Meromorphic Optimal domain of Integral Operators

TL;DR

This paper introduces and analyzes the meromorphic optimal domain of the generalized Volterra operator $T_g$ acting on Hardy spaces, providing characterizations and insights into its properties and equivalences.

Contribution

It defines the meromorphic optimal domain for $T_g$, explores its properties, and characterizes when different $g$ functions yield the same domain.

Findings

Characterization of the meromorphic optimal domain for $T_g$.

Conditions under which different $g$ functions have identical domains.

Enhanced understanding of the holomorphic optimal domain of $T_g$.

Abstract

For $g\in BMOA$, we introduce the meromorphic optimal domain $(T_g,H^p)$, i.e. the space containing the meromorphic functions that are mapped under the action of the generalized Volterra operator $T_g$ into the Hardy space $H^p$. We investigate its properties and characterize for which $g_1,g_2 \in BMOA$ the corresponding meromorphic optimal domains coincide. This investigation contributes to a more comprehensive understanding of the holomorphic optimal domain of $T_g$ in $H^p$.