The distance function and Lipschitz classes of mappings between metric spaces

作者：Marijan Marković · 发表于：Mathematika · 年份：2025 · DOI：10.1112/mtk.70038 · 研究领域：Mathematics

We investigate when the local Lipschitz property of the real‐valued function g(z)=dY(f(z),A)$g(z) = d_Y (f(z),A)$ implies the global Lipschitz property of the mapping f:X→Y$f:X\rightarrow Y$ between the metric spaces (X,dX)$(X,d_X)$ and (Y,dY)$(Y,d_Y)$ . Here, dY(y,A)$d_Y(y,A)$ denotes the distance of y∈Y$y\in Y$ from the non‐empty set ⊆Y$\subseteq Y$ . 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 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].