Geodesic
A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients
Sbornik. Mathematics, Tome 211 (2020) no. 9, pp. 1267-1309 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A natural counterpart of Vitushkin's criterion is obtained in the problem of uniform approximation of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficient on compact subsets of $\mathbb R^d$, $d\geqslant3$. It is stated in terms of a single (scalar) capacity connected with the leading coefficient of the Laurent series. The scheme of approximation uses methods in the theory of singular integrals and, in particular, constructions of certain special Lipschitz surfaces and Carleson measures. Bibliography: 23 titles.
Keywords: uniform approximation, capacities, singular integrals, Carleson measures, Vitushkin's scheme. 
@article{SM_2020_211_9_a2,
     author = {M. Ya. Mazalov},
     title = {A~criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients},
     journal = {Sbornik. Mathematics},
     pages = {1267--1309},
     year = {2020},
     volume = {211},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_9_a2/}
}
TY  - JOUR
AU  - M. Ya. Mazalov
TI  - A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 1267
EP  - 1309
VL  - 211
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_9_a2/
LA  - en
ID  - SM_2020_211_9_a2
ER  - 
%0 Journal Article
%A M. Ya. Mazalov
%T A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients
%J Sbornik. Mathematics
%D 2020
%P 1267-1309
%V 211
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2020_211_9_a2/
%G en
%F SM_2020_211_9_a2
M. Ya. Mazalov. A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients. Sbornik. Mathematics, Tome 211 (2020) no. 9, pp. 1267-1309. http://geodesic.mathdoc.fr/item/SM_2020_211_9_a2/

[1] 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 | MR | Zbl | Zbl

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

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

[4] 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

[5] M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068 | DOI | DOI | MR | Zbl

[6] J. Mateu, Yu. Netrusov, J. Orobitg, J. Verdera, “BMO and Lipschitz approximation by solutions of elliptic equations”, Ann. Inst. Fourier (Grenoble), 46:4 (1996), 1057–1081 | 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 | DOI | MR | Zbl

[8] P. V. Paramonov, “New criteria for uniform approximability by harmonic functions on compact sets in $\mathbb R^2$”, Proc. Steklov Inst. Math., 298 (2017), 201–211 | DOI | DOI | MR | Zbl

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

[10] M. Ya. Mazalov, “Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$”, Proc. Steklov Inst. Math., 279:1 (2012), 110–154 | DOI | MR | Zbl

[11] A. G. O'Farrell, “Uniform approximation by harmonic functions. Problem 12.15”, Linear and complex analysis. Problem book 3, v. II, Lecture Notes in Math., 1574, Springer-Verlag, 1994, 121 | DOI | MR | Zbl

[12] P. V. Paramonov, “Some new criteria for uniform approximability of functions by rational fractions”, Sb. Math., 186:9 (1995), 1325–1340 | DOI | MR | Zbl

[13] 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 | DOI | MR | Zbl

[14] J. Verdera, M. S. Mel'nikov, P. V. Paramonov, “$C^1$-approximation and extension of subharmonic functions”, Sb. Math., 192:4 (2001), 515–535 | DOI | DOI | MR | Zbl

[15] M. Ya. Mazalov, “On uniform approximation of harmonic functions”, St. Petersburg Math. J., 23:4 (2012), 731–759 | DOI | MR | Zbl

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

[17] 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

[18] 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

[19] M. Ya. Mazalov, “Uniform approximations by bianalytic functions on arbitrary compact subsets of $\mathbb C$”, Sb. Math., 195:5 (2004), 687–709 | DOI | DOI | MR | Zbl

[20] J. Verdera, “Removability, capacity and approximation”, Complex potential theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994, 419–473 | DOI | MR | Zbl

[21] F. John, L. Nirenberg, “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14:3 (1961), 415–426 | DOI | MR | Zbl

[22] J. Mateu, P. Mattila, A. Nicolau, J. Orobitg, “BMO for nondoubling measures”, Duke Math. J., 102:3 (2000), 533–565 | DOI | MR | Zbl

[23] G. David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Math., 1465, Springer-Verlag, Berlin, 1991, x+107 pp. | DOI | MR | Zbl