B Opic and A Kufner
Czechoslovak Academy of Sciences
Hardy-type inequalities
~ JIll ~
JIll JIll JIll JIIIJ111J111 ~...
48 downloads
1079 Views
16MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
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
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, WCIE 70P.
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
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:
-
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:
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
O
._ ..,
,~~~~~.!.~~~U-}~
.... .;:"~,_
_,,_~;:.;~~.___
-- -_.. --
n
[~~*r-p' dt f/pl
l
p
provided 1
- 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
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
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 0
such that the
a~e
diffe~ential
(4.14)
f(x) >
Further, on equa
~ Aq
K
K
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
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
_ (_ 00)
o, x
x
o,
x +
ooX
00
z
K
L q
inf f
sup O 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 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 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)
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-
~--
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
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 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
(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
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 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 .p-l,
(8.3)
~--=~-"
= .
:;''''"~
;'-:::~-F--- '~"""~=
• ,.
a
,:..~-=
~~_~~_~=
"",_"",~c,==,"""---------'-="~~-=""",,,,", __ ~ -;:c-
--
=
\11
-
~c-""",,,",,=::-~:·.-=;~--~"'~- -;;'7"~_~","~-
-.
S .9. - .
(iii-2) Therefore, there exists a subset
Min
C
Min
Let
v
u
n
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
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
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
- 1
for
13 = p
we immedi
(*)
which now replace the inequalities (8.111), (8.112). From
1
(8.136)
1
~
q < p
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
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=.=,,,".= . ~~.,,,,",,,=..,,".,,,.--=->,,,~.~,,,-,,,,-,,;:,,,,,,~=,-- .....~"'.=.:;.== .._=.--='-'--=."""-..""".,,'=.... - ;;,,.-=~,---'''' 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}
P.(Q) 1
•
consists Let
0 such that the
~n-
if and only if
u E AC. 1(Q) 1,
l/r dX~ ]
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
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
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
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
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 ]
,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
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
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
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
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
Kp ,
o
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
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
~ -~
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
q
°
we will again substantially use the inequality (6.20). To this end, let us summarize the results derived in Examples 6.8 and 8.21 (ii):
Let
°
1;;; 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 ,
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)
-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
K(p - 1)
+ KE
6
1
1
- - -- + - - - + 1 q Kp q P
~_f-+1: q Kp q
q
°
-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
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)
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
°
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
°
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
0 . Then
289
--------------
O~~=~~=
W1 ,p(lt; Ixl s - p , Ix) s) [WI,p(lt; Ixl s - p ,
C;
---'=--
-
----=---~-=--==-==-'-"'=""""=
~-----
Lq(lt; lx/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
q • Radial weights.
20.10. The case
Now we will consider imbeddings
Let
W1'P(~;vO,v1) L LQ (I1;w) 1 ~ Q < P
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
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
n
- w, v E WB(a*,oo) .
such that
N-1 N-1 w(t)t , v(t)t , q,
p)
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)
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-
0 ,
p - N . Then the following three
S p Ixi - ,
B IxI )
~ ~
w~'P(Q;
B p Ixl - ,
B IxI )
0,
B
P - N . Then the following three
>
IxI B)
0. P
KC r.
derive analogous results for the case
bounded domains. - 1 -
B [(p, 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 .
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,
ap-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
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
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) .
--