The distance function and Lipschitz classes of mappings between metric spaces

We investigate when the local Lipschitz property of the real-valued function $$ implies the global Lipschitz property of the mapping $$ between the metric spaces $$ and $$. Here, $$ denotes the distance of $$ from the non-empty set $$. 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 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 [Marković, J. Geom. Anal. 34 (2024), https://doi.org/10.48550/arXiv.2405.11509].