Criteria for $C^m$-approximability of functions by solutions of homogeneous second-order elliptic equations on compact subsets of $\mathbb{R}^N$ and related capacities

M. Ya. Mazalov ; P. V. Paramonov ; K. Yu. Fedorovskiy

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 5, pp. 847-917

Résumé

Results of the last 12 years obtained by the authors and their co-authors are discussed. The main achievement of this period of time was establishing Vitushkin-type criteria in terms of capacities for the $C^m$-approximability of functions by solutions of homogeneous elliptic equations of the second order, with constant complex coefficients on compact subsets of $\mathbb R^N$, in all dimensions $N\in\{2,3,\dots\}$ and for all smoothness exponents $m\in[0,2)$. These criteria are stated for individual functions. They yield directly the relevant criteria for classes of functions established previously by Mateu, Orobitg, Netrusov, and Verdera (1996, apart from $m=0$ and $m=1$). Another significant result established during these years was an integro-geometric description of all capacities arising in these criteria in the cases $m=0$ (Mazalov, 2024) and $m=1$ (Tolsa, 2021). In particular, these capacities were shown to be subadditive. Bibliography: 69 titles.

Détail

Keywords: homogeneous second-order elliptic operator $\mathcal L$, fundamental solution, $C^m$-approximation, Vituskin-type localization operator, Lip${}^m$-$\mathcal L$-capacity, $C^m$-$\mathcal L$-capacity, $\mathcal L$-oscillation.
Mots-clés : Hausdorff content