arXiv subjects

Kacper Kurowski

Publications and source records attributed to Kacper Kurowski.

2 recordsLinked to original sources

arXiv2026

Multipoint Characterization of Higher-Order Sobolev Spaces

In the paper, we prove a rather general characterization of higher-order Sobolev spaces. We show that the $k\mathrm{th}$-order regularity, where $k \in \mathbb{N}$, is captured via inequalities involving $2^k$-tuples of points. In fact, in full generality, the obtained results characterize higher-order Sobolev spaces based on Banach function spaces. Moreover, we show an analogous characterization of higher-order H\"{o}lder spaces. Finally, we propose a way to use the obtained results to define higher-order Sobolev and H\"{o}lder spaces on metric measure spaces.

math.FA↗

arXiv2024

New Characterizations of First Order Sobolev Spaces

We provide new characterizations of Sobolev spaces that are true under some mild conditions. We study modified first order Sobolev spaces on metric measure spaces: $\mathrm{TC}$-Newtonian space, $\hat{\mathrm{TC}}$-Newtonian space, and Gigli-like space. We prove that if the measure is Borel regular and $\sigma$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent to the Haj{\l}asz-Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Haj{\l}asz-Sobolev space.

math.FA↗