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‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
€¯à¥«ì{¨î­ì, 2002, ’®¬ 4, ‚ë¯ã᪠2

“„Š 517.98

ˆ’…ƒ€‹œ›… Ž…€’Ž›
‚‡‚…˜…Ž‰ ‘‚…’Šˆ

Œ. ‘. ¨ç¥£ªã¥¢

‚¢®¤¨âáï ª« áá ¨­â¥£à «ì­ëå ®¯¥à â®à®¢, ï¤à  ª®â®àëå ¯®à®¦¤¥­ë ®¯¥à â®à ¬¨ ¢§¢¥è¥­­®£®
ᤢ¨£ .

à¨¢®¤ïâáï ãá«®¢¨ï ¨å ®£à ­¨ç¥­­®á⨠¨ ॣã«ïà­®á⨠(¢ â¥à¬¨­ å ï¤à ) ¢ ¯à®áâ-

à ­áâ¢ å ‹¥¡¥£ ,   â ª¦¥ à¥è ¥âáï ¢®¯à®á ® á¢ï§¨ ¬¥¦¤ã â࠭ᯮ­¨à®¢ ­­ë¬ ¨ ᮯà殮­­ë¬
®¯¥à â®à ¬¨.

ˆ§¢¥áâ­®, çâ® ï¤à® ¨­â¥£à «ì­®£® ®¯¥à â®à  ᢥà⪨ ¯®à®¦¤¥­® ®¯¥à â®à®¬ ᤢ¨£  [5, 7]. à¥¤áâ ¢«ï¥â ¨­â¥à¥á á«ãç ©, ª®£¤  ï¤à® ¯®à®¦¤¥­® ®¯¥à â®à®¬ ¢§¢¥è¥­­®£® ¨«¨ ®¡®¡é¥­­®£® ᤢ¨£ .
’ ª¨¥ ®¯¥à â®àë ¢®§­¨ª îâ ¢ ⥮ਨ ¢ë஦¤ îé¨åáï í««¨¯â¨ç¥áª¨å ãà ¢­¥­¨©, £¤¥ ¢ë஦¤¥­¨¥
¯à®¨á室¨â ¯® ­®à¬ «¨ ª £à ­¨æ¥ ¨ ¬®¦¥â ­®á¨âì ¤®áâ â®ç­® ®¡é¨© (­¥á⥯¥­­®©) å à ªâ¥à [4]. ‚¥á®¢ ï äã­ªæ¨ï

 = (t) ¢ í⮬ á«ãç ¥ ¯®«ãç ¥âáï ¡®«¥¥ ®¡é¥£® ¢¨¤  ¨ ®¡« ¤ ¥â ª®­¥ç­®© £« ¤ª®áâìî

¢¯«®âì ¤® ¬­®£®®¡à §¨ï ý¢ë஦¤¥­¨ïþ. à¨ í⮬ ®­  ¤®áâ â®ç­® ¡ëáâà® ®¡à é ¥âáï ¢ ­ã«ì ­  í⮬
¬­®£®®¡à §¨¨.

L+2 = L2 (R+ ) ¨ L2 = L2 (R) | «¥¡¥£®¢ë ¯à®áâà ­á⢠ ¨§¬¥+ ¨ ¢¥é¥á⢥­­®© ¯àאַ© R, á㬬¨à㥬ëå á® á⥯¥­ìî
ਬëå ä㭪権 ­  ¯®«®¦¨â¥«ì­®© ¯®«ã®á¨ R
+
+ q
¤¢  á ­®à¬ ¬¨ k
k ¨ k
k ᮮ⢥âá⢥­­®; Lp ( ) | ¢¥á®¢®¥ ¯à®áâà ­á⢮ ‹¥¡¥£  á ­®à¬®©
‚¢¥¤¥¬ á«¥¤ãî騥 ®¡®§­ ç¥­¨ï

kf k+p;q = kq=2
f k+; q 2 R ; p > 1;
£¤¥  = (t) | ¢¥á®¢ ï äã­ªæ¨ï, 㤮¢«¥â¢®àïîé ï á«¥¤ãî騬 ãá«®¢¨ï¬:  2 C (0; 1);
0 6 (t) 6 1; (t) = 1 ¯à¨ t > d (0 < d | 䨪á¨à®¢ ­­®¥ ç¨á«®); t!
lim+0 (t) = 0;

Rd

lim ,1 ( )d < 1. ® ä㭪樨  ¯®áâந¬ äã­ªæ¨î
t!+0
t

x = '(t) =

Zd

,1 ( )d : (0; 1) ! (,1; 1):

t
Ž¡®§­ ç¨¬ ç¥à¥§
⮦¤¥á⢮¬

t = (x) äã­ªæ¨î, ®¡à â­ãî ª x = '(t),   ç¥à¥§
(t; s) äã­ªæ¨î, ®¯à¥¤¥«ï¥¬ãî
Zd


¤«ï ¢á¥å

t 2 R + ¨ s 2 R+ .

c 2002 ¨ç¥£ªã¥¢ Œ. ‘.

,1 ( )d =

Zd
t

,1 ( )d ,

Zd
s

,1 ( )d

(1)

2{18

Œ. ‘. ¨ç¥£ªã¥¢


g (x ); x 2 R

‘ ¯®¬®éìî ä㭪樨

f (t); t 2 R+

, ¨

®¯à¥¤¥«ïîâáï ®¯¥à â®àë

G;2

¨

G;2

[4],

§ ¤ ­­ë¥ ­  äã­ªæ¨ïå



Gt;!2x [f ] = G;2 [f ](x) =  2 (t)f (t)t= (x) ;

x!t [g ] = G
,1
G;
;,2 [g ](t) =  2 (t)g (x)x='(t) :
,2
q 2 [2; 1)
L+q
1

Ž¯¥à â®à

G;,2

, á«¥¤ãî騬¨ ä®à¬ã« ¬¨:

¤«ï ¢á¥å

ï¥âáï ®£à ­¨ç¥­­ë¬ ¢ ¯à®áâà ­á⢥

, ¯à¨ç¥¬ ¯à¨

q=2

|

¨§®¬¥âà¨ç¥áª¨¬ [2], â. ¥. ¨¬¥îâ ¬¥áâ® à ¢¥­á⢠

kG; [f ]k = kf k ; kG;, [g]k = kgk:
Ž¯à¥¤¥«¥­¨¥ 1 (á¬. [1, 6]).
Ž¯¥à â®à®¬ ¢§¢¥è¥­­®£® ᤢ¨£  ­ §ë¢ îâ ®¯¥à â®à, ¯à¥¤áâ ¢¨¬ë© ¢ ¢¨¤¥ Bf (x) = a(x)
f ( (x)), £¤¥
: X ! Y § ¤ ­­®¥ ®â®¡à ¦¥­¨¥,
a : X ! L(E; F ) | ­¥ª®â®à ï ®¯¥à â®à®§­ ç­ ï äã­ªæ¨ï, ¤¥©áâ¢ãîé ï ¨§ X ¢ ¯à®áâ+

2

2

+

E ¢ ¡ ­ å®¢®
B ¤¥©áâ¢ã¥â ¨§ ¯à®áâà ­á⢠ ä㭪権 ­  Y á® §­ ç¥­¨ï¬¨
¢ E ¢ ¯à®áâà ­á⢮ ä㭪権 ­  X á® §­ ç¥­¨ï¬¨ F . ‚ ç áâ­®áâ¨, ®¯¥à â®àë G;2 ¨
G;,2 ïîâáï ®¯¥à â®à ¬¨ ¢§¢¥è¥­­®£® ᤢ¨£ .
‚¢¥¤¥¬ ¢ à áᬮâ७¨¥ ®¤­®¯ à ¬¥âà¨ç¥áª®¥ ᥬ¥©á⢮ ®¯¥à â®à®¢ fTs : s 2 R + g,
®¯à¥¤¥«¥­­ëå ­  äã­ªæ¨ïå f (t); t 2 R+ , á«¥¤ãî騬 ®¡à §®¬
à ­á⢮ ®£à ­¨ç¥­­ëå «¨­¥©­ëå ®¯¥à â®à®¢ ¨§ ¡ ­ å®¢  ¯à®áâà ­á⢠

¯à®áâà ­á⢮

F.

Ž¯¥à â®à

Ts f (t) =



(
(t; s))
(t)(s)

1

2

f (
(t; s)):

(2)

Ž­® ®¡à §ã¥â ᥬ¥©á⢮ ®¯¥à â®à®¢ ¢§¢¥è¥­­®£® ᤢ¨£ , ¯à¨ç¥¬ ¯à¨

Ts = I

s=d

®¯¥à â®à

| ⮦¤¥á⢥­­ë© ®¯¥à â®à.

G;2 ¨ ä㭪樨
¨¬¥¥¬ ä®à¬ã«ã


Gt!x Gs!y T s f (t) = G;2 [f ](x , y);

ˆ§ ᢮©á⢠¯à¥®¡à §®¢ ­¨ï

;2

â. ¥. ®¯¥à â®à

;2

Ts ¯à¥¤áâ ¢«ï¥â ᮡ®©  ­ «®£ ®¯¥à â®à  ý®¡ëç­®£®þ ᤢ¨£  ­  R, ¯à¨ç¥¬

á¯à ¢¥¤«¨¢  á«¥¤ãîé ï

f; g 2 L+2 , â® ¨¬¥¥â ¬¥áâ® à ¢¥­á⢮
Z1
Z1
Ts f (t)
g(s)ds = f (s)
Ts g(t)ds:

‹¥¬¬  1. …᫨ ä㭪樨

0

ãáâì

0

k(t) | ­¥ª®â®à ï 䨪á¨à®¢ ­­ ï äã­ªæ¨ï ­  ¯®«ã®á¨ R+ .

Z1
Uk f (t) = Ts k(t)
f (s)ds

Ž¯¥à â®à

(3)

0

¡ã¤¥¬ ­ §ë¢ âì

¨­â¥£à «ì­ë¬ ®¯¥à â®à®¬ ¢§¢¥è¥­­®© ᢥà⪨.

Žâ¬¥â¨¬, çâ® ¯à¨

d < s < t ®¯¥à â®à Uk ᮢ¯ ¤ ¥â á ®¯¥à â®à®¬ ý®¡ëç­®©þ ᢥà⪨. à¨¬¥­ïï ª ®¡¥¨¬
G;2 ¨ ¤¥« ï § ¬¥­ã s = (y) ¢ ¯à ¢®© ç áâ¨, ¯®«ã稬


(4)
Uk f (t) = G;,2 G;2 [k]  G;2 [f ] ;

ç áâï¬ (3) ®¯¥à â®à

£¤¥

 | ®¯¥à â®à ý®¡ëç­®©þ ᢥà⪨ [5].
“á«®¢¨¥ ®£à ­¨ç¥­­®á⨠®¯¥à â®à 

Uk

(¢ â¥à¬¨­ å ï¤à ) ¢ ¯à®áâà ­á⢥

L+2 ¤ ¥â

ˆ­â¥£à «ì­ë¥ ®¯¥à â®àë ¢§¢¥è¥­­®© ᢥà⪨

2{19

k 2 L+1 (, 21 ), â® ¨­â¥£à «ì­ë© ®¯¥à â®à ¢§¢¥è¥­­®©
+
ᢥà⪨ (3) ï¥âáï ®£à ­¨ç¥­­ë¬ ¢ ¯à®áâà ­á⢥ L2 , ¯à¨ç¥¬
kUk jL+2 ! L+2 k 6 kkk+1;, 12 :
’¥®à¥¬  1.

…᫨ äã­ªæ¨ï

C ˆá¯®«ì§ãï à ¢¥­á⢮ (4) ¨ ­¥à ¢¥­á⢮ Œ¨­ª®¢áª®£® ¯®«ã稬
kUk f k+

Z1

G;2 [k](x
= G;2 [Uk f ] =


k

k

,1



y) G;2 [f ](y)dy


,


Z1

G;2 [k](y)
=


,1

6

Z1

,1



G;2 [k ](y )dy

Žâá, á ãç¥â®¬ ®¯à¥¤¥«¥­¨ï ¯à®áâà ­á⢠

G;2

L+1 (, 12 )

¢ë⥪ ¥â á¯à ¢¥¤«¨¢®áâì ­ è¥£® ã⢥ত¥­¨ï.

B


kG;2 [f ]k:
¨ ¨§®¬¥âà¨ç­®á⨠®¯¥à â®à 

Uk ®£à ­¨ç¥­ ¢ ¯à®áâà ­á⢥ L+2 ¨ äã­ªæ¨ï k ­¥®âà¨1
’®£¤  äã­ªæ¨ï k á㬬¨à㥬  á ¢¥á®¬ , 2 , ¯à¨ç¥¬ á¯à ¢¥¤«¨¢® à ¢¥­á⢮
kUk jL+2 ! L+2 k = kkk+1;, 21 :

’¥®à¥¬  2.

æ â¥«ì­ .


G;2[f ](x ,



y)dy


ãáâì ®¯¥à â®à

C  áᬮâਬ ¯à®¨§¢®«ì­ë¥ 䨭¨â­ë¥ ä㭪樨 f 2 L+2 ¨ g 2 L+2.

«¥¬¬¥ 1 ¨¬¥¥¬

Z1

,




Uk f (t)g(t)dt =

0

Z1  Z1

0

0

‘®£« á­®



k(s)
Ts f (t)ds g(t)dt:

Žâá ¢ ᨫã â¥®à¥¬ë ”ã¡¨­¨ ¯®«ãç ¥¬

Z1

,




Uk f (t)g(t)dt =

0

Z1

0

 Z1

k(s)

0



Ts f (t)
g(t)dt ds:

ˆá¯®«ì§ãï ­¥à ¢¥­á⢮ ƒ
¥«ì¤¥à  ¤«ï ¯à ¢®© ç á⨠ࠢ¥­á⢠ ¯®«ã稬 ®æ¥­ªã


Z1
 Z1

 k (s)
Ts f (t)



0

0


g(t)dt






ds


6 kUk jL+2 ! L+2k
kf k+
kgk+ :

¥à¥¯¨è¥¬ ­¥à ¢¥­á⢮ (5) á ãç¥â®¬ ä®à¬ã«ë (4) ¢ ¢¨¤¥


 Z1
 Z1


G;2 [f ](x
G;2 [k](y)



,1

,1

, t)
G;2[g](x)dx






dy


(5)

2{20

Œ. ‘. ¨ç¥£ªã¥¢

6 kUk jL+2 ! L+2k
kG;2 [f ]k
kG;2[g]k:
‡ ä¨ªá¨à㥬 ç¨á«® r 2 (0; 1) ¨ à áᬮâਬ ¨­â¥à¢ « Sr = (,r; r). ®«®¦¨¬
 (mes S ), 12 ¯à¨ x 2 S ;
r
r
G;2 [g](x) = G;2 [f ](x) =
0
¯à¨ x 2= Sr ;
£¤¥ mes Sr | ¬¥à  ‹¥¡¥£  ¨­â¥à¢ «  Sr . Ÿá­®, çâ® G;2 [f ] 2 L+2 ¨ G;2 [g] 2 L+2 , ¯à¨ç¥¬
kG;2 [f ]k = kG;2 [g]k = 1.
Ž¡®§­ ç¨¬ ç¥à¥§ r (y) äã­ªæ¨î, ®¯à¥¤¥«ï¥¬ãî à ¢¥­á⢮¬
r (y) =

Z1
,1

G;2 [f ](x , y)
G;2 [g](x)dx =

mes (Sr \ Sr;y ) ;
mes Sr

£¤¥ Sr;y = fx 2 R : jx , yj < rg. ”ã­ªæ¨ï r (y) ®¡« ¤ ¥â á«¥¤ãî騬¨ ᢮©á⢠¬¨:
0 6 r (y) 6 1;
‘«¥¤®¢ â¥«ì­®,

Z1

lim  (y) = 1:
r!1 r

G;p [k](y)
r (y)dy 6 kUk jL+2 ! L+2 k:

,1
¥à¥å®¤ï ª ¯à¥¤¥«ã ¯à¨ r ! 1 ¨ ¨á¯®«ì§ãï ⥮६㠔 âã, ¯®«ã稬

kkk+1;, 21 6 kU jL+2 ! L+2k;
  á ¤à㣮© áâ®à®­ë, ᮣ« á­® ⥮६¥ 1, ¨¬¥¥¬

kUk jL+2 ! L+2 k 6 kkk+1;, 21 :
ˆâ ª, á¯à ¢¥¤«¨¢®áâì âॡ㥬®£® à ¢¥­á⢠ ãáâ ­®¢«¥­ . B
Ž¯à¥¤¥«¥­¨¥ 2 (á¬. [5]). Ž£à ­¨ç¥­­ë© ¨­â¥£à «ì­ë© ®¯¥à â®à
V f (s) =

Z




k(s; t)f (t)dt

¤¥©áâ¢ãî騩 ¨§ Lp (
) ¢ Lq (
) ­ §ë¢ ¥âáï ॣã«ïà­ë¬, ¥á«¨ ¨­â¥£à «ì­ë© ®¯¥à â®à

Z

jV jf (s) = jk(s; t)jf (t)dt



â ª¦¥ ¤¥©áâ¢ã¥â ¨§ Lp (
) ¢ Lq (
) ¨ ®£à ­¨ç¥­.
ˆ§¢¥áâ­®, çâ® ­¥ ª ¦¤ë© ¨­â¥£à «ì­ë© ®¯¥à â®à ॣã«ï७ [5].

2{21

ˆ­â¥£à «ì­ë¥ ®¯¥à â®àë ¢§¢¥è¥­­®© ᢥà⪨

’¥®à¥¬  3. „«ï ⮣®, çâ®¡ë ¨­â¥£à «ì­ë© ®¯¥à â®à ¢§¢¥è¥­­®© ᢥà⪨



,
k 2 L+1 , 21 .

ॣã«ïà­ë¬, ­¥®¡å®¤¨¬® ¨ ¤®áâ â®ç­®, ç⮡ë

C  ¥ ® ¡ å ® ¤ ¨ ¬ ® á â ì ­¥¯®á।á⢥­­® á«¥¤ã¥â ¨§ ⥮६ë 2.

(3)

„ ® á â   â ® ç ­ ® á â ì. „«ï ®¯¥à â®à  (3) á ï¤à®¬ ᮤ¥à¦ é¨¬ äã­ªæ¨î
ᨫã ⥮६ë 1 ¨¬¥¥¬



Ujkj f
+

Z1



L+1 , 21 .

k

jkj, ¢

6 , 12 (t)jk(t)jdt

f
+ =
k
+1;, 21

f
+:






0

‡¤¥áì ¬ë ¢®á¯®«ì§®¢ «¨áì ­¥®âà¨æ â¥«ì­®áâìî ä㭪権

, ãáâì

¡ë«

 = (t). B

| 䨪á¨à®¢ ­­ ï ¨§¬¥à¨¬ ï äã­ªæ¨ï, ¯à¨­ ¤«¥¦ é ï ¯à®áâà ­áâ¢ã

 áᬮâਬ ®¯¥à â®à

U # f (s) =
k

ª®â®àë© ¡ã¤¥¬ ­ §ë¢ âì

Z1

0

Ts k(t)
f (t)dt;

â࠭ᯮ­¨à®¢ ­­ë¬

‘¢®©á⢠ â࠭ᯮ­¨à®¢ ­­®£® ®¯¥à â®à 

(6)

¯® ®â­®è¥­¨î ª ®¯¥à â®àã (3).

Uk#

å à ªâ¥à¨§ã¥â

,
Uk# ¤¥©áâ¢ã¥â ¢
’¥®à¥¬  4. ãáâì k 2 L1 
+
L2 , ᮢ¯ ¤ ¥â á ᮯà殮­­ë¬ Uk ª ®¯¥à â®àã (3) ¨ ॣã«ï७.
C à¨¬¥­ïï ª ®¡¥¨¬ ç áâï¬ (6) ®¯¥à â®à Gs;!2y , ¯®«ã稬
Z1
 # 


s
!
y
G;2 Uk f = Gs;!2y Ts k(t)
f (t)dt
0
Z1
G;2 [k](x , y)
G;2 [f ](x)dx:
=
,1
1

2 . ’®£¤  ®¯¥à â®à

+,

¯à®áâà ­á⢥

Žâá ¯à¨ ¯®¬®é¨ ­¥à ¢¥­á⢠ Œ¨­ª®¢áª®£® ¯®«ã稬

#
+
U f

+
2 6 kkk1;, 12 kf k :
# +
+
 +
+
+ 1.
’ ª¨¬ ®¡à §®¬, Uk : L2 ! L2 ¨
U L2 ! L2
6 kk k
1;, 2
#
+
„®ª ¦¥¬, çâ® Uk = Uk . „«ï ®£à ­¨ç¥­­ëå ä㭪権 f 2 L2
k

⥮६¥ ”㡨­¨ ¯®«ãç ¥¬ à ¢¥­á⢮

Z1  Z1

0

0



Ts k(t)
f (t)dt g(s)ds =

ª®â®à®¥ ¯¥à¥¯¨è¥¬ ¢ ¢¨¤¥

Z1  Z1

0

0



¨

g

2 L+2 ᮣ« á­®

Ts k(t)
g(s)ds f (t)dt;

hUk g; f i+ = hg; Uk# f i+;

2{22

Œ. ‘. ¨ç¥£ªã¥¢

h

i | ᪠«ïà­®¥ ¯à®¨§¢¥¤¥­¨¥ ¢ +2.

£¤¥

;

L

‘ ¤à㣮© áâ®à®­ë,

‘«¥¤®¢ â¥«ì­®, ¤«ï «î¡ëå ®£à ­¨ç¥­­ëå ä㭪権

h

g; (U

¢

L

# , U  )f i+ = 0.

f

¨

g

h

Uk g; f

i+ = h

g; U



kf

i+ .

á¯à ¢¥¤«¨¢® à ¢¥­á⢮

Žâá, á ãç¥â®¬ ¯«®â­®á⨠¬­®¦¥á⢠ ®£à ­¨ç¥­­ëå ä㭪権

k
+ k§ ª«îç ¥¬, çâ® U # = U  .
2
k
k

’ ª ª ª

Uk

| ॣã«ïà­ë© ®¯¥à â®à,   ᮯà殮­­ë©

ª ॣã«ïà­®¬ã ®¯¥à â®àã ®¯¥à â®à ॣã«ï७ (á¬. [5]), â®

 ॣã«ï७.

Uk

B

‹¨â¥à âãà 
1.

€­â®­¥¢¨ç €. .

“á«®¢¨¥ ®£à ­¨ç¥­­®á⨠¨ ­®à¬  ®¯¥à â®à  ¢­ãâ७­¥© á㯥௮§¨æ¨¨ ¢ ¯à®áâ-

à ­á⢥ ¢¥ªâ®à-ä㭪権 // Œ â. § ¬¥âª¨.|1985.|’. 45, ü 1.|‘. 3{9.
2.

¨ç¥£ªã¥¢ Œ. ‘.

Ž¡ ®¤­®¬ ª« áᥠ®¯¥à â®à®¢ ®¡®¡é¥­­®£® ¨ ¢§¢¥è¥­­®£® ᤢ¨£  ­  ¯®«ã®á¨ //

„¥¯. ¢ ‚ˆˆ’ˆ, 1994.|1411-‚-94.|27 á.
3.

¨ç¥£ªã¥¢ Œ. ‘.

ˆ­â¥£à «ì­ë¥ ®¯¥à â®àë, ¯®à®¦¤¥­­ë¥ ®¯¥à â®à®¬ ¢§¢¥è¥­­®£® ᤢ¨£  // Œ â.

§ ¬¥âª¨.|1996.|’. 59, ü 3.|‘. 452{454.
4.

ƒ«ã誮 ‚. .,

‘ ¢ç¥­ª® ž. .

‚ë஦¤ î騥áï í««¨¯â¨ç¥áª¨¥ ãà ¢­¥­¨ï ¢ë᮪®£® ¯®àï¤-

ª : ¯à®áâà ­á⢠, ®¯¥à â®àë, £à ­¨ç­ë¥ § ¤ ç¨ // ˆâ®£¨ ­ ãª¨ ¨ â¥å­¨ª¨. Œ â¥¬ â¨ç¥áª¨©
 ­ «¨§.|1985.|’. 23.|‘. 125{218.
5.

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