A generalized Hilbert operator acting on conformally invariant spaces

Abstract

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu_{n, k})_{n,k\ge 0}$ with entries $\mu_{n, k}=\mu_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of orden $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_\mu $ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.Comment: 24 page

Mean Lipschitz spaces and a generalized Hilbert operator

“…To prove this implication we need to use the integral operator I µ considered in [7,10,15,16] which is closely related to the operator H µ .…”

A Hankel Matrix Acting on Spaces of Analytic Functions

“…The question of describing the measures µ for which the operator H µ is well defined and bounded on distinct spaces of analytic functions has been studied in a good number of papers (see [8,12,21,23,30,34,38]). Carleson measures play a basic role in these works.…”

“…In the paper [23] the authors have studied the operators H µ acting on certain conformally invariant spaces such as the Bloch space, BMOA, the analytic Besov spaces B p (1 < p < ∞), and the Q s spaces. Let us introduce quickly these spaces.…”

Generalized Hilbert series operators

“…It should be pointed out that there has been a lot of work in recent years on the action of the Hilbert operator and its generalizations in different analytic function spaces. See for example [3], [4], [1], [2], [5].…”