Operators Induced by Radial Measures Acting on the Dirichlet Space
Let D be the unit disc in the complex plane. Given a positive finite Borel measure μ on the radius [0, 1), we let μ n denote the n -th moment of μ and we deal with the action on spaces of analytic functions in D of the operator of Hibert-type H μ and the operator of Cesàro-type C μ which are defined...
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| Published in: | Resultate der Mathematik Vol. 78; no. 3; p. 106 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.06.2023
Springer Nature B.V |
| Subjects: | |
| ISSN: | 1422-6383, 1420-9012 |
| Online Access: | Get full text |
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| Summary: | Let
D
be the unit disc in the complex plane. Given a positive finite Borel measure
μ
on the radius [0, 1), we let
μ
n
denote the
n
-th moment of
μ
and we deal with the action on spaces of analytic functions in
D
of the operator of Hibert-type
H
μ
and the operator of Cesàro-type
C
μ
which are defined as follows: If
f
is holomorphic in
D
,
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
(
z
∈
D
)
, then
H
μ
(
f
)
is formally defined by
H
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
∑
k
=
0
∞
μ
n
+
k
a
k
z
n
(
z
∈
D
) and
C
μ
(
f
)
is defined by
C
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
μ
n
∑
k
=
0
n
a
k
z
n
(
z
∈
D
). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in
D
. In this paper we study the action of the operators
H
μ
and
C
μ
on the Dirichlet space
D
and, more generally, on the analytic Besov spaces
B
p
(
1
≤
p
<
∞
). |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-023-01887-6 |