Operators of Hilbert type acting on some spaces of analytic functions

TL;DR

This paper studies the generalized Hilbert operator $\mathcal{H}_g(f)(z)=\int_{0}^{1} f(t) g'(tz) dt$ on a broad landscape of analytic function spaces. It provides sharp equivalences for when $\mathcal{H}_g$ is bounded or compact between Dirichlet-type spaces $\mathcal{D}^{2}_{\alpha}$ and $\mathcal{D}^{2}_{\beta}$ (for all real $\alpha$ and $\beta$), via dyadic block conditions on the Taylor coefficients of $g$, and describes the operator's range on $H^{\infty}$ in terms of the spaces $X_p$ tied to logarithmic Lipschitz/BMOA-type scales. The paper also analyzes symbols with nonnegative coefficients, establishing exact boundedness criteria between logarithmic Bloch spaces and related function spaces, in terms of partial-sum growth of $\{b_n\}$. Collectively, these results generalize the classical Hilbert operator to a wide class of analytic-function spaces, linking operator boundedness and compactness to precise coefficient-growth conditions of the symbol.

Abstract

Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator $\mathcal{H}_{g}$ is defined by $$\mathcal{H}_{g}(f)(z)=\int_{0}^{1}f(t)g'(tz)dt, \ \ z\in \mathbb{D}, f\in H(\mathbb{D}).$$ In this paper, we study the operator $\mathcal{H}_{g}$ acting on some spaces of analytic functions in $\mathbb{D}$. Specifically, we give a complete characterization of those $g\in H(\mathbb{D})$ for which the operator $\mathcal{H}_{g}$ is bounded (resp. compact) from the Dirichlet space $\mathcal{D}^{2}_α$ to $\mathcal{D}^{2}_β$ for all possible indicators $α,β\in \mathbb{R}$. We also study the action of the operator $\mathcal{H}_{g}$ on the space of bounded analytic functions $H^{\infty}$, which generalizes the known results for the classical Hilbert operator $\mathcal {H}$ acting on $H^{\infty}$. In particular, we consider the boundedness of the operator $\mathcal{H}_{g}$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.

Theorems & Definitions (43)

• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• Theorem 2.6
• proof