On the strongly subdifferentiable points in Lipschitz-free spaces

Abstract

[EN] In this paper, we present some sufficient conditions on a metric space M for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz- free space F(M) over M. Our main result reads as follows: if (M , d) is a metric space and gamma > 0, then there exists a (not necessarily equivalent) metric d(gamma) in M such that every finitely supported element in F(M, d(gamma)) is an SSD point. As an application of the main result, it follows that if M is uniformly discrete and epsilon > 0 is given, there exists a metric space N and a (1 + e)-bi-Lipschitz map Phi : M -> N such that the set of all SSD points in F(N) is dense.Christian Cobollo was supported by The grants PID2021-122126NB-C33, PID2019-105011GB- I00 and PID2022-139449NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU, Generalitat Valenciana (through Project PROMETEU/2021/070 and the predoctoral contract CIACIF/2021/378). Sheldon Dantas was supported by The Spanish AEI Project PID2019-106529GB-I00/AEI/10.13039/501100011033, Generalitat Valenciana project CIGE/2022/97 and The grant PID2021-122126NB-C33 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU. Petr Hajek was supported by GA23-04776 S and SGS24/052/OHK3/1T/13 of CTU in Prague. Mingu Jung was supported by June E Huh Center for Mathematical Challenges (HP086601) at Korea Institute for Advanced Study.Cobollo-Gómez, C.; Sheldon Dantas; Hájek, P.; Jung, M. (2025). On the strongly subdifferentiable points in Lipschitz-free spaces. Banach Journal of Mathematical Analysis. 19(1). https://doi.org/10.1007/s43037-024-00389-z19