Regularly oscillating mappings between metric spaces and a theorem of Hardy and Littlewood

TL;DR

This paper generalizes the Hardy-Littlewood theorem by establishing a relationship between growth and regular oscillation of mappings between metric spaces, extending classical results to broader contexts.

Contribution

It introduces a general framework for regularly oscillating mappings on metric spaces with weights, broadening the scope of Hardy-Littlewood type theorems.

Findings

Derived a generalized Hardy-Littlewood theorem for metric space mappings.

Extended classical results to mappings on the unit ball of normed spaces.

Established conditions linking growth and oscillation in a broad setting.

Abstract

This paper is motivated by the classical theorem due to Hardy and Littlewood which concerns analytic mappings on the unit disk and relates the growth of the derivative with the H\"{o}lder continuity. We obtain a version of this result in a very general setting -- for regularly oscillating mappings on a metric space equipped with a weight, which is a continuous and positive function, with values in another metric space. As a consequence, we derive the Hardy and Littlewood theorem for analytic mappings on the unit ball of a normed space.