Polylogarithmic Hardy space & its Nevanlinna counting function

TL;DR

This paper establishes bounds on the essential norm of composition operators on the Polylogarithmic Hardy space, linking it to the Nevanlinna counting function and extending classical results from Hardy spaces.

Contribution

It introduces the Nevanlinna counting function for PL2(D;s) and proves the Littlewood-Paley Identity for this space, providing new tools for operator analysis.

Findings

Upper bound of the essential norm of composition operators on PL2(D;s)

Connection between Nevanlinna counting function and operator norms

Extension of Hardy space results to Polylogarithmic Hardy spaces

Abstract

We present the upper bound of the essential norm of the composition operator over the Polylogarithmic Hardy space PL2(D;s).The results involve the Nevanlinna counting function for PL2(D;s). We first prove the Littlewood-Paley Identity for PL2(D;s) which leads to the Nevanlinna counting function for PL2(D;s). With all these results, not only we get the upper bound of the essential norm of the composition operator over PL2(D;s) but also we get an upper bound in terms of the angular derivative and essential norm of composition operator over the Hardy space H2.