A Characterization of the Hardy Space Associated with the Dunkl Transform
For p ≥ p 0 : = 2 λ / ( 2 λ + 1 ) with λ > 0 , the Hardy space H λ p ( R + 2 ) associated with the Dunkl transform F λ and the Dunkl operator D on the line R , where ( D x f ) ( x ) = f ′ ( x ) + λ x ( f ( x ) - f ( - x ) ) , is the set of functions F = u + i v on the half plane R + 2 = { ( x , y...
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| Vydáno v: | Complex analysis and operator theory Ročník 15; číslo 3 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
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Springer International Publishing
01.04.2021
Springer Nature B.V |
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| ISSN: | 1661-8254, 1661-8262 |
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| Abstract | For
p
≥
p
0
:
=
2
λ
/
(
2
λ
+
1
)
with
λ
>
0
, the Hardy space
H
λ
p
(
R
+
2
)
associated with the Dunkl transform
F
λ
and the Dunkl operator
D
on the line
R
, where
(
D
x
f
)
(
x
)
=
f
′
(
x
)
+
λ
x
(
f
(
x
)
-
f
(
-
x
)
)
, is the set of functions
F
=
u
+
i
v
on the half plane
R
+
2
=
{
(
x
,
y
)
:
x
∈
R
,
y
>
0
}
, satisfying the generalized Cauchy–Riemann equations
D
x
u
-
∂
y
v
=
0
,
∂
y
u
+
D
x
v
=
0
, and
sup
y
>
0
∫
R
|
F
(
x
,
y
)
|
p
|
x
|
2
λ
d
x
<
+
∞
; and the real Hardy space
H
λ
p
(
R
)
on the line
R
is the collection of boundary functions of the real parts of functions
F
∈
H
λ
p
(
R
+
2
)
. In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential
I
λ
α
associated to the Dunkl transform; and as the main result, we prove the equality
D
(
I
λ
1
f
)
=
-
H
λ
f
for
f
∈
H
λ
1
(
R
)
in a weak sense, where
H
λ
is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for
f
∈
H
λ
1
(
R
)
. |
|---|---|
| AbstractList | For
p
≥
p
0
:
=
2
λ
/
(
2
λ
+
1
)
with
λ
>
0
, the Hardy space
H
λ
p
(
R
+
2
)
associated with the Dunkl transform
F
λ
and the Dunkl operator
D
on the line
R
, where
(
D
x
f
)
(
x
)
=
f
′
(
x
)
+
λ
x
(
f
(
x
)
-
f
(
-
x
)
)
, is the set of functions
F
=
u
+
i
v
on the half plane
R
+
2
=
{
(
x
,
y
)
:
x
∈
R
,
y
>
0
}
, satisfying the generalized Cauchy–Riemann equations
D
x
u
-
∂
y
v
=
0
,
∂
y
u
+
D
x
v
=
0
, and
sup
y
>
0
∫
R
|
F
(
x
,
y
)
|
p
|
x
|
2
λ
d
x
<
+
∞
; and the real Hardy space
H
λ
p
(
R
)
on the line
R
is the collection of boundary functions of the real parts of functions
F
∈
H
λ
p
(
R
+
2
)
. In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential
I
λ
α
associated to the Dunkl transform; and as the main result, we prove the equality
D
(
I
λ
1
f
)
=
-
H
λ
f
for
f
∈
H
λ
1
(
R
)
in a weak sense, where
H
λ
is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for
f
∈
H
λ
1
(
R
)
. For p≥p0:=2λ/(2λ+1) with λ>0, the Hardy space Hλp(R+2) associated with the Dunkl transform Fλ and the Dunkl operator D on the line R, where (Dxf)(x)=f′(x)+λx(f(x)-f(-x)), is the set of functions F=u+iv on the half plane R+2={(x,y):x∈R,y>0}, satisfying the generalized Cauchy–Riemann equations Dxu-∂yv=0, ∂yu+Dxv=0, and supy>0∫R|F(x,y)|p|x|2λdx<+∞; and the real Hardy space Hλp(R) on the line R is the collection of boundary functions of the real parts of functions F∈Hλp(R+2). In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential Iλα associated to the Dunkl transform; and as the main result, we prove the equality D(Iλ1f)=-Hλf for f∈Hλ1(R) in a weak sense, where Hλ is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for f∈Hλ1(R). |
| ArticleNumber | 57 |
| Author | Li, Zhongkai Liao, Jianquan Wei, Haihua |
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| Cites_doi | 10.1090/S0002-9947-1965-0199636-9 10.1007/BF02546524 10.1360/03ys0252 10.1007/s00365-013-9179-1 10.1016/j.jfa.2013.05.024 10.1524/anly.1996.16.1.27 10.1016/j.cam.2005.02.022 10.1006/jath.1996.0061 10.1007/BF01244305 10.1090/S0002-9947-1989-0951883-8 |
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| Issue | 3 |
| Keywords | Dunkl transform 44A15 Hilbert transform 30D55 Weak Dunkl derivative 42A50 30G20 Riesz potential Hardy space |
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| References | Li (CR5) 1996; 16 Thangavelu, Xu (CR13) 2007; 199 Li, Liao (CR6) 2013; 37 Stefanov (CR10) 2002; 44 Rösler, Heyer, Mukherjea (CR9) 1995 CR3 Stein (CR11) 1970 Stein, Weiss (CR12) 1960; 103 Muckenhoupt, Stein (CR8) 1965; 118 Li, Liao (CR7) 2013; 265 Dunkl (CR2) 1989; 311 de Jeu (CR1) 1993; 113 Wang (CR14) 2005; 48 Li (CR4) 1996; 86 Z-K Li (1107_CR5) 1996; 16 Z-K Li (1107_CR6) 2013; 37 MFE de Jeu (1107_CR1) 1993; 113 Z-K Li (1107_CR4) 1996; 86 CF Dunkl (1107_CR2) 1989; 311 1107_CR3 A Stefanov (1107_CR10) 2002; 44 B Muckenhoupt (1107_CR8) 1965; 118 S Thangavelu (1107_CR13) 2007; 199 EM Stein (1107_CR12) 1960; 103 M Rösler (1107_CR9) 1995 SL Wang (1107_CR14) 2005; 48 EM Stein (1107_CR11) 1970 Z-K Li (1107_CR7) 2013; 265 |
| References_xml | – volume: 118 start-page: 17 year: 1965 end-page: 92 ident: CR8 article-title: Classical expansions and their relation to conjugate harmonic functions publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1965-0199636-9 – start-page: 292 year: 1995 end-page: 304 ident: CR9 article-title: Bessel-type signed hypergroups on publication-title: Probability Measures on Groups and Related Structures XI – volume: 103 start-page: 26 year: 1960 end-page: 62 ident: CR12 article-title: On the theory of harmonic functions of several variables, I. The theory of spaces publication-title: Acta Math. doi: 10.1007/BF02546524 – volume: 48 start-page: 448 issue: 4 year: 2005 end-page: 455 ident: CR14 article-title: A note on characterization of Hardy space publication-title: Sci. China Ser. A doi: 10.1360/03ys0252 – volume: 37 start-page: 233 year: 2013 end-page: 281 ident: CR6 article-title: Harmonic analysis associated with the one-dimensional Dunkl transform publication-title: Constr. Approx. doi: 10.1007/s00365-013-9179-1 – year: 1970 ident: CR11 publication-title: Singular Integrals and Differentiability Properties of Functions – volume: 265 start-page: 687 year: 2013 end-page: 742 ident: CR7 article-title: Hardy spaces for Dunkl-Gegenbauer expansions publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2013.05.024 – ident: CR3 – volume: 16 start-page: 27 year: 1996 end-page: 49 ident: CR5 article-title: Hardy spaces for Jacobi expansions publication-title: Analysis doi: 10.1524/anly.1996.16.1.27 – volume: 199 start-page: 181 year: 2007 end-page: 195 ident: CR13 article-title: Riesz transform and Riesz potentials for Dunkl transform publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2005.02.022 – volume: 86 start-page: 179 year: 1996 end-page: 196 ident: CR4 article-title: Conjugate Jacobi series and conjugate functions publication-title: J. Approx. Theory doi: 10.1006/jath.1996.0061 – volume: 113 start-page: 147 year: 1993 end-page: 162 ident: CR1 article-title: The Dunkl transform publication-title: Invent. Math. doi: 10.1007/BF01244305 – volume: 44 start-page: 574 year: 2002 end-page: 592 ident: CR10 article-title: Characterization of and applications to sigular integrals publication-title: Illinois J. Math. – volume: 311 start-page: 167 year: 1989 end-page: 183 ident: CR2 article-title: Differential-difference operators associated to reflection groups publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1989-0951883-8 – volume: 16 start-page: 27 year: 1996 ident: 1107_CR5 publication-title: Analysis doi: 10.1524/anly.1996.16.1.27 – volume: 37 start-page: 233 year: 2013 ident: 1107_CR6 publication-title: Constr. Approx. doi: 10.1007/s00365-013-9179-1 – volume: 311 start-page: 167 year: 1989 ident: 1107_CR2 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1989-0951883-8 – volume: 265 start-page: 687 year: 2013 ident: 1107_CR7 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2013.05.024 – volume: 86 start-page: 179 year: 1996 ident: 1107_CR4 publication-title: J. Approx. Theory doi: 10.1006/jath.1996.0061 – start-page: 292 volume-title: Probability Measures on Groups and Related Structures XI year: 1995 ident: 1107_CR9 – volume: 48 start-page: 448 issue: 4 year: 2005 ident: 1107_CR14 publication-title: Sci. China Ser. A doi: 10.1360/03ys0252 – volume: 44 start-page: 574 year: 2002 ident: 1107_CR10 publication-title: Illinois J. Math. – volume: 118 start-page: 17 year: 1965 ident: 1107_CR8 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1965-0199636-9 – volume: 199 start-page: 181 year: 2007 ident: 1107_CR13 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2005.02.022 – volume-title: Singular Integrals and Differentiability Properties of Functions year: 1970 ident: 1107_CR11 – ident: 1107_CR3 – volume: 113 start-page: 147 year: 1993 ident: 1107_CR1 publication-title: Invent. Math. doi: 10.1007/BF01244305 – volume: 103 start-page: 26 year: 1960 ident: 1107_CR12 publication-title: Acta Math. doi: 10.1007/BF02546524 |
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| Snippet | For
p
≥
p
0
:
=
2
λ
/
(
2
λ
+
1
)
with
λ
>
0
, the Hardy space
H
λ
p
(
R
+
2
)
associated with the Dunkl transform
F
λ
and the Dunkl operator
D
on the line
R
,... For p≥p0:=2λ/(2λ+1) with λ>0, the Hardy space Hλp(R+2) associated with the Dunkl transform Fλ and the Dunkl operator D on the line R, where... |
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| SubjectTerms | Analysis Hilbert transformation Infinite-dimensional Analysis and Non-commutative Theory Mathematics Mathematics and Statistics Operator Theory |
| Title | A Characterization of the Hardy Space Associated with the Dunkl Transform |
| URI | https://link.springer.com/article/10.1007/s11785-021-01107-5 https://www.proquest.com/docview/2511284079 |
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