In:
Results in Mathematics, Springer Science and Business Media LLC, Vol. 78, No. 3 ( 2023-06)
Abstract:
Let $${\mathbb {D}}$$ D be the unit disc in the complex plane. Given a positive finite Borel measure $$\mu $$ μ on the radius [0, 1), we let $$\mu _n$$ μ n denote the n -th moment of $$\mu $$ μ and we deal with the action on spaces of analytic functions in $${\mathbb {D}}$$ D of the operator of Hibert-type $${\mathcal {H}}_\mu $$ H μ and the operator of Cesàro-type $${\mathcal {C}}_\mu $$ C μ which are defined as follows: If f is holomorphic in $${\mathbb {D}}$$ D , $$f(z)=\sum _{n=0}^\infty a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n ( $$z\in {\mathbb {D}})$$ z ∈ D ) , then $${\mathcal {H}}_\mu (f)$$ H μ ( f ) is formally defined by $${\mathcal {H}}_\mu (f)(z) = \sum _{n=0}^\infty \left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) z^n$$ H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n + k a k z n ( $$z\in {\mathbb {D}}$$ z ∈ D ) and $${\mathcal {C}}_\mu (f)$$ C μ ( f ) is defined by $$\mathcal C_\mu (f)(z) = \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n$$ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n ( $$z\in {\mathbb {D}}$$ 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 $${\mathbb {D}}$$ D . In this paper we study the action of the operators $${\mathcal {H}}_\mu $$ H μ and $${\mathcal {C}}_\mu $$ C μ on the Dirichlet space $${\mathcal {D}}$$ D and, more generally, on the analytic Besov spaces $$B^p$$ B p ( $$1\le p 〈 \infty $$ 1 ≤ p 〈 ∞ ).
Type of Medium:
Online Resource
ISSN:
1422-6383
,
1420-9012
DOI:
10.1007/s00025-023-01887-6
Language:
English
Publisher:
Springer Science and Business Media LLC
Publication Date:
2023
detail.hit.zdb_id:
2043519-8
SSG:
17,1
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