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“„Š517.5

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Œ. ‘. €«¡®à®¢ 

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¯à®áâà ­áâ¢.

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1 ;:::;x

n );

~

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1 ;:::;l

n

)|¬ã«ì⨨­¤¥ªá,

l

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H

t (x)=

t l

l

1

x

1 ;:::;t

l

l

n

x

n

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n n

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; r :R n

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; r(H

t

(x))=tr(x); x2R n

;

­¥¯à¥à뢭ãî­ R n

:

˜ à®¬á業â஬¢â®çª¥xà ¤¨ãá ­ §ë¢ ¥âáשּׂ®¦¥á⢮

B

(x)=fy2R n

:r(x;y)<g:

ãáâìR n

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p ()

¯à¨­ ¤«¥¦¨âª« ááãL ~

l

p

(),¥á«¨®­ ¨¬¥¥â®¡®¡é¥­­ë¥¯à®¨§¢®¤­ë¥

D

f 2L

p

(); j:lj=1:

‡¤¥áì D

f = @

f

@x

1

1 :::@x

n

n

, = (

1 ;:::;

n

) ¨ j : lj =

1

l

1

+:::+

n

l

n

. „«ï â ª¨å ä㭪権

®¯à¥¤¥«¨¬¯®«ã­®à¬ã:

kfk

L ~

l

p ()

= X

j:lj=1 kD

fk

Lp() :

à®áâà ­á⢮¬

L ~

l

p

() ­ §®¢¥¬§ ¬ëª ­¨¥¢­®à¬¥

kfk

L ~

l

p ()

= X

j:lj=1 kD

fk

L

p ()

¬­®¦¥á⢠C 1

0

(2)

‚¢¥¤¥¬¯à®áâà ­á⢮

V

~ l

p =

\

jk:lj61

L

k p

()

;

á­ ¡¦¥­­®¥­®à¬®©

k

f

k V

~

l

p =

X

j:lj61 k

D

f

k

L()

:

‚¢¥¤¥­­ë¥¢ë襯à®áâà á⢠¨§ãç «¨áì¢à ¡®â å‚®¤®¯ìï­®¢ ‘.Š.[1{5].

ãáâì

e

2R n

|§ ¬ª­ã⮥¬­®¦¥á⢮. …¬ª®áâì®¦¥á⢠

e

­ §®¢¥¬¢¥«¨ç¨­ã

Cap

e;

L

~ l

p ()

=inf

k

u

k p

L ~

l

p ()

:

u

2M(

e;

)

;

£¤¥M(

e;

)=f

u

2

C

1

0

():

u

=1¢®ªà¥áâ­®áâ¨

e

g(á¬.[6]). ãáâì

e

|ª®¬¯ ªâ­®¥¯®¤¬­®¦¥á⢮è à 

B

:

ã¤¥¬£®¢®à¨âì,çâ®

e

|(

p;l

)-­¥áãé¥á⢥­­®¥

¯®¤¬­®¦¥á⢮

B

,¥á«¨

Cap(

e;

L

~ l

p (

B

))6

n,pl

;

n

>

pl

; p >

1¨«¨

n > l

; p

=1

;

£¤¥

|¤®áâ â®ç­®¬ « ïª®­áâ ­â ,§ ¢¨áïé ï⮫쪮®â

n

,

p

,

~l

.

‘®¢®ªã¯­®áâì¢á¥å(

p;l

)-­¥áãé¥á⢥­­ë寮¤¬­®¦¥áâ¢è à 

B

®¡®§­ ç¨¬ç¥à¥§N(

B

). €

ç¥à¥§

u

B

|á।­¥¥§­ ç¥­¨¥ä㭪樨

u

­ è à¥

B

,â.¥.

u

B

=[

m

n (

B

)]

,1 R

B

udx

.

‚¢¥¤¥¬¥é¥¯®«ã­®à¬ã

j

u

j p;l

;B =

X

0<j:lj61

l

(j:lj,1)

k

D

u

k

L

p (B

)

;

£¤¥

=(

1

;:::;

n )2N

n

:

’¥®à¥¬  1. ãáâì

e

| § ¬ª­ã⮥ ¯®¤¬­®¦¥á⢮ è à 

B

:

„«ï ¢á¥å ä㭪権

u

2

C

1

(

B

) â ª¨å,çâ®dist(supp

u;e

)

>

0¢¥à­® ­¥à ¢¥­á⢮

k

u

k Lq(B)

6

C

j

u

j p;l

;B

;

(1)

£¤¥

16

p

6

q

61

;

=

1

p

, 1

q

n

X

i=1 1

l

i 61

:

à¨

=1 (16

p

=

q <

1), ª®­áâ ­â 

C

¤®¯ã᪠¥â®æ¥­ªã

C

,p >

, np

q

c

1

Cap

e;

L

~ l

p (

B

)

:

(2)

„®ª § â¥«ìáâ¢®â¥®à¥¬ë ®á­®¢ë¢ ¥âáï­  á«¥¤ãîé¨å१ã«ìâ â å.

à¥¤«®¦¥­¨¥ 1. ‘ãé¥áâ¢ã¥â ®¯¥à â®à ¯à®¤®«¦¥­¨ï

A

:

L

~

k

p (

B

) !

L

~

k

p (

(3)

â -(1) Av=v ­ 

B

;

(2) ¥á«¨ dist(suppv;e)>0,eª®¬¯ ªâ¢ B

,â® dist(supp(Av);e)>0;

(3) kD k (Av)k Lp(B2) 6ckD k uk Lp(B)

;£¤¥ jk:lj61,16p61.

C„®ª § â¥«ìá⢮ á«¥¤ã¥â ¨§ १ã«ìâ â®¢à ¡®â[7{11] B

‹¥¬¬  1.ãáâì e| ª®¬¯ ªâ¢ B

1

. ‘ãé¥áâ¢ã¥â â ª ï¯®áâ®ï­­ ïc>1,çâ®

c ,1 Cap e; L ~ l p (B 2 )

6inffk1,uk p V ~ l p (B 1

):u2C 1 0 ( B 1 );

dist(suppu;e)>0g6c Cap e; L ~ l p (B 2 ) : (3)

Cãáâìv=A(1,u):Ž¡®§­ ç¨¬ç¥à¥§ äã­ªæ¨î¨§ C 1

0 (B

2

), à ¢­ã¨­¨æ¥¢

®ªà¥áâ­®áâ¨è à B

1

:’®£¤ 

Cap(e;B

2 )<c

Z B 2 X j:lj=1 jD (v)j p

dx=c Z B 2 X j:lj=1 X =0 C D vD , p

dx6 (4)

(â ªª ª 2C 1 0 (B 2 ),â® íâ äã­ªæ¨ï¢¬¥á⥠ᮠ᢮¨¬¨¯à®¨§¢®¤­ë¬¨ ®£à ­¨ç¥­ ) 6c Z B 2 X j:lj=1 X =0 D v p

dx6ckvk p V ~ l p (B2)

6c inf

k1,uk p V ~ l p (B 1 )

:u2C 1 0 ( B 1

); dist(suppu;e)>0

; (5)

¨ «¥¢ ï ç áâì (3) ¤®ª § ­ . „®ª ¦¥¬ ¯à ¢ãî ç áâì ®æ¥­ª¨ (3). ãáâì w 2M(e;B

2 ). ’®£¤  kwk p V ~ l p (B1) 6c X j:lj61 kD wk L p (B 2 ) 6c X j:lj=1 kD wk L p (B 2 ) : (6)

Œ¨­¨¬¨§¨àãﯮ᫥¤­îî ­®à¬ã­  ¬­®¦¥á⢥M(e;B

2 )¯®«ã稬 kwk V ~ l p (B 2 )

6cCap e; L ~ l p (B 2 ) :

Œ¨­¨¬¨§¨àãï«¥¢ãî ç áâì, § ª ­ç¨¢ ¥¬¤®ª § â¥«ìá⢮. B

’¥¯¥àì ¤®ª ¦¥¬ ⥮६ã1.

C „®áâ â®ç­® ¤®ª § âì ⥮६㠯ਠ= 1. € § â¥¬ ¢®á¯®«ì§®¢ âìáï

H

t

-®¤­®à®¤­ë¬ ¯à¥®¡à §®¢ ­¨¥¬. ãáâìN =kuk

L

p (B

1

). ’ ª ª ª dist(suppu;e)> 0;

â® ¯® «¥¬¬¥1

Cap(e; L ~ l p (B 2

))6ck1,N ,1 uk p V ~ l p (B 1 ) =cN ,p juj p p;l ;B1

+Ck1,N ,1 uk p L p (B 1 ) ; â.¥. N p Cap(e; L ~ l p (B 2

))6Cjuj p

p;l

;B

+kN ,uk p

L (B )

(4)

¥§ ®£à ­¨ç¥­¨ï®¡é­®á⨠¬®¦­®¯à¥¤¯®«®¦¨âì,çâ®u

B

1

>0:’®£¤ 

jN,u

B

1 j=

kuk

L

p (B

1 )

,ku

B

1 k

L

p (B

1 )

6ku,u

B

1 k

L

p (B

1 )

:

‘«¥¤®¢ â¥«ì­®,

kN ,uk

L

p (B

1 )

6kN ,u

B1 k

L

p (B

1 )

+ku,u

B1 k

L

p (B

1 )

62ku,u

B1 k

L

p (B

1 )

: (8)

‚ᨫã (7),(8) ¨­¥à ¢¥­á⢠ ã ­ª à¥ ¤«ï  ­¨§®âய­ëå¯à®áâà ­áâ¢

ku,u

B1 k

L

p (B

1 )

6c X

j:lj=1 kD

uk

L

p (B

1 )

;

á¯à ¢¥¤«¨¢ ®æ¥­ª 

Cap(e;

L ~

l

p (B

2 ))kuk

p

L

p (B

1 )

6juj p

p;l

;B1 :

ˆ§â¥®à¥¬¢«®¦¥­¨ï¤«ï ­¨§®âய­ëå¯à®áâà ­áâ¢(á¬. [13])¨¯®á«¥¤­¥£®

­¥à ¢¥­áâ-¢  ¯®«ãç ¥¬

kuk p

L

q (B

1 )

6c (juj p

p;l

;B

1 +kuk

p

L

p (B

1 )

)6f1+[Cap(e;

L ~

l

p (B

2 ))]

,1

gjuj p

p;l

;B

1 :

“⢥ত¥­¨¥â¥®à¥¬ë 1¤®ª § ­®. B

Žâ¬¥â¨¬,ç⮢ ¨§®âய­®¬á«ãç ¥ ⥮६ 1 ¤®ª § ­ ¢ à ¡®â¥Œ §ì¨[12] ¨ ¯à¨

¯®¬®é¨¤à㣮£®¬¥â®¤  ¯à¨p>1•¥¤¡¥à£®¬ [13].

’¥®à¥¬  2. ãáâì e | § ¬ª­ã⮥ ¯®¤¬­®¦¥á⢮

B

¨ | ç¨á«® ¨§ ¨­â¥à¢ « 

(0;1). ’®£¤  ¤«ï ¢á¥å ä㭪権¨§ ¬­®¦¥á⢠

n

u2C 1

(

B

): u

B

>0; u(x)6 ,

n

p

kuk

Lp(B)

¯à¨¢á¥åx2e o

á¯à ¢¥¤«¨¢®­¥à ¢¥­á⢮:

kuk

Lq(B)

6C juj

p;l

;B

; £¤¥ C ,p

>c

1 (1,)

, np

q

Cap

e;

L ~

l

p (B

2 )

:

C®¢â®àï冷ª § â¥«ìá⢮«¥¬¬ë 1,¯®«ãç ¥¬

c ,1

Cap(e;

L ~

l

p (B

2

))6inf

k1,uk p

V ~

l

p (B1)

:u2C 1

(

B

1

);u60 ­  e

6cCap

e;

L ~

l

p (B

2 )

:

„ «¥¥ ¨§ ­¥à ¢¥­á⢠1,N ,1

u>1, ­ e¢ë⥪ ¥â®æ¥­ª 

(1,) p

Cap

e;

L ~

l

p (B

2 )

6ck1,N ,1

uk p

V l

p (B

1 )

(5)

’¥®à¥¬  3.ãáâìn=l p; p>1. B

| è à,¤«ï ª®â®à®£®

Cap(

B

n;

L ~

l

p

())>0:

’®£¤  ¤«ï ¢á¥åä㭪権 u2D()

kuk p

L

q (B

)

6C X

j:lj=1 kD

uk p

L

p ()

; (9)

£¤¥

C6cd np=q

h

Cap(B

n;

L ~

l

p (B

2 ))

i

,1

:

C‘®£« á­®â¥®à¥¬¥ 1¨¬¥¥¬

kuk p

L

p (B

)

6cd np=q

h

Cap

B

n;

L ~

l

p (B

2 )

i

,1

juj

p;l

;B

: (10)

®áª®«ìªã ¯à¨

j:lj61 ¨ q

=

pn

n,pl

(1,j:lj)

á¯à ¢¥¤«¨¢ë ­¥à ¢¥­á⢠

kD

uk

Lp(B2) 6c

l

(1,j:lj)

kD

uk

L

q

6c l

(1,j:lj)

X

j:lj=1 kD

uk

Lp() ;

â®

kuk

p;l

;B

2 6c

X

j:lj=1 kD

uk

L

p ()

: (11)

¥à ¢¥­á⢠ (10) ¨(11) ¤ îâ®æ¥­ªã (9). B

‹¨â¥à âãà 

1. ‚®¤®¯ìï­®¢‘.Š. ƒ¥®¬¥âà¨ç¥áª¨¥á¢®©á⢠®¡« á⥩,㤮¢«¥â¢®àïîé¨åãá«®¢¨î¯à®¤®«¦¥­¨ï

¤«ï¯à®áâà ­á⢤¨ää¥à¥­æ¨à㥬ëåä㭪権//¥ª®â®à륯ਫ®¦¥­¨ïä㭪樮­ «ì­®£®

 ­ -«¨§ ª§ ¤ ç ¬¬ â¥¬ â¨ç¥áª®©ä¨§¨ª¨(’à. ᥬ¨­ à ‘.‹.‘®¡®«¥¢ ).|®¢®á¨¡¨àáª.|1984,

ü2.|‘.65{95.

2. ‚®¤®¯ìï­®¢‘.Š. Ž¯à¨­æ¨¯¥¬ ªá¨¬ã¬ ¢â¥®à¨¨¯®â¥­æ¨ « //’¥§¨á뤮ª«. XI‚á¥á®î§­®©

誮«ë¯®â¥®à¨¨®¯¥à â®à®¢¢ä㭪樮­ «ì­ëå¯à®áâà ­á⢠å. —.1|—¥«ï¡¨­áª,1986.|‘.29.

3. ‚®¤®¯ìï­®¢‘.Š. €­¨§®âய­ë¥¯à®áâà ­á⢠¤¨ää¥à¥­æ¨à㥬ëåä㭪権¨

ª¢ §¨ª®­ä®à¬-­ë¥®â®¡à ¦¥­¨ï//’¥§¨á뤮ª«. XI‚á¥á®î§­®©èª®«ë¯®â¥®à¨¨®¯¥à â®à®¢¢

ä㭪樮­ «ì-­ëå¯à®áâà ­á⢠å. —2.|—¥«ï¡¨­áª,1986.|‘.23.

4. ‚®¤®¯ìï­®¢‘.Š. ƒ¥®¬¥âà¨ç¥áª¨¥á¢®©á⢠®â®¡à ¦¥­¨©¨®¡« á⥩. Žæ¥­ª á­¨§ã­®à¬ë

®¯¥-à â®à ¯à®¤®«¦¥­¨ï//ˆáá«¥¤®¢ ­¨ï¯®£¥®¬¥âਨ¨¬ â¥¬ â¨ç¥áª®¬ã ­ «¨§ã.|®¢®á¨¡¨àáª:

 ãª ,1987.|‘.70{101.

5. ‚®¤®¯ìï­®¢‘.Š. ‘à ¢­¥­¨¥¬¥âà¨ç¥áª¨å¨¥¬ª®áâ­ëå å à ªâ¥à¨á⨪¢â¥®à¨¨ ¯®â¥­æ¨ « //

(6)

6.

€«¡®à®¢ Œ.‘.,‚®¤®¯ìï­®¢‘.Š.

“áâà ­¨¬ë¥ ®á®¡¥­­®á⨠¤«ï à¥è¥­¨ï ª¢ §¨«¨­¥©­ëå

ª¢ -§¨í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨© // ‘¨¡. ¬ â. ¦ãà­.|1992.|’. 34, ü 4, ‘. 3{14.

7.

¥á®¢Ž.‚.

à®¤®«¦¥­¨¥ ä㭪権 ¨§

L

l

p

¨

W

l

p

// ’à. Œˆ€ ‘‘‘.|1967.|’. 89.|‘. 5{17.

8.

¥á®¢Ž.‚.,ˆ«ì¨­‚..

…áâ¥á⢥­­®¥ à áè¨à¥­¨¥ ª« áá  ®¡« á⥩ ¢ ⥮६ å ¢«®¦¥­¨ï // Œ â.

ᡮ୨ª.|1968.|’. 75 (117), ¢ë¯. 4.|‘. 483{495.

9.

ˆ«ì¨­ ‚. .

ˆ­â¥£à «ì­ë¥ ¯à¥¤áâ ¢«¥­¨ï ¤¨ää¥à¥­æ¨à㥬ëå ä㭪権 ¨ ¨å ¯à¨¬¥­¥­¨ï ¢

¢®¯à®á å ¯à®¤®«¦¥­¨ï ä㭪権 ª« áᮢ

W l

p

(

g

) // ‘¨¡. ¬ â. ¦ãà­.|1967.|ü 7.|‘. 573{583.

10.

“ᯥ­áª¨©‘. ‚.,„¥¬¨¤¥­ª® ƒ.‚., ¥à¥¯¥«ª¨­‚.ƒ.

’¥®à¥¬ë ¢«®¦¥­¨ï ¨ ¯à¨«®¦¥­¨ï ª

¤¨ä-ä¥à¥­æ¨ «ì­ë¬ ãà ¢­¥­¨ï¬.|®¢®á¨¡¨àáª:  ãª , 1984.|‘. 224.

11.

¥á®¢Ž.‚., ˆ«ì¨­‚..,¨ª®«ì᪨©C.Œ.

ˆ­â¥£à «ì­ë¥ ¯à¥¤áâ ¢«¥­¨ï ä㭪権 ¨ ⥮६ë

¢«®¦¥­¨ï.|Œ.:  ãª , 1975.

12.

Œ §ìï‚.ƒ.

à®áâà ­á⢠ ‘. ‹. ‘®¡®«¥¢ .|‹.: ˆ§¤-¢® ‹ƒ“, 1985.|416 c.

13.

Hedberg L.

Two appoximation problems in function spaces // Ark. Mat.|1978.|V. 16, No. 1.|

. 51{81.

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