Generalized Hilbert operators on weighted Bergman spaces

TL;DR

This paper investigates the boundedness and compactness of generalized Hilbert operators on weighted Bergman spaces with regular radial weights, revealing how the inducing weight influences operator behavior and connecting it to Muckenhoupt conditions.

Contribution

It characterizes the boundedness and compactness of generalized Hilbert operators on weighted Bergman spaces, highlighting the role of weights and establishing decomposition norms.

Findings

Classical Hilbert operator is bounded under Muckenhoupt condition.

Boundedness depends on whether q=p or q≠p, with weight playing a crucial role.

Decomposition norms for weighted Bergman spaces are established.

Abstract

The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r<1}\bigg(\int_{r}^1(\int_t^1\om(s)ds)^{-\frac{p'}{p}}\,dt\bigg)^\frac{p}{p'} \int_{0}^r(1-t)^{-p}(\int_t^1\om(s)ds)\,dt<\infty. \tag{\dag} {equation} If $q=p$, the condition on $g$ that characterizes the boundedness (or the compactness) of $\hg: A^p_\om\to A^q_\om$ depends on $p$ only, but the situation is completely different in the case $q\ne p$ in which the inducing weight $\om$ plays a crucial role. The results obtained also reveal a natural connection to the Muckenhoupt type condition \eqref{Mpconditionaabstract}. Indeed, it is shown that the classical Hilbert operator (the case $g(z)=\log\frac{1}{1-z}$ of $\H_g$) is bounded from $L^p_{\int_{t}^1\om(s)\,ds}([0,1))$ (the natural restriction of $A^p_\om$ to functions defined on $[0,1)$) to $A^p_\om$ if and only if $\om$ satisfies the condition \eqref{Mpconditionaabstract}. On the way to these results decomposition norms for the weighted Bergman space $A^p_\om$ are established.