Generalized integral type Hilbert operator acting on weighted Bloch space

TL;DR

This paper investigates the boundedness and compactness of a generalized integral Hilbert operator on weighted Bloch spaces, extending known results for specific cases and providing comprehensive criteria for various Bloch-type spaces.

Contribution

It introduces a generalized operator acting on weighted Bloch spaces and characterizes its boundedness and compactness, broadening the understanding of integral operators in complex analysis.

Findings

Established criteria for boundedness of the operator between Bloch spaces.

Provided conditions for the compactness of the operator.

Extended results to logarithmic and other Bloch-type spaces.

Abstract

Let $\mu$ be a finite Borel measure on $[0,1)$. In this paper, we consider the generalized integral type Hilbert operator $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1}\frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t)\ \ \ (\alpha>-1).$$ The operator $\mathcal{I}_{\mu_{1}}$ has been extensively studied recently. The aim of this paper is to study the boundedness(resp. compactness) of $\mathcal{I}_{\mu_{\alpha+1}}$ acting from the normal weight Bloch space into another of the same kind. As consequences of our study, we get completely results for the boundedness of $ \mathcal{I}_{\mu_{\alpha+1}}$ acting between Bloch type spaces, logarithmic Bloch spaces among others.