The Range of the Cesàro Operator Acting on

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 < ∞ ).

show abstract

“…This was improved by Essén and Xiao who proved in [22] that C(H ∞ ) ⊂ Q p for 0 < p < ∞. This result has been sharpened in [10].…”

Section: Proof Of Theorem 5 (Ii)mentioning

confidence: 94%

“…The Cesàro operator C acting on distinct subspaces of Hol(D) has been extensively studied in a good number of articles such as [2,10,12,15,23,36,[40][41][42]44]. Let us recall that it is bounded on H p (0 < p < ∞) and on A p α (0 < p < ∞, α > −1).…”

Section: Cesàro-type Operatorsmentioning

confidence: 99%

Operators Induced by Radial Measures Acting on the Dirichlet Space

et al. 2023

View full text Add to dashboard Cite

show abstract

“…By [68], C(H ∞ ) ⊆ Q p for every 0 < p < 1. Under the condition that K is concave, we showed in [69] that C(H ∞ ) ⊆ Q K if and only if log(1 − z) ∈ Q K . Consequently, there exists a weighted function K 1 such that Q K1 is not trivial and C(H ∞ ) Q K1 .…”

Section: Distances From Bloch Functions To Q K Spacesmentioning

confidence: 99%

$${{\cal Q}_K}$$ Spaces: A Brief and Selective Survey

Bao

Wulan

2021

Acta Math Sci

Self Cite

View full text Add to dashboard Cite

show abstract

“…See [7,12,14,20,22,23] for the investigation of the Cesàro operator acting on some analytic function spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Later M. Essén and J. Xiao [14] proved that C(H ∞ ) Q p for 0 < p < 1. Recently, the relation between C(H ∞ ) and a class of Möbius invariant function spaces was considered in [7].…”

Section: Introductionmentioning

confidence: 99%