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“„Š 517.5

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p

() å à ªâ¥à¨§ãî騥áï

ª®­¥ç-­®áâìî ­®à¬ë:

kfk

L l

p ()

= X

j:lj=1 kD

fk

Lp() ;

§¤¥áì | ®âªàë⮥ ¬­®¦¥á⢮ ¢ R n

, 1 6 p 6 1, = (

1

;:::;

n

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(l

1 ;:::;l

n

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1

l

1

++

n

l

n , D

f = @

f

@x

1

1 :::@x

n

n .

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¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬ëå ­¥¯à¥à뢭ëå ä㭪権 ¢ ¯à®áâà ­á⢠å

‘®¡®-«¥¢  å®à®è® ¨§ãç¥­ë ¢ à ¡®â å ¬­®£¨å  ¢â®à®¢, á¬., ­ ¯à¨¬¥à, ‘. ‹.

‘®¡®-«¥¢[1],‚.ƒ.Œ §ìï[2],†.-‹. ‹¨®­á,.Œ ¤¦¥­¥á[3],„¦.®«ª¨­ [4],‹.

•¥¤-¡¥à£ [5].

‚  ­¨§®âய­®¬ á«ãç ¥ ¢®¯à®á ® ¯«®â­®á⨠¨§ãç «áï ¤«ï ®¡« á⥩,

㤮¢-«¥â¢®àïîé¨åãá«®¢¨îண ¨¤«ï¡«¨§ª®£®ª« áá ®¡« á⥩¢à ¡®â åŽ.‚.

¥-ᮢ , ‚.. ˆ«ì¨­ , ‘. Œ. ¨ª®«ì᪮£® [6], ‘.‚. “ᯥ­áª®£®, ƒ. ‚.„¥¬¨¤¥­ª®,

‚. ƒ. ¥à¥¯¥«ª¨­  [7], . ˆ. ‹¨§®àª¨­ , ‚. ˆ. ã७ª®¢ , ‘. Š. ‚®¤®¯ìï­®¢ 

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1. à¥¤¢ à¨â¥«ì­ë¥ ᢥ¤¥­¨ï

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| ¥¢ª«¨¤®¢® ¯à®áâà ­á⢮ â®ç¥ª x = (x

1

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n

), l =

(l

1 ;:::;l

n

) | ¬ã«ì⨨­¤¥ªá, l

i >0.

 áᬮâਬ ®¤­®¯ à ¬¥âà¨ç¥áªãî £à㯯㠯८¡à §®¢ ­¨© R n

H

t

(x)=(t l

l

1

x

1 ;:::;t

l

l

n

x

n

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Z

(x;y)6" jD

f(y)j p

dy 6M" pl

(1,j:lj) X

j:lj=1 Z

(x;y)62" jD

f(y)j p

dy (4)

¤«ï ¢á¥å x2 R n

, " > 0 ¨ ¤«ï ¢á¥å f 2 L l

p (R

n

), ª®â®àë¥ ®¡à é îâáï ¢ ­®«ì ­ 

®âªàë⮬ ¯®¤¬­®¦¥á⢥ B

" (x).

‹¥¬¬  1. ãáâì K R n

| ª®¬¯ ªâ. ‘ãé¥áâ¢ã¥â äã­ªæ¨ï '

"

(x) â ª ï,

çâ® '

"

(x) =1 ¤«ï «î¡®£® x 2K, '

"

(x)=0 ¢­¥"-®ªà¥áâ­®á⨠K ¨ ¤«ï «î¡®£®

¬ã«ì⨨­¤¥ªá  =(

1 ;:::

n )2N

n

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jD

'

"

(x)j 6K

"

,l

j:lj

: (5)

C ‡ ä¨ªá¨à㥬äã­ªæ¨î '2C 1

0 (R

n

), ®â«¨ç­ãî®â ­ã«ï¢ è à¥ (x)<1

¨ ⮦¤¥á⢥­­® à ¢­ãî ­ã«î ¢­¥ í⮣® è à . ãáâì (x) = P

'(x,), £¤¥

| ¯à®¡¥£ ¥â ¢á¥ â®çª¨ á 楫®ç¨á«¥­­ë¬¨ ª®®à¤¨­ â ¬¨ ¢ R n

. Žç¥¢¨¤­®

(x) > 0. ®«®¦¨¬

(x) =

'(x,)

(x)

. ˆ¬¥¥¬

(x) 2 C 1

(R n

),

(x) = 0 ¯à¨

(x,)> 1 ¨ ¤«ï ¢á¥å x 2 R n

¢¥à­® P

(x)=1. ãáâì ⥯¥àì h = 1

2c ", £¤¥

c | ¯®áâ®ï­­ ï ¨§ ­¥à ¢¥­á⢠ âà¥ã£®«ì­¨ª  ¤«ï ¢ë¡à ­­®£® -à ááâ®ï­¨ï.

 áᬮâਬ á¨á⥬ã ä㭪権

(H

h ,1

(x)). ãáâì fg | ¢á¥ ¢¥ªâ®àë, ¤«ï

ª®â®àëå ­®á¨â¥«ìä㭪樨

(H

h

,1(x)) ¯¥à¥á¥ª ¥â ¬­®¦¥á⢮ K. ®«®¦¨¬

'

" (x) =

X

fg

(H

h

,1(x)):

Žç¥¢¨¤­®, '

"

(x) 2 C 1

(R n

), '

"

(x) = 1 ¤«ï x 2 K, '

"

(x) = 0 ¤«ï ¢á¥å x,

«¥¦ é¨å ¢­¥ "-®ªà¥áâ­®áâ¨K ¨

jD

'

" (x)j6

K

" l

j:lj : B

Žâ¬¥â¨¬, çâ® ¤®ª § â¥«ìá⢮ «¥¬¬ë ®á­®¢ ­® ­  á奬¥, ¯à¥¤«®¦¥­­®© ¢

¨§®âய­®¬ á«ãç ¥ ž. ƒ. ¥è¥â­ïª®¬ [10] ¨ à á¯à®áâà ­¥­­®© ­ 

 ­¨§®âய-­ë© á«ãç © ‘. Š.‚®¤®¯ìï­®¢ë¬ [8].

Œë ¡ã¤¥¬ à áᬠâਢ âì ®¡« áâ¨K, 㤮¢«¥â¢®àïî騥 ãá«®¢¨î (A):

(A) ‘ãé¥áâ¢ãîâ > 0, > 0 â ª¨¥, çâ® ¤«ï «î¡ëå x;y 2 R n

nK,

㤮¢-«¥â¢®àïîé¨å ­¥à ¢¥­áâ¢ã (x;y) < ­ ©¤¥âáï á¯àשׂ塞 ï ¤ã£  R n

nK

¤«¨­®© l(), ᮥ¤¨­ïîé ï x ¨ y, ¯à¨ç¥¬ l() 6 c(x;y) ¨ ¤«ï «î¡®£® z 2

¨¬¥îâ ¬¥áâ® ­¥à ¢¥­á⢠

(z;@K)>(x;@K); (z;@K)<(y;@K);

§¤¥áì ¯®áâ®ï­­ ï c ­¥ § ¢¨á¨â ®â x ¨ y. Œ¥âਪ  ¡¥à¥âáï ¢¨¤  (2).

2. ’¥®à¥¬  ® ¯«®â­®áâ¨

’¥®à¥¬ . ãáâìK R n

|ª®¬¯ ªâ,㤮¢«¥â¢®àïî騩 ãá«®¢¨î (A).’

®£-¤  C 1

(K) ¯«®â­®(L l

)

‹¨â¥à âãà 

1.

‘®¡®«¥¢ ‘. ‹.

¥ª®â®àë¥ ¯à¨¬¥­¥­¨ï ä㭪樮­ «ì­®£®  ­ «¨§  ¢ ¬ â¥¬ â¨ç¥áª®©

䨧¨ª¥. | ‹.: ˆ§¤-¢® ‹ƒ“, 1950.|225 á.

2.

Œ §ìï ‚.ƒ.

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

3.

‹¨®­á†.-‹.,Œ ¤¦¥­¥á.

¥®¤­®à®¤­ë¥ £à ­¨ç­ë¥ § ¤ ç¨ ¨ ¨å ¯à¨«®¦¥­¨ï.|Œ.:

Œ¨à, 1971.|371 á.

4.

Polking J. C.

Approximation in

L

p

by solution of eliptic partial dierential

equa-tions //Amer. J. Math.|1972.|V. 94.|P. 1231{1244.

5.

Hedberg L.I.

Approximation in the mean by solution of eliptic equations // Duke Math.|

1973.|V. 40, No. 1.|P. 9{16.

6.

¥á®¢Ž.‚.,ˆ«ì¨­‚..,¨ª®«ì᪨©‘.Œ.

ˆ­â¥£à «ì­ë¥ ¯à¥¤áâ ¢«¥­¨ï ä㭪権

¨ â¥®à¥¬ë ¢«®¦¥­¨ï. |Œ.:  ãª .|1975.|408 á.

7.

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

’¥®à¥¬ë ¢«®¦¥­¨ï ¨

¯à¨«®-¦¥­¨ï ª ¤¨ää¥à¥­æ¨ «ì­ë¬ ãà ¢­¥­¨ï¬.| ®¢®á¨¡¨àáª:  ãª  1978.

8.

€«¡®à®¢  Œ. ‘., ‚®¤®¯ìï­®¢ ‘. Š.

“áâà ­¨¬ë¥ ®á®¡¥­­®á⨠¤«ï ®£à ­¨ç¥­­ëå

à¥-襭¨© ª¢ §¨í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨© // „¥¯. ¢ ‚ˆˆ’ˆ.|1987, B87-804.

9.

€«¡®à®¢  Œ. ‘.

¥ª®â®àë¥ ¨­â¥£à «ì­ë¥ ­¥à ¢¥­á⢠ ¨ â¥®à¥¬ë ¢«®¦¥­¨ï ¤«ï

 ­¨-§®âய­ëå ä㭪樮­ «ì­ëå ¯à®áâà ­á⢠// „¥¯. ¢ ‚ˆˆ’ˆ, 2000, 3258-‚-00.

10.

¥è¥â­ïª ž. ƒ.

Ž ¯®­ï⨨ ¥¬ª®á⨠¢ ⥮ਨ ä㭪権 á ®¡®¡é¥­­ë¬¨

¯à®¨§¢®¤­ë-¬¨ // ‘¨¡. ¬ â. ¦ãà­.|1969.|’. 10, ü 5.|C. 1109{1139.