Equivalent definition of some weighted Hardy spaces
We present the equivalent definition for spaces of functions analytic in the half-plane ${\mathbb C}_+ = \{z: Re z > 0 \}$ for which $$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\inft...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3241 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509295364276224 |
|---|---|
| author | Dilnyi, V. M. Дільний, В. М. |
| author_facet | Dilnyi, V. M. Дільний, В. М. |
| author_sort | Dilnyi, V. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:57Z |
| description | We present the equivalent definition for spaces of functions analytic in the half-plane ${\mathbb C}_+ = \{z: Re z > 0 \}$ for which
$$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\infty.$$
|
| first_indexed | 2026-03-24T02:38:50Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
V. M. Dil\nyj (Drohob. ped. un-t )
EKVIVALENTNE OZNAÇENNQ DEQKYX
VAHOVYX PROSTORIV HARDI
We present the equivalent definition for spaces of functions analytic in the half-plane C+ = z{ : Re z >
> 0} for which
sup – sin
ϕ
π
ϕ σ ϕ
<
+∞
∫ ( )
< +∞
2
0
f re e dri p p r
.
Pryvedeno πkvyvalentnoe opredelenye prostranstv analytyçeskyx v poluploskosty C+ = z{ :
Re z > 0} funkcyj, dlq kotor¥x
sup – sin
ϕ
π
ϕ σ ϕ
<
+∞
∫ ( )
< +∞
2
0
f re e dri p p r
.
Nexaj H p( )C+ , 1 ≤ p < + ∞, — prostir funkcij, analityçnyx u C+ = z{ : Re z >
> 0} , dlq qkyx
f H p = sup
–
/
π ϕ π
ϕ
2 2
0
1
< <
+∞
∫ ( )
f re dri p
p
< + ∞.
U roboti [1] pokazano, wo cej prostir zbiha[t\sq z prostorom Hardi ˜ ( )H p C+
analityçnyx u C+ funkcij, dlq qkyx
f H p˜ = sup ( )
–
/
x
p
p
f x iy dy
> ∞
+∞
∫ +
0
1
< + ∞,
i normy ⋅ H p ta ⋅ H̃ p [ ekvivalentnymy. U roboti [2] rozhlqnuto prostir
H
p
σ ( )C+ , σ ≥ 0, 1 ≤ p < + ∞, analityçnyx u C+ funkcij, dlq qkyx
f H p
σ
: = sup
–
– sin
/
π ϕ π
ϕ σ ϕ
2 2
0
1
< <
+∞
∫ ( )
f re e dri p pr
p
< + ∞.
Funkci] z c\oho prostoru magt\ majΩe skriz\ (m.4s.) na ∂C+ kutovi hranyçni
znaçennq, qki poznaça[mo çerez f iy( ) i f iy( ) e y–σ ∈ L
p( )R , pryçomu ostannq
norma dorivng[ [3] nastupnij:
f
H p
σ
× : = max
– ;
– sin
/
ϕ π π
ϕ σ ϕ
∈{ }
+∞
∫ ( )
2 2 0
1
f re e dri p pr
p
< + ∞.
Pry doslidΩenni povnoty deqkyx system funkcij u H
p
σ ( )C+ [4, 5] z’qvlq[t\sq
potreba rozhlqdaty prostory, wo vyznaçagt\sq, qk i klasyçni prostory Hardi,
© V. M. DIL|NYJ, 2008
1270 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
EKVIVALENTNE OZNAÇENNQ DEQKYX VAHOVYX PROSTORIV HARDI 1271
çerez intehruvannq po prqmyx, qki paralel\ni koordynatnym osqm. Rozhlqnemo
prostir
˜ ( )H p
σ C+ , σ ≥ 0, 1 ≤ p < + ∞, analityçnyx u C+ funkcij, dlq qkyx
f H p˜
σ
: =
max sup ( ) –
y
p p yf x iy e dx
∈
+∞
+
∫
R 0
σ ;
sup ( )
–
–
/
x
p p y
p
f x iy e dy
> ∞
+∞
∫ +
0
1
σ < + ∞. (1)
Velyçyna ⋅ H̃ p
σ
zadovol\nq[ vsi oznaky normy, zokrema nerivnist\ trykutnyka
u vypadku p = 1 vyplyva[ z nerivnosti max a{ + b ; c + d} ≤ max ;a c{ } +
+ max ;b d{ } , a dlq vypadku p > 1 takoΩ iz nerivnosti Minkovs\koho. Osnovnym
rezul\tatom ci[] statti [ nastupne tverdΩennq.
Teorema. Prostory H p
σ ( )C+ ta
˜ ( )H p
σ C+ , σ ≥ 0, 1 ≤ p < + ∞, zbihagt\sq,
pryçomu normy ⋅ H p
σ
ta ⋅ H̃ p
σ
[ ekvivalentnymy.
Zaznaçymo, wo teorema vtraça[ sylu, qkwo v (1) pid znakom maksymumu vylu-
çyty perßyj element (wo vydno na prykladi funkci] f z( ) ≡ 1) çy druhyj (ce
ma[mo z prykladu f z( ) = e zz– sinσ ). Prote iz zhadanoho rezul\tatu roboty [1]
vyplyva[, wo u vypadku σ = 0 teorema zalyßa[t\sq spravedlyvog, qkwo v (1)
vyluçyty perßyj element pid znakom maksymumu. Dlq dovedennq teoremy
rozhlqnemo prostory E
p C( , )α β[ ], 0 < β – α < 2π, 1 ≤ p < ∞, funkcij, anali-
tyçnyx u kuti C( , )α β = z{ : α < arg z < β} , dlq qkyx
f E p C ( , )α β[ ] : = sup
/
α ϕ β
ϕ
< <
+∞
∫ ( )
0
1
f re dri p
p
< + ∞.
Ci prostory vyvçalysq v [6, 7]. Tam, zokrema, pokazano, wo funkci] z cyx pros-
toriv magt\ m.4s. na ∂ α βC( , ) kutovi hranyçni znaçennq, qki teΩ poznaçatymemo
çerez f i f ∈ L
p ∂ α βC( , )[ ]. Cej prostir [ banaxovym vidnosno vkazano] normy,
qka dorivng[ nastupnij [8]:
f
E p C ( , )α β[ ]
×
: = sup
/
α ϕ β
ϕ
< <
+∞
∫ ( )
0
1
f re dri p
p
.
Rozhlqnemo takoΩ prostir
˜ ( , / )E p C 0 2π[ ], 1 ≤ p < ∞, funkcij, analityçnyx u
C( , / )0 2π , dlq qkyx
f E p˜ : = max sup ( ) ; sup ( )
/
y
p
x
p
p
f x iy dx f x iy dy
>
+∞
>
+∞
∫ ∫+
+
0 0 0 0
1
< + ∞. (2)
Ostannq velyçyna zadovol\nq[ vsi oznaky normy.
Lema 1. Qkwo f ∈ ˜ ( , / )E p C 0 2π[ ], 1 ≤ p < ∞, to f ∈ E
p C( , / )0 2π[ ] i
f E p C ( , / )0 2π[ ] ≤ c f E p1 ˜ , de stala c1 ne zaleΩyt\ vid f.
Dovedennq. Rozhlqnemo spoçatku vypadok p = 1 . Z umovy (2) vyplyva[, wo
funkciq f naleΩyt\ [9] prostoru Smirnova E1
u koΩnomu kvadrati
a = z{ :
0 < Re z < a, 0 < Im z < a} , 0 < a < + ∞. Tomu [9, s. 205] vona ma[ m.4s. na
∂a
kutovi hranyçni znaçennq, qki teΩ poznaça[mo çerez f, i
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1272 V. M. DIL|NYJ
1
2π
∂
i
f t
t z
dz
a
∫ ( )
–
=
f z z
z
a( ), ,
, arg .
∈
< <
0
2
2π π
(3)
Zafiksu[mo dovil\ne z ∈ C 0
2
; π
. Oskil\ky
1
2πi
f t
t z
dt
lk
∫ ( )
–
≤ 1
2πd
f t dt
a lk
∫ ( ) → 0, a → + ∞, k = 1, 2,
de l1 = a{ + i y : 0 < y < a} , l2 = x{ + i a : 0 < x < a} , a da — vidstan\ vid toçky
z do vidpovidno] storony ∂a , to, perexodqçy v (3) do hranyci pry a → + ∞,
ma[mo
1
2
0
2
π
∂ πi
f t
t z
dt
C ( ; )
( )
–∫ =
f z z
z
( ), ; ,
, ; .
∈
∉
C
C
0
2
0 0
2
π
π
(4)
Tomu, poznaçagçy livu çastynu ostann\o] rivnosti çerez ψ( )z , dlq z = x + i y ∈
∈
C 0
2
; π
oderΩu[mo
f z( ) = ψ( )z + ψ(– )z – ψ( )z – ψ(– )z =
–
( )
– –
;
4
4
0
2
2 2 2 2 2 2π
∂ πC
∫ ( ) +
xyt f t
t x y x y
dt ,
zvidky znaxodymo
f z( ) = 2 2
2 20
2
2 2 2 4 2π
ϕ
ϕ ϕ
+∞
∫
+( ) +
v v v
v
r f i d
r r
sin ( )
cos sin
– 2 2
2 20
2
2 2 2 4 2π
ϕ
ϕ ϕ
+∞
∫ ( ) +
ur f u du
u r r
sin ( )
– cos sin
,
z = reiϕ ∈
C 0
2
; π
.
Intehrugçy f z( ) po r ∈ ( ; )0 +∞ i vykorystovugçy teoremu Fubini, otrymu[mo
0
+∞
∫ ( )f re driϕ ≤
2 2
2 20 0
2
2 2 2 4 2π
ϕ
ϕ ϕ
+∞ +∞
∫ ∫
+( ) +
v v
v
v
r f i dr
r r
d
sin ( )
cos sin
+
+
2 2
2 20 0
2
2 2 2 4 2π
ϕ
ϕ ϕ
+∞ +∞
∫ ∫ ( ) +
f u
ur dr
u r r
du( )
sin
– cos sin
.
Vraxovugçy, wo
0
2
2 2 2 4 2
2
2 2
+∞
∫
+( ) +
v
v
r dr
r r
sin
cos sin
ϕ
ϕ ϕ
= π ϕ
2
sin ,
0
2
2 2 2 4 2
2
2 2
+∞
∫ ( ) +
ur dr
u r r
sin
– cos sin
ϕ
ϕ ϕ
= π ϕ
2
cos ,
ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
EKVIVALENTNE OZNAÇENNQ DEQKYX VAHOVYX PROSTORIV HARDI 1273
f E p C( , / )0 2π[ ] ≤ max sin cos
( ; / )
˜
ϕ π
ϕ ϕ
∈
+{ }
0 2
f E p = 2 f E p˜ .
Nexaj teper 1 < p < + ∞. Oskil\ky za umovog (2) funkciq f naleΩyt\ pros-
toru Smirnova E p � E1
u koΩnomu kvadrati
a , to zalyßa[t\sq pravyl\nog
rivnist\ (3). Z toho, wo za nerivnistg H\ol\dera
1
2πi
f t
t z
dt
lk
∫ ( )
–
≤ 1
2
1
1 1
π
l
p
p
l
q
q
k k
f t dt
t z
dt∫ ∫
( )
–
/ /
≤
≤
a
d
f t dt
q
a l
p
p
k
1
1
2
/
/
( )
π ∫
→ 0
pry a → + ∞, de k = 1, 2, a 1/ p + 1/q = 1, vyplyva[ spravedlyvist\ rivnosti (4).
Ale [10]
ϕ1( )z :4=
1
2
0
0
π
π
i
f t
t z
dt E p
+∞
∫ ∈ [ ]( )
–
( ; )C ,
ϕ2( )z :4=
1
2 2 2
0
π
π π
+∞
∫ ∈
f it
it z
dt E p( )
–
– ;C ,
tomu f = ϕ1 – ϕ 2 ∈
E p C 0
2
; π
. Oskil\ky z rezul\tativ [8] ma[mo
f E p C( , / )0 2π[ ] ≤
f
E p C( , / )0 2π[ ]
×
i za lemog Fatu
f
E p C( , / )0 2π[ ]
× ≤ 2 f E p˜ , to lemu
dovedeno.
Zaznaçymo, wo z dovedennq lemy41 vyplyvagt\ ocinky c1 ≤ 2 pry p = 1 i
c1 ≤ 2 pry 1 < p < + ∞.
Lema 2. Qkwo f ∈ E
p C( , / )0 2π[ ], 1 ≤ p < ∞, to f ∈ ˜ ( , / )E p C 0 2π[ ] i f E p˜ ≤
≤ c f E p2 0 2C( , / )π[ ], de stala c2 ne zaleΩyt\ vid f.
Dovedennq provedemo podibno do dovedennq analohiçnyx tverdΩen\ u [1,
11]. PokaΩemo, wo
sup ( )
/
x
p
p
f x iy dy
>
+∞
∫ +
0 0
1
< c f E p3 0 2C( , / )π[ ], (5)
de c3 ne zaleΩyt\ vid f. Pry vidobraΩenni w = z2
pivprqma z{ : Re z = x,
y > 0} , x > 0, perejde u hilku paraboly l x = w{ = u + iv : u = x2 – v
2 24/ x ,
v > }0 . Pry c\omu
0
+∞
∫ +f x iy dyp( ) =
l
p
x
f w
dw
w∫ ( )
2
= 1
2 1
l
p
x
f w dw∫ ( ) ,
de f w1( ) = f w1 2/( ) / w p1 2/
. DovΩyna çastyny l x , qka leΩyt\ u koΩnomu kvad-
rati ∆ ( , )u h0 = w{ = u + iv : u0 < u < u0 + h, 0 < v < }h , h > 0 u0 ∈R , ne perevy-
wu[ 2h. Tomu mira
µx D( ) =
l Dx
dw
∩
∫ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1274 V. M. DIL|NYJ
de D — dovil\nyj kompakt iz C+ : = w{ : In w> 0} , [ mirog Karlesona v C+
,
tobto µx u h∆ ( , )0( ) ≤ c h4 , de c4 ne zaleΩyt\ vid u0, h ta x. Todi [12, c. 70; 1,
s. 78] dlq koΩno] funkci] f ∈ E
p C+[ ] (tobto funkci] z klasu Hardi u C+
)
ma[mo
C+
∫ f dp
x1 µ ≤
c f
E
p
p5 1 [ ]+C
= c f E p5 0 2C( , / )π[ ] < + ∞,
wo j dovodyt\ rivnist\ (5). Nerivnist\
sup ( )
/
y
p
p
f x iy dx
>
+∞
∫ +
0 0
1
< c f E p6 0 2C( , / )π[ ]
dovodyt\sq analohiçno, a poznaçyvßy c2 = max ,c c3 6{ }, otryma[mo tverdΩennq
lemy.
Dovedennq teoremy. Qkwo f ∈ H
p
σ ( )C+ , to f z( ) ei zσ ∈
E p C 0
2
; π
. To-
mu za lemog42 f z( ) ei zσ ∈
˜ ;E p C 0
2
π
. TakoΩ f z( ) e i z– σ
∈ 4
E p C – ;π
2
0
.
Tomu, zastosovugçy lemu42 do funkci] f iz(– ) e z–σ ∈
E p C 0
2
; π
, ma[mo
f iz(– ) e z–σ ∈
˜ ;E p C 0
2
π
. Todi, vraxovugçy analityçnist\ funkci] f u C+ ,
oderΩu[mo f ∈ ˜ ( )H p
σ C+ i f H p˜
σ
≤ c f H p7 σ
, de c7 ne zaleΩyt\ vid f.
Nexaj teper f ∈ ˜ ( )H p
σ C+ . Zastosovugçy do funkci] f z( ) ei zσ
ta f iz(– ) e z–σ
analohiçnym çynom lemu41, otrymu[mo tverdΩennq teoremy.
1. Sedleckyj A. M. ∏kvyvalentnoe opredelenye prostranstv v H p
v poluploskosty y neko-
tor¥e pryloΩenyq // Mat. sb. – 1975. – 96, # 1. – S. 75 – 82.
2. Vynnyckyj B. V. O nulqx funkcyj, analytyçeskyx v poluploskosty, y polnote system πks-
ponent // Ukr. mat. Ωurn. – 1994. – 46, # 5. – S. 484 – 500.
3. Vynnyc\kyj B. V. Pro rozv’qzky odnoridnoho rivnqnnq zhortky v odnomu klasi funkcij,
analityçnyx v pivsmuzi // Mat. stud. – 1997. – 7, # 1. – S. 41 – 52.
4. Vynnyc\kyj B. V., Dil\nyj V. M. Pro neobxidni umovy isnuvannq rozv’qzkiv odnoho rivnqnnq
typu zhortky // Tam Ωe. – 2001. – 16, # 1. – S. 61 – 70.
5. Vynnyckyj B. V., Dyl\n¥j V. N. Ob obobwenyy teorem¥ Berlynha – Laksa // Mat. zametky. –
2006. – 79, # 3. – S. 362 – 368.
6. Hryhorqn Í. F. O bazysnosty nepoln¥x system racyonal\n¥x funkcyj v uhlovoj oblasty
// Yzv. AN ArmSSR. Matematyka. – 1978. – 12, # 5-6. – S. 460 – 487.
7. DΩrbaßqn M. M. Yntehral\n¥e preobrazovanyq y predstavlenyq funkcyj v kompleksnoj
oblasty. – M.: Nauka, 1996. – 672 s.
8. Martirosian V. On a theorem of Djrbashian of the Phragmen – Lindeloff type // Math. Nachr. –
1989. – 144. – S. 21 – 27.
9. Pryvalov Y. Y. Hranyçn¥e svojstva analytyçeskyx funkcyj. – M.; L.: Hostexteoryzdat,
1950. – 3364s.
10. DΩrbaßqn M. M. Bazysnost\ nekotor¥x byortohonal\n¥x system y reßenye kratnoj yn-
terpolqcyonnoj zadaçy v klassax H p
v poluploskosty // Yzv. AN SSSR. Ser. mat. – 1978. –
42, # 6. – S.41322 – 1384.
11. Vynnyc\kyj B. V. Pro nuli deqkyx klasiv funkcij, analityçnyx v pivplowyni // Mat. stud. –
1996. – 6. – S. 67 – 72.
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OderΩano 02.10.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
|
| id | umjimathkievua-article-3241 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:50Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/55/f95b3706c0a8e5b24105b333a25fa055.pdf |
| spelling | umjimathkievua-article-32412020-03-18T19:48:57Z Equivalent definition of some weighted Hardy spaces Еквівалентне означення деяких вагових просторів Гарді Dilnyi, V. M. Дільний, В. М. We present the equivalent definition for spaces of functions analytic in the half-plane ${\mathbb C}_+ = \{z: Re z > 0 \}$ for which $$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\infty.$$ Приведено эквивалентное определение пространств аналитических в полуплоскости ${\mathbb C}_+ = \{z: Re z > 0 \}$ функций, для которых $$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\infty.$$ Institute of Mathematics, NAS of Ukraine 2008-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3241 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 9 (2008); 1270–1274 Український математичний журнал; Том 60 № 9 (2008); 1270–1274 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3241/3228 https://umj.imath.kiev.ua/index.php/umj/article/view/3241/3229 Copyright (c) 2008 Dilnyi V. M. |
| spellingShingle | Dilnyi, V. M. Дільний, В. М. Equivalent definition of some weighted Hardy spaces |
| title | Equivalent definition of some weighted Hardy spaces |
| title_alt | Еквівалентне означення деяких вагових просторів Гарді |
| title_full | Equivalent definition of some weighted Hardy spaces |
| title_fullStr | Equivalent definition of some weighted Hardy spaces |
| title_full_unstemmed | Equivalent definition of some weighted Hardy spaces |
| title_short | Equivalent definition of some weighted Hardy spaces |
| title_sort | equivalent definition of some weighted hardy spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3241 |
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