View of Hausdorff h-measures and p-modules

5 Rùi_arioñs

t b

lvhere

_{I ôu}

the infimum is taken over all u e Ct, with Ll¡o:,I, yu:

âtt' \

:lñ' 'ñ)aud

^rvhere,

^{in the} particular

_casey'

:

n,, t1'e support S,,

s

^Ro).

citY.

f ^.ol

Corres

CR'

C*(F) : _linf

_{I !}

_o(¡

-

t)idp.(x)rtr¡(y)

I'

s'here

lirn

^<Þ(r)

: _+co un¿'rtr. inlinlrrn is taken over all the

me¿rsrrres

: I altd tire

sr.rppor-t .S,

C

^F.

*, L; rvill lrt, thc

vc'ctor

ipace o[

Iunctions

f(r) ^>

⁰

rf

^@

is a ker'er, then, for r<þ (

co and

EcR,,rve

crefine trre caþncity

t't'lrcre

J' = f.i

^ancl

the

convolntion

tÞxf(x)2 | _{Yx =8.}

A

Co,¡ furrctiort (È-).

\\ie / ^u'hich

remincl satisfies these

ilr"i-ir."-ì.u.,ãirtio,l

conditions

is

called

a

^test

_funclion

tor

I <Þ,rf(x)

:

₅ ^(Þ(*

-

^y)f(y)dt,.

. ^r-et

^{ü)L+(E) be}

^the

^cone

^{of all} Raclo'lneasrlres p à 0 carried

Lty E,

i'e' with ulR" -

^/'r)

-

9, .anð.

LI!E) its

s'bspac"

"o,r,por"d

of

arl r'casures

p rvith

_lirrl.l.

= ^{tíe totat} ,rariaïib,í

åì. _¡.r,

< ^o..

Fof

_¡r,

= ti (E),

we have

the

conr.olution

!*,.

^be

^the o-argebra:I:iii],:j,,iÍ;JÏÍl];abie _ror

_e\¡ery

¡.r

bero'-

gtng

to

the corre srli+ of

all posiii""

_nàãu,.l rneasrrres.

For E e ß,,

rve define^ure

,t"ot';f ¡t;, i;;;;;,t,;g

^cøþac,ity.

ca, p(EI

: _sup

_llr¿llr,

c:a,þ(E)

: hf _llfllf,

Il,fll- :

ess

stlp lÍ@)l <

_"c,

llÍllp:l\f@)þttxlþ < ut, I < j, < *,

OT

l¿

th

¡-r,(R")

1<

nreasurable

> 0

^r'r'i

14

^Þrrnu

^c¡.n¡'MeÑ '

2

1

V(i.e. for'afly) P>n- ^1, 1ôg-itt and ttt':'1,'?,"' ^anil àlso"that

'f tu

Co(Eo)

:0 with

V(i> n-l andm:1,2,'...

But I

^obtain even lrrore, i,e. sorne inclusion relatious betlveen Hausclorff h-rneasures

and clilferent

kincls

of

3essel capacities, gencralizing

in

this rvay

the

corresponcling theorems

of l.

,rr.¡tr¡ls and N. \IrtYÉRs

[2]'

^Sorne

of the

inclusions are shou'ecl

to be

^be

important ^lemma of du

Pr,DSSrs

relations,

I

deduce

the

correspon clorff h-meastlres

or

Bessel capacitie

Now,

let

us recal1 different conc Hausd.orff h-measures and.

the

diffe

'[hc

þ-moclul'us

of

^{cLn arc}

_famit'y I- o[ a

^domain

D C R"

^is

. ^Mp(l) : ttlt t

P(x)o clx

rvhere clx

is

^tjne

volum

element and

the iutimum is taken over all

Borel

,n"oiutobl"

-lunctious

p(x)Z 0

such

lhat

lpds

ì ¹ Vf ^{e l''['he}

i¡-modttltts

M(l) : M,(l) is

"tllecl ,iropty ,nod'ol,rrr.i " .

^'

À tor""omorphism f :D?D* ^{is said}

to'be:'IÇquø'sièonfornt'øl

(!'s

Sl(<oo)if

_,j

¡¿(r)

S ¡/1t*) _<

/(,1,1(t), l(

Ivhere

I is an arbitrary

arc

family

contained

in l)

^and.

I.*:,/(f).

^ì

,Û1ne Hailsd.orff k-tùeasure Hn(E)

of a

set

E C ^R'* is the

non-[egative nurnber

Hn@):lim

^iì1r

>,,,,h1d(n),,')1,' ' :

'

ù+0 _{Em}

rvhere

the

measure

function

^/ø

is

^supposed.

to

be continuous, lQn-ne.gAtiv-e,

"áä-a""t""sing in

sorneinterr.al (0,V'),

r'¡

0, ^and- such

that fu kl'¡:9,

and

where

the infimurn is taken over all

^{countable coverings}

{8,,} of

^E

by

sets E,,,having

a

d.iametei rt(É-;,,)

Í 8. " ^ì I

,

The f-cøþacily

of ø

comþøct

iet p 6 ^{R" is given} as

_i :

, capnF

:

inf

i

I,Y q(x)lþ dx,

flcÍ r S

^r,,,,

I

)'

1

fn fl-2

1 f

/u

1il-1

n

fit-I h:2

n

I

/

fn

I

^1o9,,

)'

1og,

logo logo

il-

ißforrgr,,,

1og

log

(2)

^o(z)

:

fl-2

)"-'

(ii)

cE,¿(E)

:

_[Co,

_{p(E)lþ Y}

_{ønalytdc sct E.}

(For the proof of these 3 propositions, see N.

,)rnvnns

_¡12i.¡

Corolløry.

co,p(F)

S

[Co,

o(E))o VE ç.R,.

r'deecL,

from the

precedin! 2-proposäions, we deduce

that

I

cø,p(E)

: _tîp

cø, p(F)

s

;**,¿(G) :

c6,

p(E):

[co, p(E))o ,

rvhere

.F

a1e compact and

G arc

op(11..

Now, i' ordei to obtai.

Bessél capacities, ^.rve shalr

co'sider, in

the

Í:iii:råri: ^of lhe

capacities rronr above,^the

paitic"l", L"r""i oúi:;fr)

The kernel g"(r)

>

^{0 is a}

strictly

dec'reasiug furrction oT

r :

_lø1, continrrous

outside the origin' r"u"îrrt-T),:;"t *

;;:'. ^-,

g,(*) -

^{log .1- ,}

whilc.,

as

-Í

+ co, l'l

(3) Ko*88:

^{q!,()

(cf.

lor

irrstancc N. ¡rEr¿ERS r12l).

And'

nor',

one

introdu""r

tn-å

3 ki'ds oÌ

Besscr, utþøcities

Bo,þ:

Cro,O, Ito,

l,:

^cen,p,

lo,p:lro,, ropositiol

!.

^Cafi

^E -0 iff

^(i,,c.

if

^ancl, onty

if)

thcre cxists ct

fun,ction

Í ₌ Lf

suclt, that

tie

i,ntegral

( ,Iu\,|/^=:* ^yxeE,

J lr - yl"'

taithout being,ictent,ically infinife.

2 - ^{L'analyse numérique} et Ìa théorie de l.approxibation _ Tore Z _ _{No, 1/t9?E}

0<c.3tt

v

t

(i)

cg,e(E)

:

_[Co,

t(E)]þ yn.,,

1!

q-r-l

I

e

hÈlÀrioñs 5

*

^q-n2

Proþosition.

3.

We høve

' ^g,(x): -J-

,: ^,,,*+r[]!

S"(x)

- lxl 2

_{e- ti}

,

Fot ø,

_P

> ^0,

we have also

the

relatjon

16

rvherq

¡r e Zf

_1E¡

wittr

' i

ll<Þ*¡'r'llr;<1,

,,1,

Þrrhir can¡.i'i¡.ñ

+

I | 1-1

þþ'

t: t

'tì ri' , j

c6E:cø,{E) -sup, lltrll.,, ì ,,,

l, , ¡ i,

the

supremum being taken over

all _V = ü ⁽ⁿ⁾

^such

^that

(Þ*¡r(ø)

Zl _{Vx e} R".

,

^,

Such

a

_¡.r,

is

called

a

test ,ne&sure

fx

crÛ.

v

is

called a c*-cøþøcitu'y dístributioi ¡'or E

if it is a tãst -"ur,rr,J ^Io,

co ancl llvllr:

^co(E). _{:, I ,r}

I,et us

define

for all E C

^.R',

cg(E): inf

ce(G),'

G>E

g'he¡e

G

are open sets.

For

-E

e 9r,

clefine

l}re

caþacit

Snch

a

_¡,r.

is

called

a

¿iit, rneasu,re

tor

õr,o(tl).

If

^{C is} a capacity and. &

its

domain, i.e. a class of'sribsets

of

R,'whiöh contains

the

compact sets rand

is

closed under countabie union, C

is

caliecl

an inner

caþøcity

if

E e

^€I

+

^C(E)

: sûp C(F),, ,

^¡t:t,.

' 'F , ,' ,llr.

being taken over

all

compact sets F.

C

^{E.C is cplled an outer,}

7.,p(E): sup

_llpllr,

rvhere _¡,r, varies over

the set of all _¡r e _{f-{ @)}

such

that

t'

o*(ox¡.r,)t-l

(x) Sl Yn

^{e- R".}

the supremum cøþøcity iL

. ,J

E e

^,91

+ C(E): inf

^C(G),

'G

the infirnum

being

taken over all

open G

I

^E.

Proþositiotø 1. Çø,0

is

øn outer caþøcity.

Proþosítion

2.

^co,p

is an

inner'caþøcit1t.

18

^{pErnu CÀR^MÀN}

(For the proof, see JU. c. RESÐTNJar [16], rr. 'wALrrrN 122, 231,

or

Y.

rurzlrra

_[13].)

Corollary. Cap -E

:

0

+

^{81,,, (E)}

:'

^g.

fn

^order

^to

^{irnprove some}

^oI I) ^;\r)-\lIS

^atttl

\r'e tra¡,'e

to

introclnce

the follos'ing

kentels ^:

8u, þ-\, ^rr, g(f) :

g.(r)^q'ft.r,

l'lt'

'11.,,',,,

t

¡u

to, )'!

^r,,,,

To

c1o

this, *'e

^ucecl

-t_o ìrr'estigate

the beha'ior oi the integraut

onry at

r : ₀

and

x, n. \\/t.

lravc.

6

s. P. PrlnoßleZnNsr<ri [15 l,

7

1

gþ.'@)g"þ:þ- r,^,çt(x)

- l:vj ^.ã

^{,,' I'l}

IìËLÀTIONS

(r.*^¡j, )-'[ros,,, ',

19

,x rrtivl;ls

^['2

]

^resLrlts,

0<a.<rt

þ'

þ-- ^t ut- |

À'-l

n

as I --t, ^aD

anrl.

, \ fs"{')(rog t')o '-t:l ('"- ll' 'i.'o*"'

'|)u '"' ^{t' !}

1""' 8t, þ-t, ^{r', ß}

(// .:

_I

¿>n'P-''nt"1\

t

[r,,{,')[ros

),,,,' 'ji, (t"*-,i,)' '1t"',,,;';)u t"'

^v -;> )'¡¡¡,

rvhere r.,, is here as v¡el1 as cverrvhere

in the

_L)al)cr such

that

1or{,,,

I

-'r-

L

ft

is easy

to

see that

go,¡

t,bt,() loÍ ^o.

I n

^{arc kernel:;} (:rct:orclirrg

to

thJ"'abor-e cl.efinition).

'.lhc

corresporrding capacities

u'ill

^bc

cro,þ- _r,r,,B -"= b{o,þ-1,ril, _ß')' cio,p t,tn,þ: ^13çr,¡ 'r,tt,¡"1' 0 ^¿:- _'1" S lt'

luenrrna I If _t"

e,9lL+,

d.þf n ancl ?.->þ -l,

^llten'

ancl

þ'9

Bþ"' U)g".;:r,-r,,,.8Q)

- lxl,'n' ¡rorol J-'

/tog,,

ll --

as .r

-,

^0.

^-rI

^'l.vl

/ \ ""'lrl

/

åt"1iJJt" to

st:e th¿r1 thcse

2

relations iurpl_y

tlie

prececling inccFralit\., as

Rentttrlt''llhis leltlnta g'c'c.ralizes a re,sult <¡f rlu pr,llssrs ¡1rll, ^{as l'.11 ¿rs} klnrna ÍJ.l of ì). aDAtJS

n,ù x, rrrivlins

_{i2 ].}

Propositiou _{5. tf J}

=1.þ, then,for 0,<s{n,2<.¡,{

co,

f ,,(,r) = \ --l*A2q-- _<

"o

; '

l.r- ),!" I

eueryz,lt'cre cxceþt þoss'iltl.tt'i,,

u

s¿t

Ìi

u,trticlt

is oJ

c,þttcir-tt

(:pli

- ^{0 vrj} >

=

^,,

,*

^q". zøltert Cg

:

L'u, zuitlt <Þ(r) ,_.

¡.

^(t.

(-rror ure proof, _scc

\.

rltr l)lcssis ¡14,, ilrcorcrrr {).

.\s

¿r clirect

collseclr äi"g-ì;;rr",,',u"

har_c

thc

iollo- rvirrg gerreralization

"c;,.

oI^

.tloi'y'.'t¡¡ = ïil]'l:',,,,2 < þ.i,tt,,it,t,

^ttt,tL,t.

(-'k,/(v) _,=-0Vp I

os eueíy,zptirí,

,,

> _þ - ^l-qtul 1r:.

1,,¿,

... ^st/ jì

_{zu'illt [JGp,p}

,,.,¡n'Ii).:

ror.c

it is

1¿tlsc,,

tÌrat

go x/(;v) _=-. co

irr a

bouurlccl :,,{¡-,,r,,t

let

_{¡_r.}

} 0 bc a-rìróarur" rvith

_¡-r(ll,)

:

,t

i,{"¡,

r

^{r, ¡¡, ¡}

(.r

_{-r )} r/¡r(_r

) is

lrounclcci

irr

^/1,,.

,ccer<liug proposítiori _¡rnil

-bv

rl(.ans

ol thc

pre_

thal.

\go,:,f(x)tlp.(x)

= ^llf

tt,p,1',(,,i,p.ll¿,

5

ti.l ,;pQill:,p,þ t,,,t,L\

*

_1r¡i)'

^{.,: ,.,}

cotltra<lic1 irrg

thc

hyllotht,scs.

Rett¡ttrh'

Dtr

Pióssis' _1rt'ool'

ol' lris

1h,,,r¡rcur

4

(corrcsponrli.g t"cr orrr llrccedirrg

pro¡'osition) is

incurr.cc1..

I{e

¿rsscrts

ilrat Ii

^L

i iu

^., -,,

; "

.,.i

,",t/u(l)i',' r('lltlìra l), '.r/.i:]- brri flrc ^.ico (:oîr(sr,r,rr,li'g,:*¡,."*rin,,

^{as rr} (.()rìs(.(llr(.lrt.r.

oi his

rt.urrna (gcrrcralizerr

irrr..lr..cf í,;-'iri--'ì",;íi,,,ì"i,

5r. otrr

lj,lll^

Lnts

lact - 'lt!-'" is

^.

.r,i,ñiâi ¿v(s)l" in

',r.t,,*'rrcLe.S

tlit.

_frunL, is s*PPosecl

to bc

¿r bounclecr sct, a'cl

t

e.?) Ia

^iog¡

u-7

1

1' ,,,

I ^P,

1og,,

loÍ r

^-> r,,,,

(nt.:1,2,...)

llg. x trl lo,

I

Q llgop,

p

t,zr, _B >t ¡ri lþ llttllo' Q

is

^a. cr¡trstn,nl íncleþendent o.f _V.

arñ ]- |

,l;;

:

t

Ilolcler

incqr-rality yields

g. *

p(r) :

(g, gop!p t,Dt.p ^'

glp,¡

_{t,,', I ) ':' ¡¿(5¡ 5}

_!:11

S t(gl'

'g*p!r-p,,,,s) x ¡r.(ø))þ'Lg"p,þ 1,ilr,pJt, p'(x)lþ llhrrs

þ'

llg.*rr.ll¡, f

t \et:'(x)e"p!p.-1,n,þ(x)i(x)lF llg"p,p

,,,,8 d<¡.rlllllrrlll

It

^remains

to

shou'

that

þ'

J^{s!"' (x)} eø: ^p - ^{r,,,,s (x) d} x

<.

^rr'

i

1 I

j

I

j

I I l )

)

'J

Remark.

This

theorem ,generâlizes D. ADArrs

and

N.,ùIEVERS theo- rem 3.1

in

_[2].

- ^{. l*rom the}

^{precedi4g theoreur}

^{and the corollary pf}

proposition

4,

^rve

deduce

the

^,

c 9. r. o I 1 a r

y. In

the kyþorhescs of thc þreced.ùry rrreoretn,,

ue

haae lhe inequalities

Br,,¿(E)

S

Bs,,,-r,,,,p¡(E)

<

Q "up Ë

^(rn

:,1.,2;i,.i..).

Proposition 6. VECR",

Co(E)

>0 ^or _"ö(E) >0=H,(E) :

(The

proof irr the

case

C6 is ^given bv S. _-I. ,Iaylor llgl

^theorems

.1

and 2 rvhile, for the

case

i'5,

r,vc

may

use

the métuóaò'oi

chap.

rv in carleson's book t8l ^{and the} capacitability results of n.

Fdglede

t10l).

Theorent.2.

Let 1<q <þ < ^æ _{and þ} 11, then,

_,,r,,,

B*,¡,{ H, {

^{P ¡,,p,,_ r,o,,0)}

d B1rr,ø),,(rn: I,2, ...),

l;Ve

!'a1te also Bç,,þ,-7,t,,,F¡

1

Bç,,p,Ltt,u;,g2).if nt,

1nl,

^ønd,

for mr:

^nrz

iÍ

Þr

(

_Bs'

2

capacities

rvith the

same domain,

or C is

stronger

than C')

and rvrite

C'(E) : 0.

^,

We say that C

and

ã"t

"iålu

19'

ã $;L1ï'f¿"lt",t'r-i*

n C') and write

C'

( C (or

C

>

^C').

,.n.'

Tll.ir"losion

_Bn;,6

{ II,

was established

by t.

aDÄ}rs

qnd

N. }rErrERs

t¡i '-ä", "_::,::;'"n,s¡

^{'1' Bç','t¡}

is

obvious' Norv,

let us

establish also

that

and' B¡,,,7,, -l, ^rr, þ) corresþortd,s

to

the ^'hernel, t-þ

RELATIONS

{

1',

I 9,,(Y)

9,,(r) ukcre

h(r) :

:

æ and, i,f 9l

o

À

n

ttt

log

t

)'

27

ttt-I

-"n

1og ^t

h:2

{t'*;;lJ-

[t"*, ,i,)-]

(. l\ ß-. ^: lros,,

; )

^{ror ,'}

^r

^{3_' t',,,,}

rì-0"

|Q,gr,

--,,1 ïor , ?

^1,r,..

Z0

^ÞEihU

eÄÈAÀrr.Ñ

B

ÍHEoRrìM 1"

Let L1q<q.<û; dþ:n'ønd'þ>þ.-l'

^lh'en'

VEC C

^RN,

Bø,qt(E)

3 B6,p-l,*,

pt(E) S

QB",p(E) ^(r, :2,3,

. . .),

wkere

Q'is

^ø constør,t't ind'eþend,ent

of

^E:,øn,cl tke cøþa.citi' Bç,,q¡ corresþondb lo the hernel

(4)

Bç,.¡-r,,,,,sJ(E)

!

B¡,,,p-r,,,;,ri:"¡(E)

if

^mr

1 ^m,

^an;il,

Jor m¡:

^frtz,

il _þrZ þt. ' ^'

ⁱ

First, let F

^{be a cornpact}

setwith

b6,e-r,,,81(1l)

>

^0,

If

test

^measure

ror b¡,p-t,*e1 (n),

:tl,^t:"-

the

preceding llg.srrllp,

s

Q'þ

llpllr',

^l

lor

some constaut _Q ind'ependent bf

F.

Ifence, ' v

thus, on

account

of

proposition

3, 1

^i,

'

^b¡,p'i

ti.,pl(¡-)

S QB",;(þ-)"

" I'

rr¿hich

is

clearly

true

also

Io\

þ6,p-t,,,, Bl(F)

=.0,

^a-ud; siqÇe,01,,1,-r, _ø,g) iÞì.4lr

inner ^capacity and Bs,¡

ãtr

outer'capacity, it

follolvs

that ,, I

^:

B _¡n, p

-t,,,

^Bl

(E) :

b'i,,, _p

- ^r,,,, p¡(E)

<

_Q ^B

", p(E)'

'Ihe inequalitY 'l

^\

Bç',

¡(E) 3 86,e-r,.,

pl(E)

is a

cousequenóe

of the îact that ^'

^:

71^gn,p-t,n,Bþ)

-

^g.

I 'i '"'-o 'g'' ^q(l)

Finally, inequality (4)

follows from

¡.r. is a nou-zero lemnra

ï.or

r I

^rr,

lor r S

^rr,

g",i(r)

=

Lor

b",p(F), so

that

r-

_Q.

lim

ai I ,iij

We haue ^al'so

llull' <

b",p(F).

¡+o gn,þ-l, _rnr,gr(r) I ì(-2

;) Ð'"

s,Ø(tog

s"(z)(rog

:----l'

- ^Ir

^{a test measure}

,,dlilvri=Y,,,,',,'

gn,p-t, *",ß,(,)

=

^const'

o*-" :U,[,o*o-, (^r, n^riil' _{[,o*- , (tos2} ¡,"*å)l'

=

s

const.

B,p(x)r[ _(t"r,

i,)'-" (t"*-å)u:

^const. Bar,p_z,n,e(x)

Next, on

account

of (3)

and. arguing as above,

(7)

1, s

rLj (t"*, j-l'-'1tos,,,1,u \ ^r,r* -

rùs,r¡_,t(3ld,y

s

s

const.

d ^(t"r- ¡,,)t-'(ror,,lr,lu s.nl*):

^const. Bar,p,z,n,,s(x) Combining _(S),,

(6), (7)

yietds

(B)

gu

*

Eeþ-r),t,-2,,,,8(x)

I

const. Euþ,þ.'2,u,,p(x)

for lxl ³

^2r,,

dy:I r¡

^{I u,}

d.r:

go(x

-

y)g"rp-u?)

dy

<

I

ltl

o

øt

I lyl {lvl > p, lr-yl ^{> p}}

1 i

p

I u-l,l < ^p

RELATIONS

n-l

lylør-tr ¡

lt-l hr-7

:

corst, g"1t¡

t"tl-tl

fl-1

fl

Þ:1

s

const. S,¿(p)

fl

1k-l

É const. g.,(p)p

,tr-l

11 23

!

s

dltl

0

7og^

ß

rog^

log,,

7o9,,

logo log¡

n

logo

7og,n ( 0

u(þ-tl

2

2

d(b -1\

1-t

¿:l nl

I

(5) /t ^<

const. go(p)

I ^,t"ø -

!)g,þ-r¡,p-2,,,,s,(2)

ltl > ^rro +'.,

[to*u-, [r ^l"* i)i-" [tor,,-, (zøz r)]i

=

)u=

(^rri,l'-'(

2t¡

t4)

1 \þ-2

t't)

^t

fl-l

-LJ [tor,

,ilo-'¡or. å i ^u,

^o

l'-"1

)'

ilt-1

8a*íaþ-t),þ¡z,,u,s(x):

\, ^r"@-

?)g"te_,r¡b,) lvl3¡

'''' tn

(6) t, ^< _{¿:r t} fI ^(tos h-) ^t

^\þ-2

I

const. *-l

,LI

^(t"**

í,)' ^'(

ttt_l

, 1ß

loB,,,

¡ I

Eq*

8o(þ-tl@),

^const. Sqþ,þ-2,.,8(*),

2.

^'I'he Ç6trse x ->

@.

^ì

22 ^{PETRU CARAMAN} ₁₀

Indeed, on

account

of the

preceding proposition,

? ttt

\

^{r^,,,}

:,sQ)d,[(t"*

-i

,¡'-'] =

^.o,'st,

0, _,,,,

i¡:r

^log¿

^;J-'

^{(to*"' 1'I d 1og oo}

implies the

desired inclusion.

Remarþ.

This is an exte'sior of ^theorern 9.2 inr n.

AD..\MS

a'd

N. T,TEYERS paper [2].

I,emnta2. If _"þ 3tt,

^thcn

8c'F I ^uþ - ^{1), þ} -z, r', g(x)

:

g aþ

-

^11, t, -2, ^{rr, g} 4 g"(x) _S gaþ, þ _-2, *, _ø ^(x) )

øncl

if I aþ

^the

þarticular

^cøses

þ.:2 ^ol

ⁱⁿ

:1, ,/ ^p _<

0,

th'ett'

const. Í

^{gn ,, Eaþ_- t), þ.2, ,,, ø}

^(*) :

ga(p _{t), t,}

-2, ^,,,,,

i ^{i' 'siù',}

where^Ey,Þ-2, d jionr. thr'

'exþrrit',iotr,

,i-à:", o:;,,,,,',

'[,y

_nn:;"g

þ - 'In ²

^insteø

_{order to prove the}

_relation

8a*

8aþ-t\,p-2,,,,g(x)

:

_Sotp -t\,þ .2.,,,g

x gn(x),

l

we

observe

that

ga

*

íaþ_l),þ-'',-,s(x)

:

_\ g"(x

-

^{J') 1ott,,rt,} p-t,ii,s (y)dy

: :

\ïo(p-t),¡-z,m,g(N

-

z)g"(z)d'z

--

&uØ-tr),¡-.2,,,,o* 1o(x).

u'lrete _F'or we the

used

remaining the transfomration tí 2 iuequalities, - y: clearl¡', ^z,

^hence

we

need onl1,

!: ^t; -

^z,

to verify them as

r,

-t 0 ^and

^:v

->

^oc.

I.

Suppose

first þ22 ^and, if þ:2 ^or r¡,:1,

assu¡re also

p=

O.

I.l. The

casc

^t

*0. _{Let 0} <2p - ^l:rl

^S

^{2r,,,; thet}

gq*

luþ--t)

n-z ,,,g (x)

:

I

lrl u <-- '2

&n@

--

j')gorp-n,e-t,^,s (it)d,y

¡

8,þ -

!)8oø-rt,t,-e,tn,ø(y)dy

-f

1a-e¡ al!!₂

i

{u=\'t,-rr=\¡

j

g,(x -

j)EoØ-rt,p-r,*,p

U)dl : lL + Ir*

^Ia.

I

^const.

,l], [t"*, l)'-'(^r^i)t ,-' 5

^En(x

,.!)snrp_,¡+,(y)dy

{lylÈp, 1."-rl>p} =

fi-l ,

<

const.

[ ^(t"*, ài-" (t"*.;J'

B"p+t(x)

I*l-u S

^const, Saþ,þ_2,,,,s(x).

2. The

case x

+

co.

|il_l

1. = ^{II^} flo

^{t \þ-2}

t' t

^ri

' - _Èi I gu;,)

ltos-;l go*gdþ-\(x) :

e"þ,n_2,,,,s(x).

t, = ^-E' l^r,

n!,,1'-' ¡tor,,, r!,,)u

wøl :

^goþ, _t,-z,u,,s (x).

? ot

^(þ

:2,

_þ

{ ^{0), or}

^(nl

: t,

I, - ^Iu,

allor,virrg rrs

to

corrclude

(*) S ^const.

^gaþ,þ 2,,n,s(x) ^for.

e ineclualitl.

if

1

{ _þ < ² or

(p

:

2,

lirst

ø.'þ

:

¡¡.

= j:l

'

,,o^

,1,) ^(t'r*¡îf ls-r{*) -

^{const. s"f e)}

_f

y,,-c,-.1 çty

-

-

const' 8,,-*(P)

I \r' ^'arl=

,J],

[,"r,-, ^llog

²

| logri)lj-'[,"* ,,,-,(1og2 ¡

+ bg;)

¡; ^{lr,,t*) - const.}

^{go(p)p,, ø}

^--

^{r:onst. S,,.-o}

^(p)po-rl

^à

,(z

^rog

å)]'-'['"*,,,_,(, t,r;J]' _ß,,(*) - ^const.l

^à

i)'-'(^r,, fi)'l¡t"øl - ₎ ^{s,,@fà co'st}

^Bu,

^þ

^{2,,,,e @)}

For

_ø1

<

^i0,

we

have go

*

gaþ_t),n:,_",,,,,g

(x)

>

th _l

à 1,

_{à s.(Bp) ,t=r}

ll

^l1og,

\ -'Pt l)o-"¡tor,,,i)t I ""'pl o-' !

^s,,o-,,

^u)

lylà',ry>

t/t<p À:1

8,@

-

)')goo-ttb')dy

-

l/l<p ffi-l

,tt-1 lt:1

n

til*

13 25

<p x-y

RELATIONS

lngn

1o -'(to*,,,

i)t ^t*,

^{,* goç¡-r1}

^þ) -

s"(x -

^),)Eoo-

^ubt¿yf

^¿

8¡

1r¡

-l

_pl

I

nr-- |

8e*

8uþ

1l,tr-'J,n1,A

@)21r I Il

lilrl.' 't

ì

^const.

,I ^(r"*, r

^const.

t"ll,:+

-¡I,

(t*^ );,,)o-'(^r.i)u

í'

,

5

^const. 8a, þ-2,,,,9þ)

=

^{const. gap, ¡ ,'2, nt, g(il)}

and, taking into

account

also

(3),

Thus, ^ga+ Ecþ-t),t-2,,,,g(x)

5

^const, Kop,

t

.¿,,,,ø(ri)

for _l*l > l,

^{'hehce and}

on

acCount

of

^(B),

we are

allowecl

to

couclude

that

ga 8 1uþ-tt,þ-",*,p(x)

S

co¡rst. ^gdþ,

þ ^2,,,,gþí)

et'er)'rvheLe

in the

^case

þ2 2

^and,

iÎ p:2 or nt:

^1,

Ior l3l

^0.

II. ^Noq,,

^1et

^us

^suppose

l<þ<2 or (lr:'2,f3 50), ot (nt':l;

ß <

_0).

II.1. Tke

cøse

r *

0'

,U, [t.*,-,,los;- -

^los

t)l

[1o8,,,-,[t"i

^-_

lusslJesø(4 s

E [t"*,-,(î ^t"* i ['-'¡,o*,,,-,

⁽

i ^t"* i

^)1u

^r* t't

₌

l2

g,(x - y)g"ø-ttb,)dlt I

Bop, p-z,tt,s(x)

dlyl

=

En@

- ùe"0-')(y)lYl' ^dY I

{lvl à ^p, lz-v lÈ ^p}

drS

g*(x --

y)g"tr-tt(s')dY

I

l¡ I'l<p

g"p(x)

5

coust. 8ur,ø-2,,,,g(x) lvl <p

tft- |

llla{r-t)-t l-[

PETRU CARAMAN

-'_

P 1

P

,lt-7

1r<fl

In S

const.

g"(xl - ^t',)

1

P

,il-l

1

/t¡

const. f]

À:1 ltt- 1

,t:

n

I

Þ

J

1ug

fttu

1og _, 1og,,

logo logu logo

(

I

6 ^þ

(t"*, ;l'-'(t"*. ^ùì)

,b-1

(t"u^

å r'-'

_[tos,,,

1¡r'

I

,, =T,lt"u, i)'-'(1o*- i

^lu

,-' _I

1lt I ^g.(, - t)g,p-u!)dy I

gop't,-" _,n,s(x) l'rrl ^J

J ) ln

7 \þ-2

;i

^t

þ-,

(

a(þ- ^r)

-_

I

rt--l

Ir<11

24

If. _lxl ¡ ^1,

^then

,il-l

=

IJ [t"*r,* ln-"

¡tor,,,),,|'

,r"x &..¡*ttk)- ! ^{s-t* - y)}

snrp-,tU)dy]z

= i1'1,o*o

;,f-"tt"*.,*)u ^Ls*p(x)-

^const.

^e-Ml - ^r,,,)

ra(t,-tt>¡

n-1

à fl

_frogo

_:,,f ^-' (^r-::,)'þ*øl-- ^co'st r"(;)J

=

f ^em nta 4. If _"þ I

^{n, and.}

peÐll+,

iloen,

yE C

^R,,

1

(9) lg,,t,(g"*p)o-'(x)1t,-t sQtg"p,¡

2,nt,B>ßv@),2

=/)<

^co,

^p> l)_2,

I

(10)

lg"*(gn,* tù'-'(*)lo-rzQrgaþ,þ_2,,,,exy.(x), r <.þ _<

^{2, p}

<þ _2:

iÍ _þ :2,

the 'ineoualities

(g)

anrt.

(r0)

hotd.

/or^þ ^{2 0 ott4 p} s 0,

rcsþccti- ucl!

;

_{Q,., Q,}

{ ^{0' or,} ,onitå,rx iìlilrìrclení'of i, : "

^"

Ilirst,

consider

the

case

/ > 2; by Höid.r,.

iu.ecluality,

(11)

_{Eox (Eo}

_* r)*@):

Sgn(ø

- ^{y)l\g,(y --}

^z)

d.¡t(z)lr:,

d1,

:

-I l

: \9,(x-),) s\Ø!r),r,-",,,,s(x-y¡l\g*0-z)

gaþ_ _t), t_z,n,s(x-y1a¡r"12¡1trd.y

s

þ-1 _l

sl\gli

_@

-

^y) s!-É,t,!,_2,,n,r.(x

- ^y)¿ylilí l\sq¡_n,¡_z,n,s(x _ i)

^.

lyl 4 tu lvl> t

\s"U - ^z)

^d.¡t(z)

^dylþ-' :

RELATIONS

:)

logo-1 þ-2

,fl-l

n

27

(r loge 15

I'tn

2. Tke

ca.se x

+

^cÐ

8u

*

9dþ-t),þ-z,u',8

(x) 2

^Io

:

g,(r - ^y) g"ø-u!)dy

_¿

,"r^:,,1u

lui,e,o{ò

- } r*tùlz l

_{Eaþ þ-2 ,,p(*),}

o8r, _/ut

-

1

þ-2

>n

_,t: fl-1

1

as

desired.

26 ^{PETRU CARAMI\N} 14

à

const. S.(e)

n

(logu ä)o -"

(ror,,i, )'

^o

-'

\ra(t-\ ^+at

^{etu 2}

I

coust. g.(p)

Íi (t"r-;J'-' ^(t"*. _å)u

^{po(þ-t¡ >}

il-1

à

const. e"p(p)

^I [t"*^;)'' (,"*,ii)u ]

^const' Boþ,þ-z,u',ç,(x)

2. Tke

^cq.se x

->

^co.

Eu* ^SaU,- 1),þ-2,1il,s@)

> In

>

tt. -l '"'

)

^corrst.

_,!, (t'*^

_,t,,,\n

' _(t.*,

i)ut"(3t,,,) I

t'u'(þ-t't-'

dr

>

('r,

^{-,);,,,0 '' (1og,,,} )-,,,1u 8*{',,,)'r'þ-1) >

colrs

¿

^l==l'

i;,,Y

I

^{const. tr-1}

fI

À.:1

logu

',to*"'

|,,1u

8"?"') ^à

^{const' suP'P z'n'ø(x)}

Thus

go* lloØ-tti,þ-,2,ilt,(r

(r) I

^const. 8q!,,þ-2,,,,p

(x) for

_løl

;

^1,

arrcl

this

cornpletes

the proof of oul

leruma'

Urrder ^{the^} more

restrictivc

coudition aþ

{ ^tt,[

^obtained

^the

follor'ving stronger

result

generalizing lernma

3.2. ol

t). ÀDÀIIS

and x.

IIÐYDRS pa-

per _[2]

^:

I;emma ^3. lÍ

"1,

Itø,

^th'en

&ab ^t\, þ'2,jil,þ'i' ^8o : _8o *1 $'a(l> ^t), þ-t, ^m,9 - 8aþ,þ--2,u,í)'

Ott

accotttrt

of the

lrreceding lerntna,

tt'e

^lta\te

ouly to

^prove

^that

8n ^r,, 8oþ-t),þ,2, ^kt,g

(x) I

^const

^.

^{gaþ,þ'-2, ilt,9(x)}

also

for 2<þ <

^co.

Usiug thê notation oI ^the

preceding lemtua,

u'e

^get

7. Tlte

case x

-,0.

8a'! 8nlþ-t),þ-",*,8

þt) ^> IL

^>

à

c:oust.'It]

[,"*, !^f-"(tus,,,llus.(3e)

\ratt-t)-r

^dr

:

r:1 \ P, \

o

t1l _,

:

^Çonst.

F, (t"*, i)o-'(tot,,, i')u o.'-" ^à

^corrst. 9o.þ,þ-2,,,, B (P) à

à

^const. gnþ,p-2,*,s _@).

8qþ, þ,2,., ^{p n',}

¡r(r) :

_{5 Bzo}

(x - ^{y) (ttu-} o\)'on¡)

^{S grn x v@)}

: :&*&*¡r(ru),

.rvhich completes

the proof ,of our

lenrma..

RentarÌt" This is ari extension of lernma 3.s of D. ÀD^r,rs and N.

Irrry'Rs l2l. PropositianT, _I"f ECR" is an ønølytic

set,theru

cø,p(E)

S

^C.,¿(E) S

?

õ,¿(¿).

If uc

reþlnct

ir¡, Ityiå¿,

tl,ten, the abotte,irceqn,ality hotrls

for

^{alt, sets E.}

TuEoRDlr ^,3.

Let eþ {

^{n., then,}

(13)

B("þ,ù(D)SBrop,¡-r,*,øt(E)S?iB.,r(Ë) ₍₂ <1 <q< a, _IJ>þ ^_2), ,,

^{.Bo,p(E) S} ?rBr.¡,,t,-2,.'pt(E)S

0rAi.p,or(E) (t <q<1< ^2,

_þ

_<þ

^__2) ;

tt

_*o!d

lol g)0

^anrt

^p ( 0,

^re.s-

t

of

E

ctnd tlte caþøcity Bìp,p_r,,,,gt measure for |top, r_. ,, ,,.ar(F),

by the p'o""äiåg-'íä,ilá;

from the

preceding pro-

llrrll, {

^QrT,,p(F)

sgrB,,p(F).

îhus

b(oþ, þ -r,,,,øt(F) S

?J,,¿(F) S

?rB.,p(F),

lJìl;îîîji-,þ-2,ilt,8¡ ^is tr' i''er

capacitl,,

and

Bn,,

an

orrter

ca'acity, it

Bt"p,oíE)

S

Er*r,¿-r, .,Bt(E)

is a

cli¡ect consequence

of the

relation

ri^ ^tno'o?) :

^,-.

: rlo 9cþ,þ-z,nt,g?)

,,uu"

.tn"

^second

inequarity in this lemma may

be proved

in a

similar

fìË1,{rioNs

rnequality

29

the

Bkp,p-", *,pt(E)

=

^bîop,p-2,

^.,st(E) S

?r8.,¿(E)

ú

airdif p<0

28

But,

(t2)

Þ þ-r

_-1

\ d-' ø

ela? ^tt, þ -2,th, ø(y)dy

r

0

¡ p-1 ilt-l Å:l

n

logu

PETRU CARAMÀN

r- I l-

n-'l ,:Qi<q¿

llos-..:

16

:

coilst. ,,

for

p

) þ ^{--2 and}

^p

(

r,,,.

Next, lor R 1r,,,,

þ t -1

s!,-t b,) gþ"tp' ,t, þ--2,,n,g

til-l

(y)df,

S

^cnost.

fl

h:. I

- l)-ø(l'--l) ^F,'

þ-2 ^I log,,

I

1',

p--2 -9 dy

: (

_const.

_r

f

0

æ

i (t'*-,-1rl-'

-Pæ (t"*,,

;,;

,*

\,

-2'ï rt'-2 dl'

Í

^Q'"

<

^ø¡,

heuce arrd.

takirrg iuto accorint (11), (12)

aucl lertrtna

2,

$,e deduce ^(9), rvhere Q,

: _[rnax

(Q'r, QL))þ-'.

.A.trcl no$,, consid.er

the

casc 1

<

^'þ

<

²

, ^fuon

^lernma

2

anð' Hölder itrec¡ralitS',

u,e

get

8qþ,þ z,ru,gxlr.(x)

5

const' g,lþ-t),þ-2,,,,þ*8o+

p(r) :

:

col1st, \Eotp-r¡,p ^{t,^,çt(:v}

--

)ù

gi-þ@ -- Ð el-'Ø -

^l')

ig*(J'-

z)d1t(z)d'yS

1- þ ^t

I

const.

fSST@ ^{-- y)}

g'rpL,t,þ-ù,ur,ø(*

- ^{y) dyl'-þ'}

' {\

^g"(x

-

^{y) lg,'(y}

-

z)d'1t(z)10\'

¡^'¡'-' :

1-þ 1 ^I

:

colrst.

l\ s".-'u)

s?þþ ,r,p-r,,,,ø(y)

dyl'-plg** (g,* P)'-' @)lþ-t, aud

arguirrd

as

above,

it

follorvs

that

1-þ 1

\ ^{g'*-þ (} )') s?(þ!-t\, p -r,,,,

ø(i)

^d'Y

< *,

herce, we obtain

^(10).

F'inally, in the

case _1ó

:2,

ott accouut

of (3), if. p

2-

0:

En*

go* v(tò: Bzo*p(r) :

g*p x _F@)

S

5g*p(ø

- t)(log,,¡\,\'Ortrl -

: _{8eþ, þ-2'ilt,9} * P(#)'

alg,* V)þ'',pl< 0p,, il tlogojìo, ⁰ (

p

S

^po,

ultere

Q is ø

constønt indeþendent

of p

ønel po

is

sorffic,iently s*ø11,.

Remarh.

'rhis corollary is a' exte'sion of

lemma 4.2

ir

D. ADÁ.nrs

and

N. MÞYEns'

paper l2l.

Lemma 6. Let a{n ^qnd ), =L|;

^swþþose

uhere s

>

^1,

_9,

^is _{_a} ^co,nslant 'ineteþend,cnl. o.f

p

ønd po ,is

sufficienlly

stu,oll.

Then ^go x

)'

zs

ø

boomded. fuu,ctio7.t..

Define

ll^u ::""1É ^ternr of the right part of (ra) is the convolutio' of a

boun-

rrqq

'

function rvith a

,nealure ^{^}

o{ fi'it! total variation

a*d.

is thus

arso,

uqunded,

as

d.esired.

31

.

^Bn

^{* r(s)} : _/,

,r

ì,(x) I

^(5"

- ^{!*) *}

^x(x)

1

f

1

r

,D- 1

Po

n

f

I

0

-

^const

-s

RELATIoNS

B(t,pu)

I

Po

ã"?):

{

^const.

.llog* --

,Þ: I

t

^h,ell

ß

log;

Þo

o8,,

logo -s

p p

Togo

1og,,

f

Po

Po

const

Eo@

- ^y)dt'(y) (

const.

\,"-"

^{. e\a(),,}

^x,r)

9n-l

,,-,t1,,, _"

I==t,

{t"r,

^-j )

1o9,,

g"

x

),(x)

:

_\ g"@

-

1,)ct),(y)

:

f

:

const.

fJ

-1

,ìt _1

o<PSpo(1,

I l-'q

^_-

,l

r

i;)' "'

g*(r) lor 0Sz(pu, 0 for r)

^po.

I{eu'rite

(14)

But

J-']

tù -l

u(1,

p) S 0rp"-"

_À:t

f]

,U,

(t.*-;lJ-'[,,r,, jt-"-

^const. (1og,, tt_1

<

^const.

.

*[ [t'*, jl-'

[to*,,,

*)-". *

30

^þnînu

ctn¡tMaN

¹⁸

Remark.

this ^result improves theorem 1'

except

for the

case 1

<

< þ < 2

^and

t"pt"r""îr- un-ät"osion of ^n.

^{ÀDAl\rS and, N. ¡ITIYERS' theo-}

t"ttt 3.3 io _l2l.

-""

C*ofmry.ö"pE:0+ Bnt,-r,,,,g.(E):0

VP

>.n-2 (nr':!'2' "')' iet

^us

"recafi

_some

notations' For

[l,

e ^{]ii +'}

^set

tt(çr,

x,

_P)

:

_B(r,p)

I ^,*trl

ror x =R',,0sp<co ^and

^also

u(p,

p)

:,.:tt;,

a(p.,

x,

^p)'

PropositionB.TetEçiR"bcaxl'ønal'1'l:icset'Ho(E)'0¡ÍÍ

th'eve exisîs

¡r e

&lL+, V.+

0,

suck that

u(v,

p)

Sl'a(P),0<PSPo'

^:

(For

.a proof,

see L. cÂIìT,ESoN

[8],

^chap,

II'

^theorerrl

^{1' p'}

^7')

Ärgurng ^as 1n D.

^ooiî'ä"ïi". ^*tnieo*' itttt*" 4'1 of _[2']' we obtai.

LË;; a ^5. L,et,i?-71 ^o*A

^À

^e'5ìL+;

^suþþose

tkat\9' 0< ^p

S

where

Q is

'ind'eþend,ent

of

_P.

IIence, for

^d1

:

tt'

- ^{eþ, so: Qp-} l, (h: l'

'

"'

^ttt')

dz- ^tu

^arló'

to:O (ø:7, ,.., ^{b), we obtain}

^the

Corollary. ^{Let uþ} {n

^{ønd' ¡t.}

=LI;

^{suþþose thcrt}

u(p, p)

S 0,p"-'r' ,U [a*, l)'-'r, ^{0 <}

^P

(

Po

(

^1'

tuhere

Q, is ø

conslømt ind'eþend'ent

oÍ

p'

u(P, P) S

0rPo'il

anrj

a

v.)

[t"*,+)-",0(p(po,

['or, i ^r-'*

Spo(1,

i)-'-]"0-É

u(À, p) S

0,p', -LIlt"r,;)-u,

, ^are

cotcstants 'índeþen'd'en't

of

_9'

l,

^{zae kau,}

u[(g*

* p)'

À,

p] _S

_?

[e.-"*n'rU,(t"*,

wkere O

S¿lt(n-ad'r{n

Then,,

for

⁰

^{<u <} *"*

l.f ^,

ønd' _Qt,

do - ^(tt' - (n - ^a) -- cl,

ùi iEi^AiioNs

_å3

flew

iin the ^case

^l:

<,þ <2.

^'I1his

thçorem

gerreralizes theotem

4.3

^of

D. ,{DArrs ^anc1. N. ìIDYERS

l2l.

Corollary. ^CapE : 0 +'

H,0,,,r,,,(r(li)

: 0

VP

>

!1'

- |

^(rtt'

: l, z,

^{. .}

.),

wh,ere h,,-t,,,.s(t')

:'i1i _(to*-

)

)'-

^" 1,o*,,

'

1-u

This is a

corlseqLrence

of the

precpding thcorenr

and of the

corollafy

of

proposition

4. But it na1'

be obtainecl also ftom

Proposition ^9. CapE:0+Ho@):0 for ^a'll

tneast'wefttnc-

tdonsh søtdsfydng ,

^,

for

⁰

< ^r¡ 1L safficicntly

smsll

:'

',For,

the

,proof,,

see

rr. .tr,¡rrr,rN (.l2ql,, ,theo¡ern

4.1)

or.,

D.

^,,\pAlIÍ$

([1],

theorern

2).

:' ^,

In order to ^show that thc

^precedipg

theoren is.best

^{possible, we} recall some definitions ancl previous results.

Propositio'n 10. Let:h,'be a

^inecisøre

_function

sutisfying

let us

søy

for 0Sz(J.arrrl -2

^zalnrc

0<?<"ç. If

I'-

i.srtCnntot'sel 'in R,

lken

(Ilor the proof,

see D, ÀDr\ìrs ancl'N.

ìlJr'lr{s [2], proposition

5.1.)

A set

Ë

C ^R" is

saicl

to have a

^|,otuer sþheri,cnl h.-density

at a

_þoittl

11,,,

IrirtlzlJl4

_{_-,} o.

,.-n

^h(2ri

.-,'irP

r

o

p

ll.'l'Jt¿: sct E lt.os,l,ozqtzr

sþltar

^il1, ¡¡¡

Vx =

^E,

u4e/.c.E.is sutl'tcienll! _ct

_kol.ds, satisfl,irt,g _{tø:h,cre tlte.}

0< H;,(E)<

citttstant

Q 0<r {rr(r,

^'<

^{0 <}

^{2', and,}

{'\r,s; ø:

scl tyþtt E = ), ^8,,,

^willt,

,fì,,,, ^n.,llte

^{rtitt, slt:þ,}

'n-d.itnensionnl

"irìrrrnls,

_zadth,

eclges

pf

leu,gttt, 1,,,, 2lu,.i.r

1

n,

th,e u5unl,

'øøy

snlisJl,i¡1g llre foll,ozuin,g tlou,ble inequ'a-

iir _i

_.

c| 12"h(l,n) I ,r, ci!

^cz col?,st&tlts.

3 -- L'aualyse numérique et la tbéotie de l'approximation

- ^{Tome ?} - ^{No. 1/19?E}

lim

,*7o

_l a

(16)

1

'h,(zr)

{

^Qh(r),

(15) ''

!r,

çorusistinp _of ^,,,)',u ,.!',1,,,, oitaínert ¿

tdty

B2

^þsrnu c'A'itrtiu'dÑ

Remørk. This is an ^{extension of lemma}

^4'3

N.

-

MEVERS [2].

ìiieo*dr ^{a. Let uþ} { ^n'

^Theto

Hn

l

Hnoo,p-t,rr,91 Bn,¡ ^,

zahere

h(r):/t!-aÞ(t"*il'-o Ío' q>þ ønd'r<L' ^and'whcre

^\

h,ns,p-t,n,,B(r)

: ,.-"0"#r(t"r- i)' ^{-' (t"*'} i l-u

^(m

: r' 2'

_{' '} ')

Vp > þ - ¹

^{and' 0}

<r <r,,, ^with'

^{1og,,, -1-}

7

^1'

First, ^1et E' ^{be an} analytic set ''i;i'lt

Ho*o,t-t,'',p(E')

> 0;

^then'

frorn the

^preceding proposition,

there exists rr+-ö'sui1t't1rat

^{¡.r.= 9ìL+(E)}

ancl

We niay

^asstlme

that

¡-r, has

a

compact

support'

^Then'

,by the

^preced'ing

;;;il"ú,

^rvhere

o-îi ô:i,-"''' t* lj'anci Ç")þ' it

^foilows

^that' takingì:(g"*p)p_',weafeinthelr¡,potlresesoftlreprecedirrglemrrra, which

^aliows

us to

conclucle

that

^go

*

(g"

*

p)'o-t

is

^bounded-

i'

R"

by

^a

rò-l

constant ln 1æ. ^But

^since

oir,,+ ⁰

^ancl

t.[*'* #=) ^{(ø) s} f it

B",p(E)

2 ^0,

hence Hh,þ,þ:t,n,,p1 Bo,p,

as

^desired'

'rhe inclusion Hîä-ii';:;,þ-1,u,,þ Is a direct

consequence

of

the relation

Re,tørþ.

The

assertion

of this theorem for þ >

_?.

is a direct

^con-' sequence

of

proposîiior,

-O

no.l tf,"or"ä-ã,

ftotuå"át

its contribution

^is

- ^{0 for q>}

^þ.

1

'r

i

r ^1ogo,

r+0lirn

("'i)'-'

20

of o. ¡o¡rrls

^and

(rn:1,'2,'..)

m-l a(v,

r) (

^z"-c?

_I], (t"t,

r;þ-t-þ

h(r)

^-

^1im7+0

'F,

(.,, +)'-' ^('.". i)-'