A Cesàro-like operator from a class of analytic function spaces to analytic Besov spaces

Tang, Pengcheng

Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1210-1222

Résumé

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 $B_{p}$.

Détail

DOI : 10.4153/S0008439525000402

Mots-clés : Cesàro operator, Bloch space, Besov space