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 $k\mathrm{t}\mathrm{h}$-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 Hölder spaces. Finally, we propose a way to use the obtained results to define higher-order Sobolev and Hölder spaces on metric measure spaces.