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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| Veröffentlicht in: | Complex analysis and operator theory Jg. 15; H. 3 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham
Springer International Publishing
01.04.2021
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 1661-8254, 1661-8262 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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
)
. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1661-8254 1661-8262 |
| DOI: | 10.1007/s11785-021-01107-5 |