Equivalence among various variable exponent Hardy or Bergman spaces

TL;DR

This paper investigates the conditions under which variable exponent Hardy and Bergman spaces are equivalent, revealing their relation to classical spaces and characterizing Carleson measures, with implications for operator boundedness.

Contribution

It establishes equivalence criteria among variable exponent Hardy and Bergman spaces and links them to classical spaces under specific regularity conditions.

Findings

Variable exponent Hardy spaces relate closely to classical Hardy spaces with log-Hölder continuous exponents.

Carleson measures for variable exponent Hardy spaces are characterized.

An analogue of Littlewood subordination and boundedness of composition operators are proved.

Abstract

We study the question of when two weighted variable exponent Bergman spaces or Hardy spaces are equivalent. As an application, we show that variable exponent Hardy spaces have a close relation to classical Hardy spaces when the exponent is log-H\"{o}lder continuous and has bounded harmonic conjugate (when extended from its boundary values to be harmonic in the disc). We use this to characterize Carleson measures for these variable exponent Hardy spaces. We also prove under certain conditions an analogue of Littlewood subordination and a result on the boundedness of composition operators.