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arXiv · 2606.00897

Multipoint Characterization of Higher-Order Sobolev Spaces

Kacper Kurowski
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Abstract

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.

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Kacper Kurowski. 2026-05-30. Multipoint Characterization of Higher-Order Sobolev Spaces. https://arxiv.org/abs/2606.00897

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