Best Constant in Weighted Sobolev Inequality with Weights Being

directly from

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Best Constant in Weighted Sobolev Inequality with Weights Being

Powers of Distance from the Origin

TOSHIO HORIUCHI

Department

ofMathematical Science, IbarakiUniversity, Mito, Ibaraki,310,

Japan

(Received16June 1996)

We studythe bestconstantin Sobolevinequalitywith weights beingpowersofdistancefrom theorigininn.In^thisvariationalproblem,the invariance of

n

bythe group of dilatations createssomepossible loss of compactness.Asaresultwewillseethat theexistenceof extremals and the value of bestconstantessentially dependsupon the relation among parametersinthe inequality.

Keywords: Weighted Sobolevinequality;concentration compactness.

AMSSubjectClassification: 35J70, 35J60, 35J20.

1

INTRODUCTION

We

_beginwithrecallingthe famous theorem due toGiorgioTalenti 11]"

TI-I.OREM 1.1

Let

u be any real (or complex)valued

function

ⁱⁿ

C ^(n).

Moreover,

letpbe any number such that: 1 < p < n. Then

IVul

^{pdx > S(p,q,}

ⁿ⁾ lul

^qdx

^(1.1)

Thisresearch was partiallysupported byGrant-in-Aid forScientificResearch(No. 08640163), Ministry of Education,Scienceand Culture.

275

where:

IVul

^isthelength

of

^{thegradientVu}

of

^{u, q np/(n p)and}

( - ^{;r(n) (1.2)}

Theequalitysign holdsin

(1.1) if

^{uhas the}

form:

u(x)

[a

+ blxlP/(P-1)] ^{1-n/p, (1.3)}

where

Ix (x2

^+."

+ x2)

^{anda, barepositiveconstants.}

The main purpose of the present paper is to study the best constant in the imbedding theorems for the weighted Sobolev spaces with weight functionsbeingpowers^of

Ix I.

^Namely,we are interested inthe best constant S(p,q,

, , ⁿ⁾

^inthefollowing inequality:

[VulPlxl

^{pdx >} S(p,q,

, ,

ⁿ⁾

^lulqlxl

^qdx

(1.4)

whereuisanyfunction in

C ^(Rn)

^and

1 1 1-c

+/3

n n

0<---=

</

^{< or,} l<p<

p q n q

1-c+/3

(1.5)

Fortheproofof thisinequalityand related informations, see[9;Theorem 1 in2] and [6;Theorem1 in

3].

Ifot 0

and/3

^< 0,then the best constant isalreadyobtained in 11 and[4]. Theequality signin this case also holds ifuhas thesimilarform in Theorem 1.1. Therefore we are interested in

(1.4)

whenot is apositivenumber.

In

this variationalproblem,the invarianceof

n

bythegroupof dilatations creates somepossibleloss ofcompactness.

As

aresult we show that the existence of extremal functionsessentiallydepend uponthe parameters (p, q,

or,/,

n).

For

example, there is no extremals if

ot

--/3

^{and p 2.}

Moreover

if we restrictourselves to the case when p 2, wecanmake clear the behavior of the best constant S(2,q,

or,/3, n)

rather preciselyasa function of(q,

or,/3,

n)under the condition

(1.5).

It

seems tobe worth mentioning that the equality signin

(1.4)

can not be achievedby anyfunction withcompact support.

To

see this we assume thatthereexists anextremal uhavingthesupportin aball

Br {x

E ]n..

Ix

^<

^r},

namely, the infimum is attained by u.

Here

we may assume u is nonnegative.

Moreover

it has to satisfythe Euler

Lagrange

equationin distributionsense;

div(IxlPlVulp-2vu) )lxlqu q-l,

in

Br

ulsr

^{0, u > 0 in}

Br. (1.6)

Here .

> 0 is a

Lagrange

multiplier.Then it follows from thenextlemma thatuhas to vanish almost everwhere in

Br.

LEMMA

1.2

(Pohozaev

identity)

Let

p, q, n,ot and satisfy 1 < p, 0 <

l/p- 1/q <

(1

-or

+ ^{13)/n, (1}

^-or

+

^{)p <nand}

13

^> -n/q.Assumethat u

WII ⁽ⁿ⁾

satisfy the equation

(1.6)

with Dirichletboundarycondition indistributionsense.Thenitholds that

,k[1 -ot

+/3

n(1/p l/q)]

f ^Ixlqu

^qdx

(1

1/p)

] IxlP(x, v)lVul

^p

MS, (1.7)

Br

where v isthe unit outernormalto O

Br

^{and Sis the}

⁽ⁿ

1)-dimensional Lebesguemeasure, and

W ⁽ⁿ⁾

^is

^defined

^by

^(2.2)

^and

^(2.3).

When 1/p- 1/q >

(1

-or

+ ^)/n

^itfollowsimmediatelyfrom

(1.7)

that u 0. When1/p- 1/q

(1

-or

+ ^13)/n,

^wededucefrom

^(1.7)

^thatOu ⁰

on0

Br,

and thenby

(1.6)

0

f_ div(IxlPlVulP-2Vu)dx ^; f_ ^Ixl’qu

^q-1dx,

^(1.8)

,]lYlr

thusu 0.

Proof of ^Lemma

^1.2

^By a

^standardargumentofregularization,we seethat u issmooth. Then the equalityis establishedbythe computation of divP and an integrationby partsfor

P IxI=PIVulP-Z(Vu, x)Vu. (1.9)

Fortheprecisesee[4;

Prop.

13],

[5]

and

[10].

2

WEIGHTED SOBOLEV SPACES AND

INEQUALITIES

In

this section weshallmodifythe classical Sobolevspacessothat^wecan treat the variationalproblemsin thesubsequentsections.Tothisend we recall theweighted inequality of^Sobolevtype.

LEMMA

2.1 Letpsatisfy 1 < p <

+cxz

and letn satisfyn >_ 2.Suppose (1-o+)p<n, O< 1/p-1/q

(1-c + ^)/n

^{and-n/q <}

3

^{< o,}

then there is apositive numberCsuch that

for

^anyu

C ^(]n),

(L ^lulqlxlq

^dx

)l/q

^{< C}

_(f ^[Vu[P[x[P

^{dx (2.1)}

Ou Ou Ou

Here, Vu

(-x, Ox- OXn)

^and

^IVul (Y--1 I’ff ^[2)

^1/2

Theproofof thisis seen inmany places,forexamplein Maz’ja’s book

[9;

Theorem 1 and its corollaries in

2].

Thisresult is also obtained as a corollaryto the more general imbeddingtheorem in the author’spaper [6;

Theorem 1 in

3].

This lemma naturally leads us to define the following spaces:

Let

1 < p <

+o

and or, be real numbers >

-niP. ^Let L(JR n)

denote thespace of Lebesguemeasurable functions, defined on

Rn,

for which Ilu;

LP(In)II lulPlxl

^{tpdx <}

^+cx.

^(2.2)

W; ^(IRn)

^{is definedby}

WI[ff ^{(IR n) {u tq(P)(IRn) lVu LP(IRn)}, (2.3)}

where

1 1 1-cr

+/3

^np

or q(p)

p q(p) n n (1-ot +/3)p

(2.4)

equip

W:ff ^(]Rn)

with the norm

We

Ilu;

wli()ll-

^Ilu;

L(P)(IRn)II ^{4-IIIVul; LP(n)II., (2.5)}

We

also set

1,p 1,p

R, ^{(]R ) {u} W, ⁽⁾

^uis a radialfunction

},

1,p 1,p

Ilu;

R,(n)ll-

^Ilu;

W, ^{(I)ll. (2.6)}

Under these notations we prepare a compactness proposition ^{for the} imbeddingand restrictionoperators

W21 ^(IRn) L ^(B)

^foranyballB.

PROPOSITION2.2

Let

_psatisfy 1 < p <

+cx

and letnsatisfyn > 2.

By B

wedenoteanarbitraryballin

N n.

(1) Assume

^that

(1

c

+ ¹³⁾

p < n, 0 < 1/p 1

/

r <

(1

ot

+ /

^{n and}

-n/q <

13

^{<or,then thefollowing}restrictions

of

the mapping^arecompact;

Wlot; ^{(]n) _.} Lr(B),

^{p <r < q(p) np/[n p(1-or}

+/3)]. (2.7) (2) Assume

^that

(1

ot

+

^)p < n, 0 < lip

1/r

<

(1

^ot

+ ^)/n

^and

-n/q <

,

^thenthefollowing imbeddingmappingsarecompact:

RI’P( n) -- ^L(B),

^{p <_r <q(p) np/[n p(1-or}

+/3)]. (2.8) In

the assertion

(2)

of thisproposition,^rmayexceed the so-called Sobolev exponent provided/3 > or, because elements in

R,/(R

1,p

ⁿ⁾

^are essentially dependent uponone variable. Andtheproofiselementary bythe use of the polarcoordinate system.

For

the detail see[4;

Lemma

10]for instance.

3

MAIN RESULTS

We

shall studythe followingvariationalproblems.

Assume

that p, q,n,ot

and/3

satisfy

1

<p<+cx,

(1-c+fl)p<n, O< 1/p-1/q

(1-ot + ^fl)/n

n>2,

and

-n/q <

t3

^<

Under theseassumptionsweset

(3.2)

(P) In

thefollowing

problem

weassume insteadofthe inequality

(3.2)

-n/q

</3. (3.3)

SR(p,q, or,

fl, n)

^inf

IVulPlxl

^{p’dx u}

Ro,(IRn), ^Ilu LII

¹

(Pn)

In

theproblem (PR),if we make achangeof variables definedby

Ixl

^r

1/h,

h

(1

-ot

+ ^fl)(n

^p

+

^pc)

n-p(1 -or

+fl) (3.4)

wegetanequivalentvariationalproblem

(P)

^forv

^{C1 (]+)}

SR(p,q,or,

, ⁿ⁾

^C(p,q,

^k)

inf

Iv ^(r)lpr

n/(1-+fl)-Idr

]v(r)lqr

^n/(1-+)-1dr 1

I,

(PR)

where

C(p,q,h)

IS ^n-111-p/q

^hp-I+p/q

^(3.5)

and

sn-ll

^isthe area ofn 1-dimensional unitsphere.Thisproblemwas solvedbyTalenti using the notion ofHilbertinvariant integral. Namelyit follows from

Lemma

2 in 11 that the infimum is achievedbyfunctionsof the form

hp

v(r)

[a

+ blxlT:--

^p(1---+fl)

^(3.6)

Then with somewhat more calculations we see

LEMMA

3.1

Assume

that

(3.1)

^and

(3.3).

^Thenwehave St p q,or,

fl ⁿ⁾

PY

(3.7)

wherey 1 ot

+ ft. ^In

^particular

if

^or

^fl,

^thenwehave

Sn(p’q’t’t’n) S(p’q’n)

(n- ^p+

^p

Pt)

^p

^(3.8)

Therefore weimmediately get

LEMMA

3.2

(1) Assume

that1/p- 1/q

1/n,

1 < p < n andn > 2. Thenwehave S(p,q,

n)

< SR(p, q,or, or,n),

S(p,q,

n)

> SR(p, q,or, or,n),

ifot

^>

^O

(3.9) ifot ^<O

(2) Assume

that

(3.1)

and

(3.3).

Thenwehave

S(p,

q,or,

, ⁿ⁾

^{k1-p-p/qS(p,q,}

^O,

^or,n),

^(3.10)

where k _n--p+otp^n-p

From

thislemmaitseemsthat ifot < 0,

S

_e(p,q,

or,/3, n)

^isalso the best constant fortheproblem (P),and inthe subsequent argumentthisprovesto be true. Thefollowinglemmaispartiallyproved

by

TalentiandEgnell in the case thatot

0,/3

< 0

).

LEMMA

3.3

Assume

thatp, q, or,

13,

n satisfy

(3.1)

and

(3.2). Assume

^that

<t<O,

then

S p, q, or,

fl, n)

Sg(p,q, or,

13, n). (3.11)

Proof of ^Lemma

^3.3

^By

^{apolarcoordinatesystem,we rewrite}

^(P)

^toobtain

I ^{fs fo} [lku12)p/2raP+n-1

inf

n--1

(lOrul2 _{+ --7} g--

^drdSo,

lulqr

^q+n-1drdSo, 1

n-1

(P’)

where

So,

is an 1-dimensionalLebesguemeasure and

A

istheLaplace Beltrami operator ontheunit sphere S

n-1.

Making a changeof variables definedby

r

pk,

k ⁿ P

(3.12)

n p+otp wehave

[

inf

kl-p-p/q

,-

(lOpvl2

^._[_k2

IAvlZ]P/2102

^!

^pn-1

^dpdSo,

u

wl;(n), ^ivl qp(n-p)q/p-1

dpdS 1

n-1

(P’)

wherev(p

co)

k^-1/qu(p^k

co).

Sinceot _< 0bytheassumption,we see k > 1.Therefore we see

S(p,q, or,

8, n)

^> k^1-p-p/q S(p,q,

O, ,8

or,n),

(3.13)

where we used (n p)q/p 1 q(/3 -or)

+

^{n 1.}

Since/3

^{-ot < 0,} the assertion

(4.4)

inLemma4.1 and thespherically symmetric decreasing rearrangementofvleads us to

S(p,q,O,

, n)

Sic(p, q,

O,

or,

n). (3.14)

Therefore the assertionfollows fromLemma3.2.

Now

we are in apositiontostate our mainresult.

THEORE 3.4

(1) Assume

thatO < ot

--/3, 1/2-

1/q

1/n,n

> 2. Thenitholdsthat S(2,q,or, or,

n)

S(2,q,0, 0,

n)

S(2,q,

n). (3.15)

Moreover there exists no extremal

function

^{which attains the}

infimum

ⁱⁿ

(2) Assume

thatot >

O,

ot >

,

^{0 <} lip- 1/q

(1

-ot

+ ^)/n,

^{n > 2}

n Then the

infimum

^{S(p q or,}

13, n)

^{is attained}byan and l < p <

_

+

extremal

function

^{u in}

W; ^{(an) anc}

^thisu

^satisfies

ⁱⁿdistributionsensethe equation:

-div(lxlPlVulP-2Vu)

^I.

Ixlqlulq-2u, _(3.16)

whereI is a

Lagrange

multiplier.

Remark Inthe assertion(1),the best constantS (p,q,ct,or,

n)

isnot known unless p 2. Becausetheproofin thispaper essentiallyusethelinearlity of the Euler

Lagrange

equation.

But

atleast we seethat S(p,q, or,or,

n)

<

S (p,q,n)intheproofof the assertion

(1).

And the bestconstantin assertion (2)isalso unknown forthepresent.

We

also note that if^wereplacetheweight function

Ix

^by

IXn ^I,

^{we can}show a similar result.

4

PROOF OF THE ASSERTION

1

First weprovethe assertion 1.Let beadomainof

R n.

Foranonnegative function

f

⁶

C()

with having acompact support, we denote by

S(f)

the spherically symmetric decreasing rearrangement of

f

^{(the Schwarz}

symmetrization of

f).

^{That is:}

S(f)(x)

sup{t

/x(t)

>

IS-l. Ixl ^},

^/z(t)

^{I{x f(x)}

^>

^}l.

(4.1)

We preparethefollowinglemmas. The first one iswell-known for theproof see 11; Lemma1]for instance

).

LEMMA

4.1

Let S(f)

^{be the}sphericallysymmetricdecreasing rearrange- ment

of

^a nonnegative

function f C(f2)

with a compactsupport. Let g ⁶

C((0, cx))

be a nonnegative decreasing

function.

^Then,

for

^every

exponent p > 1, the followings hold:

f ^s(f)P

^dx

f ^fP

^dx,

^(4.2)

f ^lS(f)lP

^dx

^<- f ^{I flp (4.3)}

f S(f)Pg(lxl)dx

>_

ffP

^(4.4)

The next one isavariantof theHardy-Sobolev inequality.

LEMMA

4.2

Assume

that

f C2(f2),

u 6

C(),

^C

^R

^{n (n > 2). Let}

us set

v(x) S(If" ul)(x).

Thenitholds that

[Vv[ 2dx +

- ^{[A(f 2)- 2lVfl}

^{dx <_}

^IVu

^dx.

^(4.5)

Admitting this in the present we shall establish the assertion

(1)

in Theorem3.4.

Proof of

^theassertion

^{(1) By}

^{the use}of these lemmas for

f Ix ^1’,

^f2

R n,

weseethat

S(2,q,n)

Ivl

^qdx

+

^oe(o

+

ⁿ

²⁾ u2lxl

^2(c->dx

[ IVulZlxI

^2tdx.

^(4.6)

Here 1/2

1/q

l/n,

n > 2.

Hence

if there exists anextremal function u 6

W ^(R"),

then we have

S(2,q,

n)

4-or(or4-n 2)

f ^tZlx]

^{2(c-1)dx <_ S(2,q, o,c,}

^n).

^(4.7)

Thisimplies

S(2,

q, oe,c,

n)

>

S(2,

q,

n),

andobviouslythe equality sign holds in

(4.7)

onlyifu 0.Therefore it suffices to see theoppositeinequality.

Letusset, fory 6

I

ⁿ

\ {0},

S(y) inf{

f. ^[Vul2lyl2

^dx

^[ulqlYl=q

^{dx 1,u 6}

^C(n)}. _(4.8)

Onethencheckseasilythat if wereplaceuby

,-n/qu(.

y/F.),⁸ >0,q

2n/(n 2)

andlet^etend to0,we have

S(2,

q, or, or,

n)

^< S(y). Onthe other hand,it holds

S(y)=inf{f. ^ivul2

^dx.

lulqdx=l}=S(2,

q,n).

(4.9) So

that we seeS(2,q, or, or,2) S(2,q,

n).

Proof of ^Lemma

^{4.2 Firstwehave}

fo

IV(f U)[

²dx

[[Vul2f

²

-+- ^{U2IVf[ 2]}

^dx

+

- ^{VU2 v f2}

^dx

lvul2 f

^{2 l}

f. ^{U2[A(/2) 21Vfl 2]}

^dx.

^(4.10)

Then, from

Lemma

1 we can show the desired result.

Here

we note that this proofstill works if we put either

f(x) Ix c,

^{(or > 0,n >}

²⁾

^or

f ^(x) Ixnl ^’,

^{(o >_}

^1/2,

^{n >}

^2).

5

PROOF OF THE ASSERTION

2

us setforu 6

W;ff ^(]n)

Let

J(u) f,

^lulqlxlqdx,

(5.1) E(’) finn ^IVulPlxl

^pdx

Here0 < (1-or

+)/n

l/p- 1/q,ot > > -n/q, p(1-or

+/3)

< n.

Wealso set for 0 < ⁾ < 1

Sz inf[E(u)

J(u)

),u 6

W ^{(IR n) (5.2)}

Assume

that

{uj}

^C

WI? ^(IRn)

^{isaminimizingsequencesuch that}

lim

E(uj)

^{S=_ S(p,q,ot,}

, ⁿ⁾ J(uj) I

(j-1,2,3

).

j+

(5.3)

In

order toprovethe existence of the extremal function in

W: ^(n),

^{first we}

show thetightnessof thesequenceconsidered.Letusalso set pj

--IVujlPlxlCp + ]ujlqlx] q,

Qj(R) I

_JBR(O)^{pjdx (j 1,2,3}

^{). (5.4)}

For

6 1 ot n/p ande >0,^wesetu^e

eu(x/e).

Then we see

J

(u)

J

(u

e)

and E

(u)

E

(ue). (5.5) Hence

wemayassumefrom the first

Qj(1) ,

1 ^{j 1, 2,3}

^(5.6)

Then we see

LEMMA

5.1 Foranarbitrary ^e > O, thereexistssomepositive numberR suchthatwehave

dx < (j 1, 2, 3

(5.7)

n\B(O)

P

Proof of ^Lemma

^{5.1 Firstwenote}that for somepositive^numberL lim f

I

pjdx L >_

I +

^S.

^(5.8)

j--+^cx:)

JRn

According to the argumentin [8; Theorem 1 in part 1], we just have to show that dichotomy cannot occur. Tothis end we assume thatdichotomy occurs.Then, extracting subsequencefrom

{uj

^ifnecessary,we see that for anarbitrarye > 0,there existpositivenumbers

A

6 (0,L), Rand asequence ofpositive^numbers

Rj

^{such that:}

A-- fB

^{(0)pjdx <}

’ fBj

^(0)\BR(0)

^pjdx

^{<e, and j--+lim}

^Rj

^=cx.

^(5.9)

Let f

^{and g be}nonnegativesmooth functions such that 1,

Ixl

^{< 1}

{

^1,

^Ixl

^{> 1}

f

0,

Ixl

^{> 2, g}

o, _Ixl

<

1/2. ^(5.10)

Let

us set

ul(x) f(x/R)’u,

u2(x) g(x/Rj)

^u.

Then,forany^e > 0there is apositive integer

N()

such that

(5.11)

[IV f (x/R)lPlujl

^p

+ IVg(x/Rj)lPlujlP]lxl

^pdx <

et (5.12)

for j >

N (e). In

fact we see

IP[x[

^pdx

_<lxl_<2R

)P/q

<

luj

^q

Ix

^qdx

_<lxl_<2R

<

CRPe.

Ixl

^{1-- dx}

1<2R

(5.3)

And in a similarway,

dx

CRPe. (5.14)

lujlPlxl

^p

j/2<lxl<Rj

Here

Cisapositivenumberindependentof eachRand_uj.Thereforewe see

(5.12).

Ontheotherhand,wemayassumethat for some numberss, 6 [0, 1]

lq

Ix ^I/q

^dx

--

^s,

f lulqlxl

^{q dx --+ t,}

^(5.15)

I1 ^(s + ^t)l

^{_< e.}

From

Sobolevinequality,there is apositivenumbercsuch that

f ^lvujlPlxl

^{pdx >c.}

^(5.16)

When e tendsto0,we mayassumethats

s(e)

alsoconvergestosome number g 6 [0, 1].

In

casethatg _0or 1, then we see S >_ c/

S

efor any^e > 0, and this contradictstoSobolevinequality. Therefore,^{g 6} (0,

1).

Since

(e)

-+ 1

g,

wehave

S > S --1-S^1---

[P/q

^--[-.

(1 -)P/q]s

>

S, (5.17)

and thisisa contradiction.

Afterallwe seethat under the condition

(5.6)

^theminimizing sequence

{u}.

j=l

c W,

^1,p

^(IRn)

^and

{PJ}-I

^{aretight in}

L(IR ⁿ⁾

^{and a space of all}

bounded measures on

R

ⁿ respectively. To^seethe existence ofextremals,we need^an apparentvariant of the concentration compactness lemma due to Lions in[7]and [8].

For

thesakeof self-containedness we state it here. The proofis omitted.

Let {uj

^{be a}minimizing sequence satisfying

(5.3).

CONCENTRATION COMPACTNESS

LEMMA

^5.2 Let

{uj

^beaboundedsequencein

WI; ^(an)

^{convergingweaklytosome u}and such that

]Vuj IPINI

^paconverges

weakly_{to lZ}and

luj

^q

Ix

^{q convergestightlytovwhere}Ix^{andv arebounded} nonnegativemeasures on

N n.

Then we have

(1)

There^{existsome at} most countablesetJandtwo

families {xj}jej of

distinctpointsin

IR n, _{vj}jej

in (0,

cx)

such that:

lulqlxl

^q

, _{+ 2_.,} ,a,, ,

>_

+ 2..,,a,,

J J

(5.18)

for

^{sometzj >} O. Moreoveritholds that

V;/q

^<

^lZJ.

_S

^(5.19)

(2) If

^v

WI(Nn)

^and

^IV(uj

^q-

^v)lPlxl

^{p converges weakly to some}

measure-if, then -/z L

(R).

(3) If

^{u =_0 and}

^lz(IRn)

^<_

^{Sv(ln) p/q,}

^{thenJ isa}singleton andwehave 1

13

’(Xo

S,p/q

^lz,

^(5.20)

for

^{someg > 0and}

xo

^]n.

ENDOFPROOF^OFTHEOREM. Fromthis lemma we mayassumethatthere is aweak limit u

WII ^(Nn)

^{of the} minimizing sequence

{uj}.

^Therefore

it sufficesto show that_uj converges stronglyto u

7

⁰ identically under the condition o > >_ 0.

From

the assumption we

see/x(Rn)

^S and

v(Rn)

1(tightness).

Here

wenotethatthe lack ofcompactnesscan occur only atthe origin.

Because

the weightfunction vanishesonly there.

More

precisely,ifD isboundedopen subsetof

R

ⁿ having ^apositive distanceto theorigin,^thentheimbedding operator

WII ^{(Rn) Lq(D)}

^iscompact

under the conditions 1/p 1/q

(1

^oe

+ fl)/n,

^c >

fl

^{> -n/q and}

1 < p <

n/(1

^ot

+/3). Now

ifu 0,then from

Lemma

5.2andthe above

+/-/z 3o ^But

remark we seev s

1

Qj(1) >_

^1(o)

lujlqlxl

^qdx

--

^1.

^(5.21)

This isa contradiction.

Next

we see_uj converges stronglytou.Letus set a

f, lu

^q

Ix

^qdxand assume that 0 <a < 1. Then from the lemma we have

vo

^{1 -a,} lzo

S13g/q, f ^IVulPlxl

^{pdx <S-lzo.}

^(5.22)

Hence

we see

lVulPlxl

^{pdx S} lzo

S(1 P/q)

<

S(1 vo)

^p/q

Sa

^p/q

(5.23)

Onthe otherhand,itholds

lvulPlxl

^{pdx > Sa Sa}

^{p/q. (5.24)}

Sowereachacontradiction.

Appendix

In

this section wecalculatethe best constantS(2, q(ot),

or,/3(o0,

n), where or,

fl(ot),

q(ot)andn satisfythe relations;

2or

fl

^{q ot n >2,q ot 2(n}

+ 200

ⁿ

+

^{2t 2}

fl(ot)-

^or.

(A.1)

n

+

^2or

^2’

ⁿ

+

^2or

PROPOSITION

A.

1

In

additiontothese assumptionswe assume that 2or isa positive integer. Thenitholdsthat

S(2, q(ot),or,

fl(ot), n) Sn(2,

q(ct),or,

fl(ot), n)

=S(2 2n/(n 2),

n -k-

2ot)

yr-2a/(n+2O

(I’((n ^{1-’(n/2) + 2t)/2)} )

^2/(n+2)

_(A.2)

Proof

Then

S > 2,--inf

IVu Ixl

^dx"

lu Ix[

^{dx 1}

Note

that

fv l7ul2lxl

^2tdx

Z

^Ca

fv ^IVu[

2"2al 2a2^{1 62}

^Xn

2a"

dx’

We

abbreviate

S(2,

q(ot),or,

fl(ot), n)

toS.

Let

usset

V

[0,

)n.

(A.3)

(A.4)

! Then we see where^0- (0-1,0"2

o"n)

and

ca

S _> 2.+2 inf

ca

^(6a

/ Ca )2/q

^(or)

y

e=l,e>_0lal=

[fv

^2..2a

-2a"dx" fvlV,lq(’)2’ ^-2a"dx 1].

inf

IXTva

^{.x ...x}n

(A.5)

Weneeds more notations.

Z

(Z

Z² Z

n)

j (./J

.zJ ^,rJ

^]I?2aj+l

.’1’"2’ "2aj+l

ka IS

^2a

sZazl ^Szan

Va(z) v(lzll, Iz21 ^Iznl)

Under these notations

(A.6)

(A.7)

Here

weprepare

elementary

lemmas.

290 T. HORIUCHI

LEMMAA.2 Under thesamenotations, itholds that Ccr

r

^8or

^kcr ]2/q((*)

inf

-

^/

^C

ye=l,e>0[

1-2/q((*)

(A.8)

LEMMA

A.3

Ca

Isn-ll

¹

r(-)

E _k-- 2nlSn+2(*-I

2nrr^(*

r() ^(A.9)

Proof of ^Lemma

^A.2

^Let

^{us set}

F({ea})a)=(*, ^be) E

^{)7 be}

E ^a

¹

^(A.10)

lal--(* Irl--(*

Applying^themethod ofa

Lagrange

multiplierto

F({e}ll--(*,/z)

^{under the}

restriction

lrl--(*

^{8r 1,8or _>}0, the infimum in

Lemma

is attainedwhen

q(ot)

q(a)

(

²

^)2-q()

¹

11.2-q()

q

(or) lcrl=(* ^{cr /}

Therefore the assertion is now obvious.

Proof ofLemma

^{A.3 In placeof ea, weset}

(A.12)

wherea (0-1,0"2

ng, _lal

_0-1

+

^0"2

-+- +

^{0"n.Thenweshow}

n,a 1.

To

thisend let us set that

lal--(*

8cr

’.’(*

^l(n,

^a) Dn,c, Dn

^{a Fin}_j--l\crjl

^[2crj’,

andl(n,

0)

^1.

(A.13)

We

shall show the assertioninductively. Sowe assume that

e ’

^1,

^(A.14)

Icr=crl-q-... +Crk=/3

forany nonnegative integers k

and/3

satisfying k ^< n 1

and/3

< or, and weconsider thecasek n.

By

thehypothesisof the induction, we see

Dn,cr Dn-l,a

l(n 1,^ot k)

-1.

Irl=o ^k--0 Icrl=c-k ^k=0

(A.15)

Then it suffices to show that

=o ()

^{l(n 1, k)}

-

^l(n,

^)-,

namely

LEMMAA.4 Forn,ot >O, weset

e(n, or)

Z

⁽ⁿ

¹⁾⁽ⁿ

^{-4- 1)...}

⁽ⁿ

³

^{+ 2(or}

^k))

^(A.16)

Then

n(n +

^{2)... (n}

+

^2or

²⁾

e(n,

ot)

(A.

17)

Proof ofLemma

^{A.4 Againwe}make use of the inductiononthe values of

ot

+

^n.Weassume that

(A.18)

holdswhen^ot

+

^{n <m.Nowwe}assume that

ot

+

^{n rn}

+

1 First we see that forany nonnegative integersot andn P(n,^ot

+ ¹⁾

^2P(n,

^or) +

^{P(n 2,oe}

+

^1).

^(A.18)

Then the desiredequality

(A.18)

easilyfollows from these relations.

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