Geodesic
On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in $\pmb{\mathbb R}^2$
Sbornik. Mathematics, Tome 213 (2022) no. 6, pp. 831-843 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For certain capacities that were used previously to formulate criteria for the uniform approximability of functions by solutions of strongly elliptic equations of the second order on compact subsets of $\mathbb R^2$, a number of metric properties are established. New, more natural criteria for individual approximability are obtained as consequences. Unsolved problems of interest are stated. Bibliography: 13 titles.
Keywords: strongly elliptic equations of the second order in $\mathbb R^2$, $C$-capacity, Vitushkin-type localization operator, subadditivity problem for capacity.
Mots-clés : Hausdorff content 
@article{SM_2022_213_6_a4,
     author = {P. V. Paramonov},
     title = {On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in $\pmb{\mathbb R}^2$},
     journal = {Sbornik. Mathematics},
     pages = {831--843},
     year = {2022},
     volume = {213},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_6_a4/}
}
TY  - JOUR
AU  - P. V. Paramonov
TI  - On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in $\pmb{\mathbb R}^2$
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 831
EP  - 843
VL  - 213
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_6_a4/
LA  - en
ID  - SM_2022_213_6_a4
ER  - 
%0 Journal Article
%A P. V. Paramonov
%T On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in $\pmb{\mathbb R}^2$
%J Sbornik. Mathematics
%D 2022
%P 831-843
%V 213
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2022_213_6_a4/
%G en
%F SM_2022_213_6_a4
P. V. Paramonov. On metric properties of $C$-capacities associated with solutions of second-order strongly elliptic equations in $\pmb{\mathbb R}^2$. Sbornik. Mathematics, Tome 213 (2022) no. 6, pp. 831-843. http://geodesic.mathdoc.fr/item/SM_2022_213_6_a4/

[1] 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

[2] P. V. Paramonov, “New criteria for uniform approximability by harmonic functions on compact sets in $\mathbb R^2$”, Complex analysis and applications, Tr. Mat. Inst. Steklova, 298, MAIK “Nauka/Interperiodika”, Moscow, 2017, 216–226 ; English transl. in Proc. Steklov Inst. Math., 298 (2017), 201–211 | DOI | MR | Zbl | DOI

[3] M. Ya. Mazalov, “A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients”, Mat. Sb., 211:9 (2020), 60–104 ; English transl. in Sb. Math., 211:9 (2020), 1267–1309 | DOI | MR | Zbl | DOI

[4] L. Hörmander, The analysis of linear partial differential operators, v. I, Grundlehren Math. Wiss., 256, Distribution theory and Fourier analysis, Springer-Verlag, Berlin, 1983, ix+391 pp. | DOI | MR | Zbl

[5] 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

[6] P. V. Paramonov, “Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of $\mathbb R^2$”, Mat. Sb., 212:12 (2021), 77–94 ; English transl. in Sb. Math., 212:12 (2021), 1730–1745 | DOI | MR | Zbl | DOI

[7] M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations”, Mat. Sb., 199:1 (2008), 15–46 ; English transl. in Sb. Math., 199:1 (2008), 13–44 | DOI | MR | Zbl | DOI

[8] M. Ya. Mazalov, “Uniform approximation of functions by solutions of second order homogeneous strongly elliptic equations on compact sets in ${\mathbb{R}}^2$”, Izv. Ross. Akad. Nauk Ser. Mat., 85:3 (2021), 89–126 ; English transl. in Izv. Math., 85:3 (2021), 421–456 | DOI | MR | Zbl | DOI

[9] P. V. Paramonov and K. Yu. Fedorovskiy, “Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations”, Mat. Sb., 190:2 (1999), 123–144 ; English transl. in Sb. Math., 190:2 (1999), 285–307 | DOI | MR | Zbl | DOI

[10] N. S. Landkof, Foundations of modern potential theory, Nauka, Moscow, 1966, 515 pp. ; English transl., Grundlehren Math. Wiss., 180, Springer-Verlag, New York–Heidelberg, 1972, x+424 pp. | MR | Zbl | MR | Zbl

[11] V. Ya. Èiderman, “Estimates for potentials and $\delta$-subharmonic functions outside exceptional sets”, Izv. Ross. Akad. Nauk Ser. Mat., 61:6 (1997), 181–218 ; English transl. in Izv. Math., 61:6 (1997), 1293–1329 | DOI | MR | Zbl | DOI

[12] A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory”, Uspekhi Mat. Nauk, 22:6(138) (1967), 141–199 ; English transl. in Russian Math. Surveys, 22:6 (1967), 139–200 | MR | Zbl | DOI

[13] G. M. Goluzin, Geometric theory of functions of a complex variable, 2nd ed., Nauka, Moscow, 1966, 628 pp. ; English transl., Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969, vi+676 pp. | MR | Zbl | MR | Zbl