B Opic and A Kufner Czechoslovak Academy of Sciences
Hardytype inequalities
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Scientific &
 Technical
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B Opic and A Kufner Czechoslovak Academy of Sciences
Hardytype 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, 3334 Alfred Place, London, WC1E 7DP. First published 1990 AMS Subject Classification: 26D10, 46E35 ISSN 02693674
British Library Cataloguing in Publication Data Kufner, Alois, 1934Hardytype inequalities 1. Mathematics. differential inequalities 1. Title II. Opic, B. 515.3'6
ISBN 0582051983 Library of Congress CataloginginPublication Data Kufner, Alois. Hardytype inequalities / A. Kufner and B. Opic. p. cm. (Pitman research notes in mathematics series, ISSN 02693674 ; 219) ISBN 0470215844 (Wiley) 1. Inequalities (Mathematics) I. Opic, B. II. Title. III. Series. QA295.K87 1990 512.9'dc2O
8914502
CIP
Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
Contents
Introduction Chapter 1.
1
The onedimensional 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 (Ndimensional) 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 top 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 (onedimensional) 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) =
tEp
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
1ep+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
,
xep 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 functionsnonnegative 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)
v1(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
eBt2/(P1)
(eSt2/(P1) 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;v1P
)
b
G(u) = Jg() u(x) dx for every u e LP(a,b;v) a Moreover, G II
=
,
Ilg
.
p,,(a,b),v1P Consequently, (7.6) 74
[LP(a,h;v)]*

Lp'(a,b;v 1p?
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 v1p' (x)
ge (t)
J[ 1
dx
4 p,
Ig
lP
II
q',(a,b),wq r
=
IUJ
vlp (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
v1/PII
P',(Yk'yk1)
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'yk1)
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
21/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
 PL
[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
kw
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 = vlp, 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
dtl1 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
dsIpliq1 v(1p')q(x)
V'p'(s)
= C {J fn (x) vp'(x) an
x
lJ
an
r
I
lan
1/p
p(q1) v(1p')(1q)(x)
fn(s) dsJp/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
1p,
p/q' v
(x) dx
J
x
p/q
x
fn(s) ds
}(1)/P V'p' (s) dsJp v1p,(x) dx
=
I
an
an
an
139
bn
x
P/q
rq/P = C An
j
fn(s) dsJ
IJ
I
(1q)/p
_
w(x) dx
}
flan flan
where
x IP
(9.36)
vlpa(s) dsv1pI(x)
w(x) a
x E (an,bn)
,
n
Since, in view of (9.29), bn
x dsllP
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) < ( xm1(s 1 1

t)n1 dt
xm1 sn n
0
147
S
m1
(
xm+1 
s n K1(x,s) = s
I(1 J0
x
s t
(1  S
1
=
s
n1
(1  5
dt
m+n2
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
m1
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(m1)gw(x)
v(x)
r
(r
m1
x
IJ
n s
and with the weight functions
f(s) = snf(s)
f(s) ds
,
v(x) = xnpv(x)
.
Indeed,
w(x) dx = I
0f
0
. =
(
J
lJ
0
and 148
X
fr
°(D
x I
sn f(s) ds q x(m1)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 xnp 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) wq/P(x) w(x) dx
(I(o,w,w,q,P))
(pq)/p
J Q
(12.12)
I = I(0,w,w,q,p) =
f
w4/(qP)(x) wP/(Pq)(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 'onedimensional' 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
,
RN1 ,
i.e.
.
w,v1,...,vN E W(0,1) p = q
For
= (x11x2,...,xN1) 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 onedimensional Hardy inequality, we can estimate the inner integral on the righthand 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
IYIp1
i
and consequently
the inequality p
r
N
LNP1
(14.22)
N 1 gulp IQYIP
v +
lax.l
v
IYIP 1 Iulp1
p ax.
with
yi
from (14.20) holds a.e. in
sgn u v yi] ? 0
12
Since au u p1
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
Np1
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 Ip2 N
ax,
L
1
y
ax
i=1
Y
y ax
P2
ax.
r
YI1P sgny]I
=
1

2
NC
IvYlp2
1=1
212
IYIp2
y
i
+
C
i=1
(1  P)IYIP ('Y ) axi
v =
w+(1 P)
ivy
P I
v
and this formula together with (14.23) yields N Np1
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)]1p
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
1PI
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 nonlinear 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
vlp, (xW
vlp'(t) dtJp
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 onedimensional 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) 1P
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
P1
(div
I
i=l[axi Ip)1
g)1/p Ii=1
vl/(P1)1/P 1
1
Using also the formula (14.35), we have N Iulp1
P
J
f
lulp1
Igil dx = J P
axiI
i=1
N (1Lllax gi
12
n
n N
lulp1
(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))1P
I
Y
1=1
ac
ax.
x
P/(P1)lp1
)
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/(P1) lp1
N``
r (AG(x))1P
ax (x)
p/(P1)P1
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
Ep I
x  xO
and
E E R ,
E
p  N
,
put
(14.44)
vi(x) = Ix  x
Ix i  x Oi1
le
2p
(
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 < limsup
aQ
alp1 sgn a
x0IEp
Iu(x)Ip Ix 
J
n N
(xi  x01) vni(x) I ds L
i1
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 ith 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  x0IEp 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/qN/p+1(rp
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/qN/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)
c1 < 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;
sP,
x
G Lq(52; Ix!a)
,
1x13P, 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
aS0, 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 nonweighted 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
xlRp,
Ixl8)(; Lq(12;
xla)
if and only if
NN+1 q p
0, qas+NN+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}; IxI8p,
IxI8)
and
are welldefined 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';
lxlsp,
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, t0R} 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
tN1
J lu(t,0)Iq w(t) q,0,w = J S1 a*
llullq
dt dO
by the onedimensional 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)tN1,
.4 = 4 (H,
(20.33)
Let
.
v(t)tN1,
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
aB+NN+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
xlsP
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/pN
,
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 (iii1)
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} (iii2)
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