Generalized Hilbert operators on weighted Bergman spaces

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 $A_{\text{om}}^p$, where the weight function $\text{om}$ belongs to the class $\mathbb{R}$ 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 $\text{hg}:A_{\text{om}}^p\to A_{\text{om}}^q$ depends on $p$ only, but the situation is completely different in the case $q\mathrm{\ne }p$ in which the inducing weight $\text{om}$ 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 $L_{{\int }_t^1\text{om}(s)ds}^p([0,1))$ (the natural restriction of $A_{\text{om}}^p$ to functions defined on $[0,1)$) to $A_{\text{om}}^p$ if and only if $\text{om}$ satisfies the condition \eqref{Mpconditionaabstract}. On the way to these results decomposition norms for the weighted Bergman space $A_{\text{om}}^p$ are established.