Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into $H^\infty$

TL;DR

This paper characterizes when an integral operator with a symbol having non-negative Maclaurin coefficients is bounded or compact from various analytic function spaces to $H^$, based on conditions on the coefficients.

Contribution

It provides a comprehensive description of boundedness and compactness criteria for integral operators induced by such symbols across multiple classical function spaces.

Findings

Characterization of boundedness conditions

Criteria for compactness of the operator

Applicable to Hardy, Dirichlet, Bloch, and BMOA spaces

Abstract

For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(\zeta)g'(\zeta)\,d\zeta$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.