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A Cesàro-like operator from a class of analytic function spaces to analytic Besov spaces
Published online by Cambridge University Press: 07 April 2025
Abstract
Let 
$\mu $ be a finite positive Borel measure on 
$[0,1)$ and 
$f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} \in H(\mathbb {D})$. For 
$0<\alpha <\infty $, the generalized Cesàro-like operator 
$\mathcal {C}_{\mu ,\alpha }$ is defined by 
$$ \begin{align*}\mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, \end{align*} $$
where, for 
$n\geq 0$, 
$\mu _n$ denotes the nth moment of the measure 
$\mu $, that is, 
$\mu _n=\int _{0}^{1} t^{n}d\mu (t)$.
For 
$s>1$, let X be a Banach subspace of 
$H(\mathbb {D})$ with 
$\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$. In this article, for 
$1\leq p <\infty $, we characterize the measure 
$\mu $ for which 
$\mathcal {C}_{\mu ,\alpha }$ is bounded (resp. compact) from X into the analytic Besov space 
$B_{p}$.
MSC classification
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
The author was supported by the Scientific Research Fund of Hunan Provincial Education Department (No. 24C0226).
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