The distance function and Lipschitz classes of mappings between metric spaces

TL;DR

This paper explores conditions under which local Lipschitz properties of a distance-based function imply the global Lipschitz property of the original mapping between metric spaces, extending classical results to more general settings.

Contribution

It establishes new criteria linking local and global Lipschitz properties for mappings between metric spaces, including analytic functions on uniform domains.

Findings

Local Lipschitz property of $g(z)$ implies global Lipschitz of $f$ under certain conditions

Characterization of Lipschitz classes for analytic functions on uniform domains

Extension of Hardy-Littlewood theorem to metric space mappings

Abstract

We investigate when the local Lipschitz property of the real-valued function $g(z) = d_Y (f(z),A)$ implies the global Lipschitz property of the mapping $f:X\to Y$ between the metric spaces $(X,d_X)$ and $(Y,d_Y)$. Here, $d_Y(y,A)$ denotes the distance of $y\in Y$ from the non-empty set $\subseteq Y$. As a consequence, we find that an analytic function on a uniform domain of a normed space belongs to the Lipschitz class if and only if its modulus satisfies the same condition; in the case of the unit disk this result is proved by K. Dyakonov. We use the recently established version of a classical theorem by Hardy and Littlewood for mappings between metric spaces. This paper is a continuation of the recent article by the author [14].