Capacities commensurable with harmonic ones

M. Ya. Mazalov

Sbornik. Mathematics, Tome 215 (2024) no. 2, pp. 250-274

Résumé

Let $\mathcal L$ be a second-order homogeneous elliptic differential operator in $\mathbb R^N$, $N\ge3$, with constant complex coefficients. Removable singularities of $\mathrm L^{\infty}$-bounded solutions of the equation $\mathcal Lf=0$ are described in terms of the capacities $\gamma_{\mathcal L}$, where $\gamma_{\Delta}$ is the classical harmonic capacity from potential theory. It is shown for the corresponding values of $N$ that $\gamma_{\mathcal L}$ and $\gamma_{\Delta}$ are commensurable for all $\mathcal L$. Some ideas due to Tolsa are used in the proof. Various consequences of this commensurability are presented; in particular, criteria for the uniform approximation of functions by solutions of the equation $\mathcal Lf=0$ are stated in terms of harmonic capacities. Bibliography: 19 titles.

Détail

Keywords: homogeneous elliptic equation with complex coefficients, capacity, energy, singular integral.

Bibliographie

[1] P. V. Paramonov and K. Yu. Fedorovskii, “Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations”, Sb. Math., 190:2 (1999), 285–307 | DOI | MR | Zbl

[2] P. V. Paramonov and K. Yu. Fedorovskiy, “Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities”, Sb. Math., 214:4 (2023), 550–566 | DOI | MR

[3] R. Harvey and J. C. Polking, “A notion of capacity which characterizes removable singularities”, Trans. Amer. Math. Soc., 169 (1972), 183–195 | DOI | MR | Zbl

[4] M. Ya. Mazalov, “A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients”, Sb. Math., 211:9 (2020), 1267–1309 | DOI | MR | Zbl

[5] L. Carleson, Selected problems on exceptional sets, Van Nostrand Math. Studies, 13, D. Van Nostrand Co., Inc., Princeton, NJ–Toronto, ON–London, 1967, v+151 pp. | MR | Zbl

[6] M. V. Keldyš, “On the solvability and stability of the Dirichlet problem”, Amer. Math. Soc. Transl. Ser. 2, 51, Amer. Math. Soc., Providence, RI, 1966, 1–73 | DOI | MR | Zbl

[7] M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations”, Sb. Math., 199:1–2 (2008), 13–44 | DOI | MR | Zbl

[8] P. V. Paramonov, “On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in ${\mathbb R}^2$”, Sb. Math., 213:6 (2022), 831–843 | DOI | MR | Zbl

[9] M. Ya. Mazalov, “Uniform approximation of functions by solutions of second order homogeneous strongly elliptic equations on compact sets in ${\mathbb{R}}^2$”, Izv. Math., 85:3 (2021), 421–456 | DOI | MR | Zbl

[10] A. L. Volberg and V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026 | DOI | MR | Zbl

[11] X. Tolsa, “Painlevé's problem and the semiadditivity of analytic capacity”, Acta Math., 190:1 (2003), 105–149 | DOI | MR | Zbl

[12] P. V. Paramonov, “Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of $\mathbb R^2$”, Sb. Math., 212:12 (2021), 1730–1745 | DOI | MR | Zbl

[13] J. Verdera, “$C^m$-approximation by solutions of elliptic equations, and Calderón–Zygmund operators”, Duke Math. J., 55:1 (1987), 157–187 | DOI | MR | Zbl

[14] N. N. Tarkhanov, The analysis of solutions of elliptic equations, Math. Appl., 406, Kluwer Acad. Publ., Dordrecht, 1997, xx+479 pp. | DOI | MR | MR | Zbl | Zbl

[15] R. Harvey and J. Polking, “Removable singularities of solutions of linear partial differential equations”, Acta Math., 125 (1970), 39–56 | DOI | MR | Zbl

[16] A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory”, Russian Math. Surveys, 22:6 (1967), 139–200 | DOI | MR | Zbl

[17] N. S. Landkof, Foundations of modern potential theory, Grundlehren Math. Wiss., 180, Springer-Verlag, New York–Heidelberg, 1972, x+424 pp. | MR | Zbl

[18] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970, xiv+290 pp. | MR | Zbl

[19] P. V. Paramonov, “Criteria for the individual $C^m$-approximability of functions on compact subsets of $\mathbb R^N$ by solutions of second-order homogeneous elliptic equations”, Sb. Math., 209:6 (2018), 857–870 | DOI | MR | Zbl