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Jan Bíma

Publications and source records attributed to Jan Bíma.

3 recordsLinked to original sources
arXiv2025

Lipschitz free $p$-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction

The geometric analysis of non-locally convex quasi-Banach spaces presents rich and nuanced challenges. In this paper, we introduce the Schur $p$-property and the strong Schur $p$-property for $0 < p \leq 1$, providing new tools to deepen the understanding of these spaces, and the Lipschitz free $p$-spaces in particular. Moreover, by developing an adapted version of the compact reduction principle, we prove that Lipschitz free $p$-spaces over discrete metric spaces possess the approximation property, thereby answering positively a question raised by Albiac et al. in arXiv:2005.06555v2.

math.FA↗
arXiv2024

Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces

Given two metric spaces $\mathcal N \subseteq \mathcal M$ in inclusion and $0<p\leq 1$, we wish to determine the smallest constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ such that any Lipschitz map $f: \mathcal N \to Z$ into any $p$-Banach space $Z$ can be extended to a Lipschitz map $f' : \mathcal M \to Z$ satisfying $\operatorname{Lip} f' \leq \mathfrak{t}_p (\mathcal N, \mathcal M)\cdot \operatorname{Lip} f$. In this article, we prove that if $\mathcal N$ has finite Nagata dimension at most $d$ with constant $γ$, then $\mathfrak{t}_p (\mathcal N, \mathcal M) \lesssim_p γ\cdot (d+1)^{1/p -1} \cdot \log (d+2)$ for all $0<p\leq 1$. We show that examples of spaces with finite Nagata dimension include doubling spaces, as well as minor-excluded metric graphs. We also establish that the constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ generally increases as $p$ approaches zero.

math.FA↗