On the boundedness of generalized integration operators on Hardy spaces

TL;DR

This paper characterizes exactly which functions make a generalized integration operator bounded between different Hardy spaces, resolving a previous conjecture and advancing understanding of operator behavior in complex analysis.

Contribution

It provides a complete characterization of symbols for the boundedness of the generalized integration operator between Hardy spaces, confirming a prior conjecture.

Findings

Complete characterization of symbols g for boundedness

Affirmative answer to the conjecture on operator boundedness

Advances understanding of generalized integration operators

Abstract

We study the boundedness and compactness properties of the generalized integration operator $T_{g,a}$ when it acts between distinct Hardy spaces in the unit disc of the complex plane. This operator has been introduced by the first author in connection to a theorem of Cohn about factorization of higher order derivatives of functions in Hardy spaces. We answer in the affirmative a conjecture stated in the same work, therefore giving a complete characterization of the class of symbols $g$ for which the operator is bounded from the Hardy space $H^p$ to $H^q, \, 0<p,q<\infty.$