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, Dn
!1
b) ’ ª ª ª ¬ ªá¨¬ «ì­ ï äã­ªæ¨ï M (x) ¯®«ã­¥¯à¥à뢭  á­¨§ã, â® ¬­®¦¥á⢮
Et ®âªàë⮥.
c) ’ ª ª ª ¬­®¦¥á⢮ Et ®âªàëâ® ¤«ï «î¡®£® t > 0, â® MD | ¡®à¥«¥¢áª ï äã­ªæ¨ï. B
à¥¤«®¦¥­¨¥ 2.
 q
D
X
U D
t>0
ãáâì

|

-ª¢ §¨ ¤¤¨â¨¢­ ï äã­ªæ¨ï, ®¯à¥¤¥«¥­­ ï ­  ®â-

ªàëâëå ¯®¤¬­®¦¥áâ¢ å ®¡« áâ¨

®âªàë⮣® ¬­®¦¥á⢠

®¤­®à®¤­®£® ¯à®áâà ­á⢠

.

’®£¤  ¤«ï «î¡®£®

¨ «î¡®£®

,

,







 x 2 U : MU (x) > t

6 q ct2 (U ):

„«ï ¤®ª § â¥«ìá⢠ ¯à¥¤«®¦¥­¨ï 2 ­ ¬ ¯®âॡã¥âáï á«¥¤ãîé ï
‹¥¬¬  2 [4, c. 12].
E
X
ãáâì

­¥­¨¥¬ ª®­¥ç­®£® ­ ¡®à  è à®¢
­¥¯¥à¥á¥ª îé¨åáï è à®¢

| ¨§¬¥à¨¬®¥ ¯®¤¬­®¦¥á⢮

fBj g

B1 ; : : : ; Bm
m
X
k=1

, ïî饥áï ®¡ê¥¤¨-

. ’®£¤  ¬®¦­® ¢ë¡à âì ª®­¥ç­ë© ­ ¡®à ¯®¯ à­®
¨§

fBj g

â ª®©, çâ®

1
(Bk ) > c,
2 (E ):

¥à¥©¤¥¬ ª ¤®ª § â¥«ìáâ¢ã ¯à¥¤«®¦¥­¨ï 2.
C Ž¯à¥¤¥«¨¬ ¬­®¦¥á⢮ Et á«¥¤ãî騬 ®¡à §®¬:


,






Et = x 2 U : MU (x) > t ;

¨ ¯ãáâì E ¡ã¤¥â ¯à®¨§¢®«ì­®¥ ª®¬¯ ªâ­®¥ ¯®¤¬­®¦¥á⢮ ¬­®¦¥á⢠ Et . ˆ§ ®¯à¥¤¥«¥­¨© ¬ ªá¨¬ «ì­®© ä㭪樨 MU ¨ ¬­®¦¥á⢠ Et á«¥¤ã¥â, çâ® ¤«ï «î¡®© â®çª¨ x 2 E
áãé¥áâ¢ã¥â è à B x  U , ᮤ¥à¦ é¨© â®çªã x, â ª®©, çâ®
(B x ) > t(B x ):

(3)

’ ª ª ª x 2 B x , â® ¢ ᨫ㠪®¬¯ ªâ­®á⨠¬­®¦¥á⢠ E ¬ë ¬®¦¥¬ ¢ë¡à âì ª®­¥ç­ãî
ᮢ®ªã¯­®áâì â ª¨å è à®¢, ¯®ªà뢠îéãî ¬­®¦¥á⢮ E . ® «¥¬¬¥ 2 ¨§ ­¥¥ ¬®¦­®
¢ë¡à âì ª®­¥ç­ë© ­ ¡®à ¯®¯ à­® à §«¨ç­ëå è à®¢ B1 ; : : : ; Bm â ª®©, çâ®
(E ) 6 c2

m
X
k=1

(Bk ):

(4)

à¨¬¥­ïï ª ª ¦¤®¬ã ¨§ è à®¢ Bk (3) ¨ § â¥¬ (4), ¯à¨å®¤¨¬ ª ­¥à ¢¥­á⢠¬
X
P

k

(Bk ) > t

m
X
k=1

1
(Bk ) > tc,
2 (E ):

ˆá¯®«ì§ãï ®æ¥­ªã (Bk ) 6 q(U ), ¨¬¥¥¬ (E ) 6 q ct2 (U ) ¤«ï ¯à®¨§¢®«ì­®£® ª®¬k
¯ ªâ­®£® ¯®¤¬­®¦¥á⢠ E  Et . ¥à¥å®¤ï ª â®ç­®© ¢¥àå­¥© £à ­¨ ¯® ¢á¥¬ â ª¨¬
E  Et , ¯®«ãç ¥¬ âॡ㥬®¥ ã⢥ত¥­¨¥. B

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‘. Š. ‚®¤®¯ìï­®¢, €. „. “å«®¢

Bepå­ïï ¨ ­¨¦­ïï ¯à®¨§¢®¤­ë¥ q-ª¢ §¨ ¤¤¨â¨¢­®© ä㭪樨 ¬­®¦¥á⢠, § ¤ ­­®© ­  ®âªàëâëå ¬­®¦¥áâ¢ å ¨§ D, ®¯à¥¤¥«ïîâáï á«¥¤ãî騬 ®¡à §®¬:

B )
0
inf (B ) ;
0 (x) = hlim
sup (
!0  tg:

0 (x) < +1

¤«ï ¯®ç⨠¢á¥å

x2D

.

| ¯à®áâà ­á⢮ ®¤­®à®¤­®£® ⨯  á ®¯à¥¤¥«¥­­®© ­ 

®âªàëâëå ¯®¤¬­®¦¥áâ¢ å ®¡« áâ¨

DX q
U D

-ª¢ §¨ ¤¤¨â¨¢­®© ä㭪樥© ¬­®¦¥á⢠

Z 0(x) d(x) 6 c q (U ):



.

’®£¤  ¤«ï «î¡®£® ®âªàë⮣® ¬­®¦¥á⢠

2 2

U

C à¨ 1 < t < 1 ¬­®¦¥á⢮ A = fx 2 U : 0 < 0 (x) < 1g ¯à¥¤áâ ¢¨¬®0 ¢ ¢¨¤¥ ®¡ê¥¤¨­¥­¨ï ­¥¯¥à¥á¥ª îé¨åáï ¨§¬¥à¨¬ëå ¬­®¦¥á⢠Pn = fx 2 A : tn <  (x) 6 tn+1 g,
n 2 Z.  áᬮâਬ ¯à®¨§¢®«ì­®¥ ª®¬¯ ªâ­®¥ ¬­®¦¥á⢮ Fn  Pn ¨ ¤«ï 䨪á¨à®¢ ­­®£® ª®­¥ç­®£® ­ ¡®à  K  Z à áᬮâਬ ¯à®¨§¢®«ì­ë¥ ®âªàëâë¥ ¯®¯ à­® ­¥¯¥à¥á¥ª î騥áï ¬­®¦¥á⢠ Un  Fn , Un  U , n 2 K . ’®£¤ , ¯à¨¬¥­ïï á«¥¤á⢨¥ 2 ª
¬­®¦¥á⢠¬ Fn ¢¬¥áâ® Et ¨ Un ¢¬¥áâ® U , ¯®«ãç ¥¬ ­¥à ¢¥­á⢠

Z
X
X
X
1
1
,
t (F ) >
t
q(U ) > (U ) >
0 (x) d(x):
2

n K

n

c2 q n2K

n

n

c2 q n2K

1

Fn

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Ž¯¥à â®àë á㯥௮§¨æ¨¨ ¢ ¯à®áâà ­áâ¢ å ‹¥¡¥£ 

’ ª ª ª ¬­®¦¥á⢠
Fn  Pn , n 2 Z, ­ ¡®à K  Z ¨ ç¨á«® t > 1 ¯à®¨§¢®«ì­ë,  
0
(fx 2 U :  (x) = 1g) = 0, ⮠⥮६  ¤®ª § ­ . B
‘«¥¤á⢨¥ 3 (⥮६  ‹¥¡¥£ ).
X
D
X
f
L1;loc (D)
ãáâì

« áâì ¢
¢á¥å

.

x2D

à¥¤¯®«®¦¨¬, çâ® äã­ªæ¨ï

1



| ®¤­®à®¤­®¥ ¯à®áâà ­á⢮,

¯à¨­ ¤«¥¦¨â

.

| ®¡-

’®£¤  ¤«ï ¯®çâ¨

Z

jf (y) , f (x)j d(y) = 0:
!0lim
; B 3x (B )

(5)

B

C „«ï ä㭪樨 g 2 L1;loc(D) ¨ ®âªàë⮣® ¬­®¦¥á⢠ U  D ¯®«®¦¨¬
(U ) =

Z

g(y) d(y):

U

’ ª ª ª  | 1-ª¢ §¨ ¤¤¨â¨¢­ ï äã­ªæ¨ï ¬­®¦¥á⢠, â® ¯® ⥮६¥ 1 ¤«ï ¯à®¨§¢®«ì­®£® ®âªàë⮣® ¬­®¦¥á⢠ U  D ¢ë¯®«­¥­® ­¥à ¢¥­á⢮
Z

g(y) d(y) 6 c2

Z

g(y) d(y) = c2 (U );

U

U

£¤¥
g (x) = lim

1

sup
h!0 x2B ; t > 0
0

, â® ¤«ï «î¡®£® ®âªàë⮣® ¬­®¦¥á⢠

’®£-

¢ë¯®«­ï¥âáï ­¥à ¢¥­á⢮

,

, ¨¬¥¥¬

e(E ) 6 qt (U ):

(7)

C „¥©á⢨⥫쭮, ¢®§ì¬¥¬ ¯à®¨§¢®«ì­ãî â®çªã x 2 E . ’®£¤  ­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì è à®¢ Bkx  U , x 2 Bkx , à ¤¨ãáë ª®â®àëå áâ६ïâáï ª ­ã«î, â ª ï, çâ®
­¥à ¢¥­á⢮
(Bkx ) > t > 0
(B x )
k




,S

S

¢ë¯®«­¥­® ¯à¨ ¢á¥å k 2 N . ‘¥¬¥©á⢮ x E k N Bkx ®¡à §ã¥â ¯®ªàë⨥ ¬­®¦¥á⢠
E ¢ á¬ëá«¥ ‚¨â «¨, §­ ç¨â, ­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì
¯®¯ à­® ­¥¯¥à¥á¥ª îé¨åáï
S
è à®¢ Bm ¨§ í⮣® ᥬ¥©á⢠ â ª ï, çâ® e (E n m Bm ) = 0.
ˆ¬¥¥¬
X
e (E ) 6 (Bm );
2

2

m

¨ t(Bm ) < (Bm ) ¯à¨ ª ¦¤®¬ m. ‘«¥¤®¢ â¥«ì­®,

te (E ) 6 t

X

m

(Bm ) 6

X

m

(Bm ) 6 q(U )

¨ ¯®í⮬㠭㦭®¥ ­¥à ¢¥­á⢮ ãáâ ­®¢«¥­®. B
’¥®à¥¬  3. ãáâì
¬­®¦¥á⢠



X

| ®¤­®à®¤­®¥ ¯à®áâà ­á⢮,  

q

-ª¢ §¨ ¤¤¨â¨¢­ ï äã­ªæ¨ï

®¯à¥¤¥«¥­  ­  ®âªàëâëå ¯®¤¬­®¦¥áâ¢ å ®¡« áâ¨

«î¡®£® ®âªàë⮣® ¬­®¦¥á⢠

U D
Z

U

D

X.

’®£¤  ¤«ï

 (x) d(x) 6 q(U ):

(8)

0

€­ «®£¨ç­®¥ ­¥à ¢¥­á⢮ ¤«ï ­¨¦­¥© ¯à®¨§¢®¤­®©  ¢¬¥áâ®
¢¥àå­¥©  ¡ë«® ãáâ ­®¢«¥­® ¢ [8, «¥¬¬  2.3] ¯à¨ ¯à®¨§¢®«ì­®¬ q ¢ ¯à¥¤¯®«®¦¥­¨¨,
çâ® q-ª¢ §¨ ¤¤¨â¨¢­ ï äã­ªæ¨ï ¬­®¦¥á⢠ ®¯à¥¤¥«¥­  ­  ¡®à¥«¥¢áª¨å ¯®¤¬­®¦¥áâ¢ å ¥¢ª«¨¤®¢  ¯à®áâà ­á⢠ Rn .
C à¨ 1 < t < 1 ¬­®¦¥á⢮ A = fx 2 U : 0 <  (x) < 1g ¯à¥¤áâ ¢¨¬® ¢ ¢¨¤¥
®¡ê¥¤¨­¥­¨ï ­¥¯¥à¥á¥ª îé¨åáï ¬­®¦¥á⢠Pn = fx 2 A : tn <  (x) 6 tn+1 g, n 2 Z.
 áᬮâਬ ¯à®¨§¢®«ì­®¥ ª®¬¯ ªâ­®¥ ¬­®¦¥á⢮ Fn  Pn ¨ ¤«ï 䨪á¨à®¢ ­­®£® ª®­¥ç­®£® ­ ¡®à  K  Z à áᬮâਬ ¯à®¨§¢®«ì­ë¥ ®âªàëâë¥ ¯®¯ à­® ­¥¯¥à¥á¥ª î騥áï ¬­®¦¥á⢠ Un  Fn , Un  D, n 2 K . B®§ì¬¥¬ ¯à®¨§¢®«ì­ãî â®çªã x 2 Fn . ’®£¤ 
­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì è à®¢ Bkx  Un , x 2 Bkx , à ¤¨ãáë ª®â®àëå áâ६ïâáï ª
­ã«î, â ª ï, çâ® ­¥à ¢¥­á⢮
0

‡ ¬¥ç ­¨¥ 1.
0

0

0

(Bkx ) > tn (Bkx ) > 0

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S

¯à¨
¢á¥å k 2 N . ãáâì F = n Fn ; n 2 K . Žç¥¢¨¤­®, ç⮠ᥬ¥©á⢮ è à®¢
x
x2F k2N Bk ®¡à §ã¥â ¯®ªàë⨥ ¬­®¦¥á⢠ F ¢ á¬ëá«¥ ‚¨â «¨. ‘«¥¤®¢ â¥«ì­®,
­ ©¤¥âáï ¯®á«¥¤®¢ â¥«ì­®áâì
¯®¯ à­® ­¥¯¥à¥á¥ª îé¨åáï è à®¢ Bm ¨§ í⮣® ᥬ¥©á⢠
S
â ª ï, çâ®
(F n m Bm ) = 0. ‡ ¬¥â¨¬, çâ® ¢ á¨«ã ¢ë¡®à  è à®¢ á¯à ¢¥¤«¨¢® ᢮©á⢮
S
(Fn n fBm : Bm \ Fn 6= ?g) = 0 ¤«ï «î¡®£® n 2 K . „ «¥¥ ¨¬¥¥¬
¢ë¯®«­¥­®
,S
S

q (U )

>

X

m2N

(Bm ) =

X

X

n2K Bm \Fn 6=?

(Bm ) >
=

X

n2K
X

n2 K

tn





X

Bm \Fn 6=?

tn (Fn ) >

(Bm )

X

n2K

t,1

Z

Fn

0 (x) d(x):

’ ª ª ª ¬­®¦¥á⢠ Fn  Pn , n 2 Z, ­ ¡®à K  Z ¨ ç¨á«® t > 1 ¯à®¨§¢®«ì­ë, ­¥à ¢¥­á⢮ (8) ¤®ª § ­®. B
‘«¥¤á⢨¥ 4. ãáâì
æ¨ï ¬­®¦¥á⢠
¯®ç⨠¢á¥å

x



2X

X

| ®¤­®à®¤­®¥ ¯à®áâà ­á⢮,  

®¯à¥¤¥«¥­  ­  ®âªàëâëå ¯®¤¬­®¦¥á⢠å

q -ª¢ §¨ ¤¤¨â¨¢­ ï äã­ª®¡« á⨠D
. ’®£¤  ¤«ï

X

0 (x) 6 q0 (x):

C ˆ§ ­¥à ¢¥­á⢠ (8) á«¥¤ã¥â, çâ® ¤«ï ¯à®¨§¢®«ì­®£® è à  B  D á¯à ¢¥¤«¨¢®
1

(B )

Z

B

1 (B ):
0 (x) d(x) 6 q (B

)

¥à¥å®¤ï ¢ ¯®á«¥¤­¥¬0 ­¥à ¢¥­á⢥ ª ­¨¦­¥¬ã ¯à¥¤¥«ã ¯à¨  ! 0 ¨ ¯à¨¬¥­ïï á«¥¤á⢨¥ 3, ­ å®¤¨¬, çâ®  (x) 6 q0 (x). B
ˆ§ ⥮६ë 3 ¨ á«¥¤á⢨ï 4 ¢ë⥪ îâ ᢮©á⢠ (ª¢ §¨) ¤¤¨â¨¢­®© ä㭪樨, ®¯à¥¤¥«¥­­®© ­  ®âªàëâëå ¯®¤¬­®¦¥áâ¢ å ®¤­®à®¤­®£® ¯à®áâà ­á⢠ X, ®¡®¡é î騥 ᮮ⢥âáâ¢ãî騥 १ã«ìâ âë ¨§ [3], ¢ ç áâ­®áâ¨, ᮮ⭮襭¨ï (1) ¨ (2) ¯à¨ X = Rn .
‘«¥¤á⢨¥ 5.
X
1(
)

DX
a)
x2D
ãáâì

æ¨ï ¬­®¦¥á⢠

| ®¤­®à®¤­®¥ ¯à®áâà ­á⢮,  

- ª¢ §¨  ¤¤¨â¨¢­ ï äã­ª-

®¯à¥¤¥«¥­  ­  ®âªàëâëå ¯®¤¬­®¦¥áâ¢ å ®¡« áâ¨

¢ ¯®ç⨠ª ¦¤®© â®çª¥

. ’®£¤ 

áãé¥áâ¢ã¥â ª®­¥ç­ ï ¯à®¨§¢®¤­ ï

lim (B ) = 0 (x);
!0; B 3x (B )
b)

¤«ï «î¡®£® ®âªàë⮣® ¬­®¦¥á⢠

Z

U

U

D

á¯à ¢¥¤«¨¢® ­¥à ¢¥­á⢮

0 (x) d(x) 6 (U ):

‡ ¬¥â¨¬, ç⮠ᮮ⭮襭¨¥ ¬¥¦¤ã ¢¥àå­¥© ¨ ­¨¦­¥© ¯à®¨§¢®¤­ë¬¨ q-ª¢ §¨ ¤¤¨â¨¢­®© ä㭪樨 (á«¥¤á⢨¥ 4) ¬®¦¥â ¡ëâì ¤®ª § ­® ­¥§ ¢¨á¨¬ë¬ ®¡à §®¬. à¨¢®¤¨¬®¥ ­¨¦¥ ¤®ª § â¥«ìá⢮ ®¡®¡é ¥â à áá㦤¥­¨ï ¨§ [3, ⥮६  5, c. 207; 9, á. 33{35].

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Ž¯¥à â®àë á㯥௮§¨æ¨¨ ¢ ¯à®áâà ­áâ¢ å ‹¥¡¥£ 

à¥¤«®¦¥­¨¥ 4. ãáâì



q

|

-ª¢ §¨ ¤¤¨â¨¢­ ï äã­ªæ¨ï, ®¯à¥¤¥«¥­­ ï ­  ª®-

DX
X
0 (x) 6 q0 (x)
x2D
C à¥¤áâ ¢¨¬ ¬­®¦¥á⢮ A = fx 2 D : 0 (x) > q0(x)g ¢ ¢¨¤¥ A = S Ars , £¤¥
r;s
®¡ê¥¤¨­¥­¨¥ ¡¥à¥âáï ¯® ¢á¥¬ ¯ à ¬ à æ¨®­ «ì­ëå ç¨á¥« r > s,   Ar;s = fx 2 D :
0 (x) > r > s > q0 (x)g. ”¨ªá¨à㥬 ¯ àã ç¨á¥« r > s ¨ ¤®ª ¦¥¬, çâ® (Ars ) = 0.
ãáâì ¯®á«¥¤®¢ â¥«ì­®á⨠è à®¢ Bk 3 x; Bk  D ¨ Bk 3 x; Bk  D ¢ë¡à ­ë â ª, çâ®
(Bk ) ¨ 0 (x) = lim (Bk ) :
0 (x) = rad(lim
Bk )!0 (Bk )
rad(Bk )!0 (Bk )
’®£¤ 


 ) 
(
B
)
(
B
k
k
Ars = x 2 D : 9k0 (x) 8k > k0 (x) (B ) > r > s > q (B  ) :
k
k
‚®§ì¬¥¬ ¯à®¨§¢®«ì­®¥ " > 0 ¨ â ª®¥S®âªàë⮥
¬­®¦¥á⢮
G
,
ᮤ¥à¦ é¥¥
A
rs , çâ®
S 
Bk ®¡à §ã¥â ¯®ªàë⨥ ‚¨â «¨ ¬­®¦¥á⢠
(G) 6 e(Ars ) + ". ‘¥¬¥©á⢮ B =
x2Ars k
Ars . à¨¬¥­¨¬ ª ᥬ¥©áâ¢ã B ¨ ¬­®¦¥áâ¢ã Ars «¥¬¬ã 3, ¨á¯®«ì§ãï ⮫쪮 ⥠è àë
Bk, ª®â®àë¥ á®¤¥à¦ âáï
¢ G.
®«ã稬 ¯®á«¥¤®¢ â¥«ì­®áâì ­¥¯¥à¥á¥ª îé¨åáï è à®¢
,
Sk , ¤«ï ª®â®à®© e Ars n S Sk = 0. Ÿá­®, çâ®
­¥ç­ëå ®¡ê¥¤¨­¥­¨ïå ®âªàëâëå è à®¢ ¨§ ®¡« áâ¨
¤«ï ¯®ç⨠¢á¥å

à ­á⢮. ’®£¤ 

k

‡ ¬¥â¨¬, çâ®

(U ) >

X

k

, £¤¥

| ®¤­®à®¤­®¥ ¯à®áâ-

.

X
(Sk ) > qs (Sk ):

e(Ars ) 6

k

X

k 2N

e(Ars \ Sk):

®í⮬ã, ¯à¨¬¥­ïï ¯à¥¤«®¦¥­¨¥ 3 ª ¬­®¦¥áâ¢ã Ars [ Sk ¢¬¥áâ® E ¨ Sk ¢¬¥áâ® U ,
¨¬¥¥¬
X
X
e(Ars ) 6 e(Ars \ Sk ) 6 qr (Sk) 6 rs (G) 6 rs (e(Ars ) + "):
ˆâ ª,

k

k 2N

e(Ars ) 6 " r ,s s :
®áª®«ìªã " ᪮«ì 㣮¤­® ¬ «®, ¯®«ãç ¥¬ (Ars ) = 0. B

‡ ¬¥ç ­¨¥ 2. ¥§ã«ìâ âë ¯ à £à ä  1 á¯à ¢¥¤«¨¢ë â ª¦¥ ¨ ¢ ⮬ á«ãç ¥, ¥á«¨ ¢¬¥áâ® ¬¥âà¨ç¥áª®£® ¯à®áâà ­á⢠ (X; d) à áᬠâਢ ¥âáï ¬­®¦¥á⢮ X â ª®¥, çâ®
¤«ï ª ¦¤®© â®çª¨ x 2 X ®¯à¥¤¥«¥­  á¨á⥬  fB (x)g ­¥¯ãáâëå ®£à ­¨ç¥­­ëå ¯®¤¬­®¦¥á⢠¬­®¦¥á⢠ X, ¯ à ¬¥âਧ®¢ ­­ëå ; 0 <  < 1, â. ¥. § ¤ ­  á¨á⥬ 
fB = B (x)g ®âªàëâëå è à®¢ á 業â஬ ¢ â®çª¥ x à ¤¨ãá  . à¥¤¯®« £ ¥âáï,
çâ® è àë ¬®­®â®­­ë ®â­®á¨â¥«ì­® : B1 (x)  B2 (x) ¯à¨ 1 < 2 , 㤮¢«¥â¢®àïîâ
᢮©á⢠¬ 4, 5, 6,   â ª¦¥ ¤¢ã¬ ä®à¬ã«¨àã¥¬ë¬ ­¨¦¥ âॡ®¢ ­¨ï¬.
T
B (x) = fxg S B (x) = X
‘¢®©á⢮ 7.



¨



.

‘¢®©á⢮ 8. „«ï ª ¦¤®£® ®âªàë⮣® ¬­®¦¥á⢠

x 7! (B (x) \ U )

­¥¯à¥à뢭 .

U

¨ ª ¦¤®£®

>0

äã­ªæ¨ï

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(Bkx ) < tn+2 (Bkx )
S

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tn+2 (Fn ) 6

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t2

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n2K Fn

 (x) d(x) 6 t2
0

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k
X
i=1

(Ui ) 6 q(U ):

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kf j Lp (E )k =

¨

Z

E

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jf jp d(x)

< +1; 1 6 p < 1;

kf j L1 (E )k = ess sup jf (x)j < +1:

x2E

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¥á«¨ áãé¥áâ¢ã¥â ¯®áâ®ï­­ ï K < 1 â ª ï, çâ®
' f j Lq (D)
6 K
f j Lp (De )
¤«ï
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f~(y) = 0 ¢­¥ Ae. ’®£¤  ' f (x) = (f  ')(x) = ' f~j',1 (Ae) (x), x 2 ',1 (Ae), ¨ ¯®í⮬ã ' |
®£à ­¨ç¥­­ë© ®¯¥à â®à.
‹¥¬¬  5.
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' : Lp (De ) ! Lq (D) 1 6 q < p 6 1
2.1.

ãáâì ®â®¡à ¦¥­¨¥

¢«®¦¥­¨ï

(Ae) =

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sup

f 2Lp (Ae)

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ï¥âáï ®£à ­¨ç¥­­®© ¬®­®â®­­®© áç¥â­®- ¤¤¨â¨¢­®© ä㭪樥©, ®¯à¥¤¥«¥­­®© ­ 

Ae  De e(Ae) > 0
C Žç¥¢¨¤­®, çâ® (Ae1 ) 6 (Ae2 ), ¥á«¨ Ae1  Ae2 .
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1
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¡®à¥«¥¢áª¨å ¬­®¦¥á⢠å

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fi j Lp (A
ei )
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i=1
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à â®à ¢«®¦¥­¨ï

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. ’®£¤  ¤«ï ¢á类£® ¡®à¥«¥¢áª®£®

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(',1 (Ae)) 1q 6 (Ae) 1 ~(Ae) p1 ; 1 6 q < p < 1;
¨

(',1 (Ae)) 6 K p ~(Ae); 1 6 q = p < 1:
C ’ ª ª ª ®¯¥à â®à ¢«®¦¥­¨ï ' : Lp (De ) ! Lq (D) ®£à ­¨ç¥­, â® ¤«ï ¢á类£®
¡®à¥«¥¢áª®£® ¬­®¦¥á⢠ Ae  De ¢ë¯®«­ïîâáï ­¥à ¢¥­á⢠
k' f j Lq (',1 (Ae))k 6 (Ae) 1 kf j Lp (Ae)k 6 (De ) 1 kf j Lp (Ae)k

1{26

‘. Š. ‚®¤®¯ìï­®¢, €. „. “å«®¢

¯à¨ 1 6 q < p < , ¨
1

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' f Lq (',1 (Ae)) 6 K f Lp (Ae)
j

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f (y) = Ae(y) ¬­®¦¥á⢠ Ae, ¯®«ãç ¥¬ âॡ㥬®¥ ã⢥ত¥­¨¥. B
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