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‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
ˆî­ì{ᥭâï¡àì, 2001, ’®¬ 3, ‚ë¯ã᪠3

“„Š 517.5

’…Ž…Œ€ Ž ‹Ž’Ž‘’ˆ
Œ. ‘. €«¡®à®¢ 

‘ä®à¬ã«¨à®¢ ­  ¨ ¤®ª § ­  ⥮६  ® ¯«®â­®á⨠¯à®áâà ­á⢠ ¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬ëå ä㭪権 ¢  ­¨§®âய­ëå ¯à®áâà ­áâ¢ å ‘®¡®«¥¢  ¯à¨ ­¥ª®â®àëå ãá«®¢¨ïå
­ «®¦¥­­ëå ­  ®¡« áâì.

‚ ­ áâ®ï饩 à ¡®â¥ ¨§ãç ¥âáï ¢®¯à®á ® ¯«®â­®á⨠¯à®áâà ­á⢠ ¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬ëå ä㭪権 ¢  ­¨§®âய­ëå ¯à®áâà ­áâ¢ å ‘®¡®«¥¢ . Œë ¡ã¤¥¬ à áᬠâਢ âì ¯à®áâà ­á⢠ Llp (
) å à ªâ¥à¨§ãî騥áï ª®­¥ç­®áâìî ­®à¬ë:
D f Lp (
) ;
f Llp (
) =
k

X

k

j

:lj=1

k

k

§¤¥áì
| ®âªàë⮥ ¬­®¦¥á⢮ ¢ R n , 1 6 p 6 ,  = (1; :: : ; n ) ¨ l =
f  .
(l1; : : : ; ln ) | ¬ã«ì⨨­¤¥ªáë,  : l := l11 + + lnn , D f = @x1 1@:::@x
nn
‚ ¨§®âய­®¬ á«ãç ¥ à §«¨ç­ë¥  á¯¥ªâë § ¤ ç¨ ® ¯«®â­®á⨠¯à®áâà ­á⢠
¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬ëå ­¥¯à¥à뢭ëå ä㭪権 ¢ ¯à®áâà ­áâ¢ 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¨å ãá«®¢¨î ண  ¨ ¤«ï ¡«¨§ª®£® ª« áá  ®¡« á⥩ ¢ à ¡®â å Ž. ‚. ¥á®¢ , ‚. . ˆ«ì¨­ , ‘. Œ. ¨ª®«ì᪮£® [6], ‘. ‚. “ᯥ­áª®£®, ƒ. ‚. „¥¬¨¤¥­ª®,
‚. ƒ. ¥à¥¯¥«ª¨­  [7], . ˆ. ‹¨§®àª¨­ , ‚. ˆ. ã७ª®¢ , ‘. Š. ‚®¤®¯ìï­®¢ 
¨ ¤à.

1

j






j

1. à¥¤¢ à¨â¥«ì­ë¥ ᢥ¤¥­¨ï

ãáâì R n | ¥¢ª«¨¤®¢® ¯à®áâà ­á⢮ â®ç¥ª x = (x1; : : : ; xn ), l =
(l1; : : : ; ln ) | ¬ã«ì⨨­¤¥ªá, li > 0.
 áᬮâਬ ®¤­®¯ à ¬¥âà¨ç¥áªãî £à㯯㠯८¡à §®¢ ­¨© R n
l

l

Ht (x) = (t l1 x1 ; : : : ; t ln xn ) (t 2 R + );

c 2001 €«¡®à®¢  Œ. ‘.

(1)

3{4

Œ. ‘. €«¡®à®¢ 

P

£¤¥ l1 = n1 ni=1 l1i , ¨ £« ¤ªãî Ht -®¤­®à®¤­ãî ¬¥âਪã, ®¯à¥¤¥«ï¥¬ãî ¢¥ªâ®à®¬
l 2 N n ¯® ä®à¬ã«¥
!1
(x; y ) =

Rn .

n
X

i=1

jxi , yi j2li

2l

(2)

­¥¯à¥à뢭ãî ­ 
˜ à®¬ á 業â஬ ¢ â®çª¥ x à ¤¨ãá  r ­ §ë¢ ¥âáï, ª ª ®¡ëç­®, ¬­®¦¥á⢮
Br (x) = fy 2 R n : (x; y ) < rg:
ãáâì
 R n | ®âªàë⮥ ¯®¤¬­®¦¥á⢮, p > 1. ã¤¥¬ £®¢®à¨âì, çâ®
äã­ªæ¨ï f 2 Lp (
) ¯à¨­ ¤«¥¦¨â ª« ááã Llp (
), ¥á«¨ äã­ªæ¨ï ¨¬¥¥â ®¡®¡f  ,
饭­ë¥ ¯à®¨§¢®¤­ë¥ D f 2 Lp (
), j : lj = 1. ‡¤¥áì D f = @x1 1@:::@x
nn



n
1
 = (1 ; : : : n ) ¨ j : lj = l1 +


+ ln . „«ï â ª¨å ä㭪権 ®¯à¥¤¥«¨¬ ¯®«ã­®à¬ã
X 
kf kLlp(
) =
kD f kLp(
) :
(3)
j:lj=1



à®áâà ­á⢮¬ Llp (
) ­ §®¢¥¬ § ¬ëª ­¨¥ ¢ ­®à¬¥ (3) ¬­®¦¥á⢠ C01 (
)
¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬ëå ä㭪権 á ­®á¨â¥«¥¬ ¢
.
ãáâì K | ¯®¤¬­®¦¥á⢮ R n . Ž¡®§­ ç¨¬ ¬­®¦¥á⢮ ä㭪権 ¨§ Llp (R n )

¨¬¥îé¨å ª®¬¯ ªâ­ë¥ ­®á¨â¥«¨ ¢ K ç¥à¥§ (Llp)K .
ãáâì e  R n | § ¬ª­ã⮥ ¬­®¦¥á⢮. …¬ª®áâìî ¬­®¦¥á⢠ e ­ §®¢¥¬
¢¥«¨ç¨­ã:

cap (e; Llp ) = inf fkU kpLlp : U 2 N(e)g;
£¤¥ N(e) = fU 2 C01 : U = 1 ¢ ®ªà¥áâ­®á⨠eg (á¬. [8]).
‚¢¥¤¥¬ ¥é¥ ¯®«ã­®à¬ã:

jU jp;l;Br =

X

0 0
ãáâì

ä㭪権

£¤¥

| § ¬ª­ã⮥ ¯®¤¬­®¦¥á⢮ è à 

â ª¨å, çâ®

.

„«ï ¢á¥å

¢¥à­® ­¥à ¢¥­á⢮

kU kLq (Br ) 6 C jU jp;l ;Br ;
P
1 6 p 6 q 6 1  = ( p1 , q1 ) ni=1 l1i 6 1
=1 16p=q r, q ; cap (e; Llp (B2r )):

,

.

’¥®à¥¬  ® ¯«®â­®áâ¨
‘«¥¤á⢨¥ 1. ‘ãé¥áâ¢ã¥â ¯®áâ®ï­­ ï M â ª ï, çâ®

Z  p
Z  p
 (1,j :lj) X
pl
jD f (y)j dy 6 M"
jD f (y)j dy
j:lj=1(x;y)62"

(x;y)6"
¢á¥å x 2 R n ,

3{5
(4)

¤«ï
" > 0 ¨ ¤«ï ¢á¥å f 2 Llp (R n ), ª®â®àë¥ ®¡à é îâáï ¢ ­®«ì ­ 
®âªàë⮬ ¯®¤¬­®¦¥á⢥ B"(x).
n | ª®¬¯ ªâ. ‘ãé¥áâ¢ã¥â äã­ªæ¨ï '" (x) â ª ï,
‹¥¬¬  1. ãáâì K  R
çâ® '" (x) = 1 ¤«ï «î¡®£® x 2 K , '" (x) = 0 ¢­¥ "-®ªà¥áâ­®á⨠K ¨ ¤«ï «î¡®£®

¬ã«ì⨨­¤¥ªá   = (1; : : : n ) 2 N n ¨¬¥¥â ¬¥áâ® ®æ¥­ª 
jD'" (x)j 6 K
",l j:lj :
(5)
C ‡ ä¨ªá¨à㥬 äã­ªæ¨î ' 2 C01 (R n ), ®â«¨ç­ãî ®â ­ã«ïP¢ è à¥ (x) < 1
¨ ⮦¤¥á⢥­­® à ¢­ãî ­ã«î ¢­¥ í⮣® è à . ãáâì (x) =  '(x ,  ), £¤¥
 | ¯à®¡¥£ ¥â ¢á¥ â®çª¨ á 楫®ç¨á«¥­­ë¬¨ ª®®à¤¨­ â ¬¨ ¢ R n . Žç¥¢¨¤­®
(x, ) . ˆ¬¥¥¬  (x) 2 C 1 (R n ),  (x) = 0 ¯à¨
(x) > 0. ®«®¦¨¬  (x) = '(


x) P
n
(x ,  ) > 1 ¨ ¤«ï ¢á¥å x 2 R ¢¥à­®   (x) = 1. ãáâì ⥯¥àì h = 21c ", £¤¥
c | ¯®áâ®ï­­ ï ¨§ ­¥à ¢¥­á⢠ âà¥ã£®«ì­¨ª  ¤«ï ¢ë¡à ­­®£® -à ááâ®ï­¨ï.
 áᬮâਬ á¨á⥬ã ä㭪権  (Hh,1 (x)). ãáâì f g | ¢á¥ ¢¥ªâ®àë, ¤«ï
ª®â®àëå ­®á¨â¥«ì ä㭪樨  (Hh,1 (x)) ¯¥à¥á¥ª ¥â ¬­®¦¥á⢮ K . ®«®¦¨¬
X
'" (x) =  (Hh,1 (x)):
f g

Žç¥¢¨¤­®, '" (x) 2 C 1 (R n ), '" (x) = 1 ¤«ï x 2 K , '" (x) = 0 ¤«ï ¢á¥å x,
«¥¦ é¨å ¢­¥ "-®ªà¥áâ­®á⨠K ¨
jD '" (x)j 6 "lK 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 ­¥­­®© ­   ­¨§®âய­ë© á«ãç © ‘. Š. ‚®¤®¯ìï­®¢ë¬ [8].
Œë ¡ã¤¥¬ à áᬠâਢ âì ®¡« á⨠K , 㤮¢«¥â¢®àïî騥 ãá«®¢¨î (A):
(A) ‘ãé¥áâ¢ãîâ  > 0,  > 0 â ª¨¥, çâ® ¤«ï «î¡ëå x; y 2 Rn n K , n㤮¢«¥â¢®àïîé¨å ­¥à ¢¥­áâ¢ã (x; y) <  ­ ©¤¥âáï á¯àשׂ塞 ï ¤ã£ 
 R n K
¤«¨­®© 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¨

ãáâì K  R n
1
¤  C0 (K ) ¯«®â­® (Llp )K .

’¥®à¥¬ .

| ª®¬¯ ªâ, 㤮¢«¥â¢®àïî騩 ãá«®¢¨î (A). ’®£-

3{6
çâ®

Œ. ‘. €«¡®à®¢ 

C ˆá¯®«ì§ãï á«¥¤á⢨¥ 1, ¬ë ¢¨¤¨¬, çâ® áãé¥áâ¢ã¥â ¯®áâ®ï­­ ï M â ª ï,
Z

(x;y)6"

jD f (y)jpdy 6 M"pl (1,j:lj)

Z

X

j:lj=1(x;y)62"

jDf (y)jpdy

(6)

¤«ï ¢á¥å f 2 (Llp )K , x 2 @K ¨ ¤®áâ â®ç­® ¬ «®¬ ".
 áᬮâਬ äã­ªæ¨î '"(x) ¨§ «¥¬¬ë 1. ˆá¯®«ì§ãï ª« áá¨ç¥áª¨© ¬¥â®¤,
¤®áâ â®ç­® ¤®ª § âì, çâ® D0 (K ) \ Llp (Rn ) ¯«®â­® ¢ (Llp)K . ãáâì f 2 (Llp)K
¨ ¯ãáâì f" = '"
f . ®ª ¦¥¬, çâ® fDf" g, ®£à ­¨ç¥­­®¥ ¬­®¦¥á⢮ ¢ Lp ¤«ï
0 6 j : lj 6 1 ¨ çâ® ff"g á室¨âáï ª f ¢ Lp . ’®£¤  áãé¥áâ¢ã¥â ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì ff"j g á« ¡® á室ïé ïáï ¢ Llp ª ä㭪樨 f . ® ⥮६¥  ­ å  | ‘ ªá 
á« ¡® á室ïé ïáï ¯®á«¥¤®¢ â¥«ì­®áâì ff"j g ᮤ¥à¦¨â ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì,
ᢥà⪨ ª®â®à®© ᨫ쭮á室ïâáï
ª f ¢ Llp .

’ ª ª ª D f" = P  D, '"
D f , â® ¤®áâ â®ç­® ¯®ª § âì, çâ® fD
'"

D f g ®£à ­¨ç¥­  ¢ Lp , j
+  j = jj, j : lj = 1. à¨
= 0, ã⢥ত¥­¨¥
®ç¥¢¨¤­®. ®«®¦¨¬
6= 0. ãáâì (x) | à ááâ®ï­¨¥ ®â â®çª¨ x ¤® K , (x) =
inf f(x; y) y 2 K g. ”㭪樨 D
'" ¨¬¥îâ ­®á¨â¥«¨ ¢ ¬­®¦¥á⢥ L" = fx :
 (x) 6 "g. ‘«¥¤®¢ â¥«ì­®,
Z
Z
 j
:lj


p
l
jD f jpdx:
(7)
jD '" (x)D f (x)j dx 6 c
"
L"

®ªà®¥¬ R n è à ¬¨ B"(x). ‚ ᨫã ãá«®¢¨ï (A) áãé¥áâ¢ã¥â ¯®áâ®ï­­ ï N â ª ï, çâ® ª ¦¤®¥ x 2 R n ¯à¨­ ¤«¥¦¨â ­¥ ¡®«¥¥ 祬 N è à ¬. ãáâì fzk g |
­ã¬¥à æ¨ï 業â஢ è à®¢, ª®â®àë¥ ¯¥à¥á¥ª îâ L". ’®£¤  ¤«ï ª ¦¤®£® k à ááâ®ï­¨¥ ®â zk ¤® @K ­¥ ¡®«ìè¥ ç¥¬ 2", â ª¨¬ ®¡à §®¬ ­ ©¤¥âáï â®çª  xk 2 @K
â ª ï, çâ® (xk ; zk ) 6 2" ¨ B2"(zk )  B3C"(xk ). ˆ, á«¥¤®¢ â¥«ì­®, è àë
fB3C"(xk )g ¯®ªà뢠îâ L" . ˆá¯®«ì§ãï (6) ¨¬¥¥¬
Z
Z
X

p
jD f (x)jpdx
jD f j dx 6 M
k (x;xk )63C"
L"
Z
 (1,j :lj) X X
pl
(8)
6 M"
jD f (x)jpdx
n j:lj=1(x;xk )63C"

6 M"pl (1,j:lj)kf kLlp :

¥à ¢¥­á⢠ (7) ¨ (8) ¯®ª §ë¢ îâ, çâ® kf"kp;l 6 M kf kp;l ¤«ï ¢á¥å ¤®áâ â®ç­®
¬ «ëå
". Žª®­ç â¥«ì­® ®â¬¥â¨¬, çâ® f , f" ¨¬¥¥â ­®á¨â¥«ì ¢ L" ¨ kf , f" kp;l <

M"l kf kp;l , â ª¨¬ ®¡à §®¬ f" ! f ¢ Lp . B

’¥®à¥¬  ® ¯«®â­®áâ¨
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

3{7

‹¨â¥à âãà 

¥ª®â®àë¥ ¯à¨¬¥­¥­¨ï ä㭪樮­ «ì­®£®  ­ «¨§  ¢ ¬ â¥¬ â¨ç¥áª®©
䨧¨ª¥. | ‹.: ˆ§¤-¢® ‹ƒ“, 1950.|225 á.
Œ §ìï ‚. ƒ. à®áâà ­á⢠ ‘®¡®«¥¢ .|‹.: ˆ§¤-¢® ‹ƒ“, 1985.|416 á.
‹¨®­á †.-‹., Œ ¤¦¥­¥á . ¥®¤­®à®¤­ë¥ £à ­¨ç­ë¥ § ¤ ç¨ ¨ ¨å ¯à¨«®¦¥­¨ï.|Œ.:
Œ¨à, 1971.|371 á.
p by solution of eliptic partial dierential equaPolking J. C. Approximation in L
tions //Amer. J. Math.|1972.|V. 94.|P. 1231{1244.
Hedberg L. I. Approximation in the mean by solution of eliptic equations // Duke Math.|
1973.|V. 40, No. 1.|P. 9{16.
¥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.
“ᯥ­áª¨© ‘. ‚., „¥¬¨¤¥­ª® ƒ. ‚., ¥à¥¯¥«ª¨­ ‚. ƒ. ’¥®à¥¬ë ¢«®¦¥­¨ï ¨ ¯à¨«®¦¥­¨ï ª ¤¨ää¥à¥­æ¨ «ì­ë¬ ãà ¢­¥­¨ï¬.| ®¢®á¨¡¨àáª:  ãª  1978.
€«¡®à®¢  Œ. ‘., ‚®¤®¯ìï­®¢ ‘. Š. “áâà ­¨¬ë¥ ®á®¡¥­­®á⨠¤«ï ®£à ­¨ç¥­­ëå à¥è¥­¨© ª¢ §¨í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨© // „¥¯. ¢ ‚ˆˆ’ˆ.|1987, B87-804.
€«¡®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.
¥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.
‘®¡®«¥¢ ‘. ‹.

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‘â âìï ¯®áâ㯨«  20 ᥭâï¡àï 2001