Cesàro-like operators acting on spaces of analytic functions

Abstract

Let $${\mathbb {D}}$$ D be the unit disc in $${\mathbb {C}}$$ C . If $$\mu $$ μ is a finite positive Borel measure on the interval [0, 1) and f is an analytic function 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 ), we define $$\begin{aligned} {\mathcal {C}}_\mu (f)(z)= \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n,\quad z\in {\mathbb {D}}, \end{aligned}$$ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where, for $$n\ge 0$$ n ≥ 0 , $$\mu _n$$ μ n denotes the n-th moment of the measure $$\mu $$ μ , that is, $$\mu _n=\int _{[0, 1)}t^nd\mu (t).$$ μ n = ∫ [ 0 , 1 ) t n d μ ( t ) . In this way, $${\mathcal {C}}_\mu $$ C μ becomes a linear operator defined on the space $${\mathrm{Hol}}({\mathbb {D}})$$ Hol ( D ) of all analytic functions in $${\mathbb {D}}$$ D . We study the action of the operators $${\mathcal {C}}_\mu $$ C μ on distinct spaces of analytic functions in $${\mathbb {D}}$$ D , such as the Hardy spaces $$H^p$$ H p , the weighted Bergman spaces $$A^p_\alpha $$ A α p , BMOA, and the Bloch space $${\mathcal {B}}$$ B .

Carleson measures and the range of a Cesàro-like operator acting on $H^\infty$

“…Note that Q p = B for any p > 1. Theorem 3.1 generalizes Theorem 5 in [16] from the Bloch space B to all Q p spaces. For p = 1, Theorem 3.1 gives an answer to a question raised in [16, p. 20].…”

“…Recently, P. Galanopoulos, D. Girela and N. Merchán [16] introduced a Cesàro-like operator C µ on H(D). For nonnegative integer n, let µ n be the moment of order n of a finite positive Borel measure µ on [0, 1); that is,…”

“…If dµ(t) = dt, then C µ = C. In [16], the authors studied the action of C µ on distinct spaces of analytic functions.…”

“…It is quite natural to study C µ (H ∞ ). In [16] the authors characterized positive Borel measures [16, p. 20], the authors asked whether or not µ being a Carleson measure implies that C µ (H ∞ ) ⊆ BMOA. In this paper, by giving some descriptions of s-Carleson measures on [0, 1), for 0 < p < 2, we show that C µ (H ∞ ) ⊆ Q p if and only if µ is a Carleson measure, which giving an affirmative answer to their question.…”

Pseudo-Carleson measures and generalized Ces`aro-like operators

“…Recently, Galanopoulos et al [12] defined a Cesàro-like operator on H(D). If µ is a finite positive Borel measure on the interval [0, 1), they defined…”