RN \ K

K C RN

.

We will mainly deal with the following special cases: K = G

where

K = 0

,

K = {0}

is a bounded domain. Then

G

Q = RN

or St = RN \ {01

or

0 E , . The role of the set

Let

such that

SZ

=

RN \ G

.

from Subsection 17.2 will be

Qn

played by

Stn = {x E S2; IxI < ni

(20.4)

and we denote (20.5)

Stn

int

=

(S2

\

52

Again we have 0n

S2

n+1 z

moreover, according to (20.1), for complement of the closed ball

20.3. The function a function

c r s'

1

r = r(x)

r

.

the set

Stn

coincides with the

B(0,n)

S2 E

defined on

2)

Stn

.

We will suppose that there exists see (20.1)] and a constant

[for

such that

(20.6)

r(x) < 3 IxI

(20.7)

c-1 < r(y)
n

r

for a.e.

r(x) ` c r

x E Stn

for a.e.

,

x E Stn

and

y E B(x,r(x))

20.4. Remark.

If we compare the assumptions about

and

0

with those

r

of Section 18, we see that there are certain differences - in the classes of domains considered,

- in the definition of the sets - in one property of

r(x)

that the ball

xl/3

[see (18.4) and (20.4),

[see (18.7) and (20.6)].

The important auxiliary function by the function

0n

r = r(x)

is now 'controlled' from above

which together with the

B(x,r(x))

belongs to

n

relation

12 E 0 ensures

provided

B(x,r(x)) n 03n x 0 This is the situation which occurred in Section 18 due to the condition (18.7) [see Lemma 18.5], and we may introduce the following and

n = max (n,2)

.

convention:

20.5. Convention. All assertions formulated in Subsections 18.6 to 18.12 remain true if we suppose that 1 E 2 [instead of (18.2), define 0 by n (20.4) [instead of (18.4)] and assume that r = r(x) satisfies (20.6)

[instead of (18.7). All other assumptions (about the weight functions v1 , w

,

numbers

about the auxiliary functions d^)

n

63n ,

r

,

b0

,

b1

,

b0

b1

,

,

v0

about the

) remain unchanged [compare also the identical conditions

(20.7) and (18.8)].

The proofs of these 'new' theorems are literally the same as those of the 'old' ones, and therefore, the formulation as well as the proofs are left to the reader.

Now, we will give some examples in which we will use the following notation: For (20.8)

x E '0

,

put

a* = inf {IxI; x E c}

and denote by (20.9)

¶

0,1

the set of all

0 E 2 such that

12 = RN \-G

with

G E C

0,1

.

Theorems 18.11, 18.12 together with Convention 20.5 imply the following results.

20.6. Example.

Let

1 < p < q < -

,

a, B E R ,

12 C 0

0,1 ,

aX > 0

.

Then

289

W1,P(c2;

s-P,

x

G Lq(52; Ix!a)

,

1x13-P, lxl

W1'P('R;

C

XIm)

if and only if N

- P + 1? 0,

Q -

- N+1 >0

a

P+

qN

N

p+

1

S 0

- N+

1

< 0 ].

-

q N

q

[Here we set

r(x) = lxl/3

20.7. Example.

For

,

p

Let

1

q

,

-

S

+ N q

p

p

b0(x) = lxlm

< p = q
0 p

q

q

p

a-S0, q q p p [Here we set

r(x) =

20.9. Remark.

1

b0(x) = ealxl

,

,

b1(x) = eSIxI

]

Let us go back to Example 20.6. The condition

a*

H E 0

(together with the assumption

0,1

) has guaranteed the validity of

the 'local' imbeddings (18.18), (18.26) since the corresponding weighted spaces

and

W1'P(12n;v0,v1)

Lq(S2n;w)

are for

a;;

> 0

isometrically

isomorphic to the corresponding non-weighted spaces and we can use the classical Sobolev (Kondrashev) imbedding theorems. This approach fails if

a

= 0

.

Nevertheless, for

can use the results from Section 18 since

d(x) = dist (x,M) = IxI obtain for

:

S2 = RN \ {0}

W1,P(Q;

Q = RN \ {0}

we

(see (18.2)) and

iU = {0} x 0

according to Example 18.15 (and Lemma 19.14) we that

xlR-p,

Ixl8)(; Lq(12;

xla)

if and only if

N-N+1 q p

0, qa-s+N-N+1=0. p p q

The same result obviously holds if we take a certain difference: while the spaces W1'P(RN \ (0};

Ixls p,

lxls)

Q = RN

.

However, there is

W1'P(RN \ {0}; IxI8-p,

IxI8)

and

are well-defined since the conditions (16.18)

291

and (16.19) are satisfied for every =

lxls -p

vl(x) =

,

xls

S E R , in the case

the conditions (16.18) are satisfied for S q

S > p - N

lxls)

,

.

Therefore, when

W0'p(R';

lxls-p,

lxls)

.

Radial weights.

.

v0(x) =

S2 = RN,

Now we will consider imbeddings

of the type

Wl,p(Q;v0,vl) L Lq(S;w) for

< q < p
0

.

Then there exists a partition of unity

R

=

with the following properties: R,

2 E CW(RN)

supp 0R C B(0, R + 4)

,

RN \ B(0,R)

,

supp 42 C R

on RN

(iv)

0 < 0i < 1

(v)

01(x) + 02(x) = 1

(vi)

there exists a constant

,

i = 1,2

for

x E RN K > 0

independent of

R

such that

i (x)

ax

for

s K

i = 1,2

j

,

j

The proof is standard and is left to the reader.

Let

20.12. Theorem.

1

< p < -

that there exist a constant (20.12)

0 (t)

2 E `0

,

and a number

k > 0

to > a*

t > t0

for a.e.

->- k v1(t) t -P

v0, vl E WC(a*,-)

,

.

Suppose

such that

.

Then the set

Cbs (2) = {g E

(20.13)

is a dense subset of Proof.

W1'P(SZ;v0,v1)

u

where

and fix

u E W1 'P(52;v0,v1)

Let

function

supp g ( S2

is bounded} vi(x) = vi(IxI)

c > 0

,

i = 0,1

Then there exists a

.

,

E

uEE Cm(S2) (-" W1,P(S2;v0)v1) = V

(20.14) such that

(20.15)

lu - uEI1 1,p,S2,v0'v1


0

see (20.1)) and for

lx

for

t

>_

7/4

denote

R}

Fh(x) = fh

J

,

x E RN

I

Further, for

s > 0

denote

Sts = Ix E 0; The function

Fh

xl < s}

,

Sts = int

from (20.17) belongs to

(S2

\ Sts)

C'(RN)

.

and satisfies

293

0 i Fh(x) < Fh(x) =

for x E

1

for

1

RN ,

x E 12R+5h/4 U a0

'

(20.18)

x E RN

for

< cf h

supp Fh C B(0, R + 2h)

j

,

= 1,2,...,N

.

If we define (20.19)

with

u

(20.20)

ue,h(x) = uE(x) Fh(x)

E C- (R)

ue,h

,

from (20.17), then obviously

Fh

from (20.14) and

x E S1

,

supp uc,h C B(0, R + 2h)

,

1R+h ) C C 12

supp (uE -

.

These properties together with (20.18) and (20.12) imply that for

h > max {R, t0-R} 1/p

I1/p

(20.21)


a*

and let lim a(t) = 0

(20.29)

t-*

Then

(20.30) with

Lq(S2;wA)

c

a(x) = A(Ixl)

.

Proof. Using the density argument we can consider

by zero to the whole R

N

.

Extending

and introducing the spherical coordinates

g(t) = u(t,O) E C0(a -)

we have that

u E C0(0)

for every fixed

0

.

u

(t,0),

Due to

(20.27), we can estimate the inner integral in

tN-1

J lu(t,0)Iq w(t) q,0,w = J S1 a*

llullq

dt dO

by the one-dimensional Hardy inequality according to Theorem 8.17 and arrive finally at the estimate (20.31)

uj

au

q,0,w < c 1

c2IIuII1,p,0,vCv1

at p,O,v1

Consequently, we have proved (20.28). In order to obtain (20.30) it suffices to show proof of Theorem 20.13 (20.32)

Take

lim

ueX

hull

IjullX`1

n>H

,

I Ullq

q,51n ,wa

= 0 , where

q,S2n,w.1

X = WO'p(c2'v0,v1)

Then

.

=

J l_(x) lq w(x) A(lxl)

dx

2n

,1(n) J lu(x) lq w(x) dx =`

pn

298

similarly as in the

that

-

sup

n,

-

a(n) c2

jujjq

in view of the monotonicity of

and of (20.31). The condition (20.32)

A

now follows by (20.29).

20.16. Theorem.

w, v E WB(a*,-)

Let

< q < p < -

1

0 C RN

be unbounded,

Suppose that there exists a number

.

w(t)tN-1,

.4 = 4 (H,

(20.33)

Let

.

v(t)tN-1,

q, p)


a*

such that

°°

Then

(20.34) with

W0'p(c2;v,v) S, Lq(0;w)

v(x) = v(Ixl)

Moreover, let

w(x) = w(lxl)

,

A E WB(a*,-)

satisfy the assumptions of Theorem 20.15.

Then

(20.35)

W0'p(E2;v,v) C C> Lq(0;wa)

a(x) = A(lxl)

with

First we will prove (20.34). According to Theorem 17.10 it suffices

Proof.

to verify that lim

(20.36)

sup

nQn = Qn+5

where

Let

q,Q

Ilull 0

(1

then we can give necessary and sufficient conditions:

S2 E CO'1

21.5. Theorem.

C

K

a-B+N-N+1>0 q p p q

,

or 1

r}

In the case of the space

W1'P(Q;v0,v1)

,

it can be even

.

Using Theorems 21.7, 21.8, Lemma 16.12 and the results from Section 20 (see Examples 20.6 - 20.8, Remark 20.9, Examples 20.19 - 20.21) we immediately obtain necessary and sufficient conditions for the Hardy inequality 308

1/q (21.14)

w(x) dxj

u(x)Iq

K =

to be valid on the class

lll 1/p

C[J

s

X(0)

V u(x)Ip v1(x) dx

, which will be specified in the

following examples.

21.10. Example.

Let

w(x) = Let

(i)

< p,q < W

I

a

x

S E R

xls-P

v0(x) =

0E£

a,

,

v1(x) = Ixls

,

[o C ,V0'11

a* >

,

and put ,

0 / p - N [g > p - N]

0 ,

Then the Hardy inequality (21.14) holds with a finite constant

class

K(0) = W01'p(0;v0,v1) 1

<m

p = q

[ K(0) = W1'P(Q;v0,v1) N

N+1

q

p

if and only if either

p

q

on the

C

- B+N - N+1

a

0

.

q

0

p

or

H = RN \ {0}

Let

(ii)

p - N

S

,

L 0 > p - N [. Then the Hardy in-

equality (21.14) holds with a finite constant W0'P(Q;v0,v1)

(21.15)

[ K(Q) = W1'P(H;v0,v1)

`= p

1

-

0

4 - P + Q

,

.

p + 1 = 0

Then the Hardy inequality

on the class

C

-

K(Q) = WO'P(H;v0,v1)

if and only if the condiiton (21.15) is satisfied.

< p,q < W

S/p-N

,

H t .)

[ Q C

0'1 ],

a* >

1

,

[ S > p - N [ and put

Ixla

lnYlixl,

v0(x) =

lix

-P In Ixl,

v1(x) =

Then the Hardy inequality (21.14) holds with a finite constant

class

K(Q) _

if and only if

p - N < 0 < Np - N

(21.14) holds with a finite constant or

1

on the class

C

K(H) = WO'P(H;v0,v1)

[ K(52) = W1'P(H;v0,v1) ]

x10 lndlxI. C

on the

if and only if one of

the following two conditions is satisfied:

(i)

1

`p=q
0

S2 E IZ

a* > 0

p- N

,

a
p- N, a


1

;

and either

p

or > p

a=

N

p,

< d

or

p - N

,

a < - N

,

p - N

,

a = - N

,

> p- 1

or d

> p -

,

y K d- p.

For the weight functions

(iii)

v0(x) = ealx

,

vl(x) =

the norms (21.9) and (21.10) are equivalent (iii-1)

312

1

on

if

a,

S E R ,

d E R

SZ E 9)

a* > 0

,

a=S

,

(a;S) g, (0;0)

or

=

RN \ {0}

,

a0

,

a<S

,

p>N

or

0 = RN \ {01 or

O = RN \ {0} (iii-2)

on

0 E 43* >0

W1'p(S2;v0,v1)

a

1

1< p < N if

N>

1

if

a* > 0 or

,

p=

a

1

In this section we have been in fact concerned

with two special types of weights depending on d(x) = disc (x,252)

or on Ix

= disc (x,{0})

.

It is possible to extend many of the foregoing results to the more general case of weights of the type (21.17)

where

v(x) = v(dM(x)) v E W(O,m)

and

dM(x) = dirt (x,M)

M C M C Q and

,

mN(M) = 0

.

(See also Example 12.10 where

M was its edge, i.e.

M C 20

but

0

was a polyhedron

M x 2Q .)

One can expect that some of the general theorems from Section 18 can be used with an auxiliary function

r = r(x)

of the type

r(x)

1

.

Some results concerning the case

p = q

can be found in

314

,

A. KUFNER

[2]; as concerns the approach described in Sections 17, 18, cf. B. OPIC, J. RAKOSNIK [1], where also further references can be found.

0

Appendix

22. LEVEL INTERVALS AND LEVEL FUNCTIONS In this additional section, we will give the proof of HALPERIN's Theorem 9.2 which is a fundamental tool for the proof of the Hardy inequality with

0 < q