Conditions for $C^m$-approximability of functions by solutions of elliptic equations

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

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 6, pp. 1023-1068

Résumé

This paper is a survey of results obtained over the past 20–30 years in the qualitative theory of approximation of functions by holomorphic, harmonic, and polyanalytic functions (and, in particular, by corresponding polynomials) in the norms of Whitney-type spaces $C^m$ on compact subsets of Euclidean spaces. Bibliography: 120 titles.

Détail

Keywords: $C^m$-approximation by holomorphic, harmonic, and polyanalytic functions; $C^m$-analytic and $C^m$-harmonic capacity; $s$-dimensional Hausdorff content; Vitushkin localization operator; Nevanlinna domains; Dirichlet problem.

Bibliographie

[1] C. Runge, “Zur Theorie der eindeutigen analytischen Functionen”, Acta Math., 6:1 (1885), 229–244 | DOI | MR | 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] T. W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969, xiii+257 pp. | MR | Zbl | Zbl

[4] F. Hausdorff, Mengenlehre, Walter de Gruyter Co., Berlin, Leipzig, 1935, 306 pp. | MR | Zbl

[5] X. Tolsa, “The semiadditivity of continuous analytic capacity and the inner boundary conjecture”, Amer. J. Math., 126:3 (2004), 523–567 | DOI | MR | Zbl

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

[7] 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 | MR | Zbl

[8] P. V. Paramonov, “On harmonic approximation in the $C^1$-norm”, Math. USSR-Sb., 71:1 (1992), 183–207 | DOI | MR | Zbl

[9] E. P. Dolzhenko, “Ob approksimatsii na zamknutykh oblastyakh i na nul-mnozhestvakh”, Dokl. AN SSSR, 143 (1962), 771–774

[10] M. S. Melnikov, “Otsenka integrala Koshi po analiticheskoi krivoi”, Matem. sb., 71(113):4 (1966), 503–514 | MR | Zbl

[11] M. S. Melnikov, X. Tolsa, “Estimate of the Cauchy integral over Ahlfors regular curves”, Selected topics in complex analysis, Oper. Theory Adv. Appl., 158, Birkhäuser, Basel, 2005, 159–176 | DOI | MR | Zbl

[12] M. Ya. Mazalov, “Kriterii ravnomernoi priblizhaemosti garmonicheskimi funktsiyami na kompaktakh v $\mathbb R^3$”, Tr. MIAN, 279 (2012), 120–165

[13] M. V. Keldysh, “On the solvability and stability of the Dirichlet problem”, Amer. Math. Soc. Transl. Ser. 2, 51 (1966), 1–73 | MR | Zbl

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

[15] J. Verdera, M. S. Mel'nikov, P. V. Paramonov, “$C^1$-approximation and extension of subharmonic functions”, Sb. Math., 192:3-4 (2001), 515–535 | DOI | MR | Zbl

[16] A. G. O'Farrell, “Uniform approximation by harmonic functions”, Linear and complex analysis. Problem book 3, part 2, Lecture Notes in Math., 1574, Springer-Verlag, Berlin, 1994, 121 (problem 12.15) | DOI | MR | Zbl

[17] J. Deny, “Systèmes totaux de fonctions harmoniques”, Ann. Inst. Fourier (Grenoble), 1 (1949), 103–113 | DOI | MR

[18] M. V. Keldysh, “O razreshimosti i ustoichivosti zadachi Dirikhle”, Dokl. AN SSSR, 18:6 (1938), 315–318 | Zbl

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

[20] M. Labréche, De l'approximation harmonique uniforme, Thèse, Université de Montréal, 1982

[21] A. A. Gonchar, “O ravnomernom priblizhenii nepreryvnykh funktsii garmonicheskimi”, Izv. AN SSSR. Ser. matem., 27:6 (1963), 1239–1250 | MR | Zbl

[22] Yu. A. Lysenko, B. M. Pisarevskii, “Instability of harmonic capacity and approximations of continuous functions by harmonic functions”, Math. USSR-Sb., 5:1 (1968), 53–72 | DOI | MR | Zbl

[23] A. Debiard, B. Gaveau, “Potentiel fin et algèbre de fonctions analytiques. I”, J. Funct. Anal., 16:3 (1974), 289–304 | DOI | MR | Zbl

[24] P. M. Gauthier, S. Ladouceur, “Uniform approximation and fine potential theory”, J. Approx. Theory, 72:2 (1993), 138–140 | DOI | MR | Zbl

[25] A. G. O'Farrell, “Metaharmonic approximation in Lipschitz norms”, Proc. Roy. Irish Acad. Sect. A, 75:24 (1975), 317–330 | MR | Zbl

[26] 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

[27] 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

[28] 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

[29] 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. | MR | MR | Zbl | Zbl

[30] J. Mateu, J. Orobitg, “Lipshitz approximation by harmonic functions and some applications to spectral synthesis”, Indiana Univ. Math. J., 39:3 (1990), 703–736 | DOI | MR | Zbl

[31] L. Carleson, “On the connection between Hausdorff measures and capacity”, Ark. Math., 3:5 (1958), 403–406 | DOI | MR | Zbl

[32] M. S. Mel'nikov, “Metric properties of analytic $\alpha$-capacity and approximation of analytic functions with a Hölder condition by rational functions”, Math. USSR-Sb., 8:1 (1969), 115–124 | DOI | MR | Zbl

[33] A. G. O'Farrell, “Estimates for capacities and approximation in Lipschitz norms”, J. Reine Angew. Math., 311/312 (1979), 101–115 | DOI | MR | Zbl

[34] A. G. Vitushkin, “Primer mnozhestv polozhitelnoi dliny, no nulevoi analiticheskoi emkosti”, Dokl. AN SSSR, 127:2 (1959), 246–249 | Zbl

[35] A. G. O'Farrell, “Rational approximation and weak analyticity. II”, Math. Ann., 281:1 (1988), 169–176 | DOI | MR | Zbl

[36] A. G. O'Farrell, “Rational approximation in Lipschitz norms. II”, Proc. Roy. Irish Acad. Sect. A, 79:11 (1979), 103–114 | MR | Zbl

[37] A. G. O'Farrell, “Hausdorff content and rational approximation in fractional Lipschitz norms”, Trans. Amer. Math. Soc., 228 (1977), 187–206 | DOI | MR | Zbl

[38] J. Verdera, “On $C^m$-rational approximation”, Proc. Amer. Math. Soc., 97:4 (1986), 621–625 | DOI | MR | Zbl

[39] Xuan Uy Nguyen, “An extremal problem on singular integrals”, Amer. J. Math., 102:2 (1980), 279–290 | DOI | MR | Zbl

[40] M. Ya. Mazalov, “Kriterii priblizhaemosti garmonicheskimi funktsiyami v prostranstvakh Lipshitsa”, Issledovaniya po lineinym operatoram i teorii funktsii. Vyp. 40, Zap. nauchn. sem. POMI, 401, POMI, SPb., 2012, 144–171

[41] A. Ruiz de Villa, X. Tolsa, “Characterization and semiadditivity of the $C^1$-harmonic capacity”, Trans. Amer. Math. Soc., 362:7 (2010), 3641–3675 | DOI | MR | Zbl

[42] 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

[43] J. L. Walsh, “The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions”, Bull. Amer. Math. Soc., 35:4 (1929), 499–544 | DOI | MR | Zbl

[44] S. N. Mergelyan, “Uniform approximations to functions of a complex variable”, Amer. Math. Soc. Transl., 101, Amer. Math. Soc., Providence, RI, 1954, 294–391 | MR | MR | Zbl

[45] P. V. Paramonov, “$C^m$-approximations by harmonic polynomials on compact sets in $\mathbb R^n$”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 231–251 | DOI | MR | Zbl

[46] P. V. Paramonov, “On approximation by harmonic polynomials in the $C^1$-norm on compact sets in $\mathbf R^2$”, Russian Acad. Sci. Izv. Math., 42:2 (1994), 321–331 | DOI | MR | Zbl

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

[48] H. Lebesgue, “Sur le problème de Dirichlet”, Palermo Rend., 24:1 (1907), 371–402 | DOI | Zbl

[49] E. H. Johnston, “The boundary modulus of continuity of harmonic functions”, Pacific J. Math., 90:1 (1980), 87–98 | MR | Zbl

[50] S. J. Gardiner, Harmonic approximation, London Math. Soc. Lecture Note Ser., 221, Cambridge Univ. Press, Cambridge, 1995, xiv+132 pp. | MR | Zbl

[51] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, P. Noordhoff, Ltd., Groningen, 1963, xxxi+718 pp. | MR | MR | Zbl

[52] M. B. Balk, Polyanalytic functions, Math. Res., 63, Akademie-Verlag, Berlin, 1991, 197 pp. | MR | Zbl

[53] E. P. Dolzhenko, “On boundary properties of the components of polyharmonic functions”, Math. Notes, 63:5-6 (1998), 724–735 | DOI | DOI | MR | Zbl

[54] E. P. Dolzhenko, V. I. Danchenko, “On the boundary properties of solutions to the generalized Cauchy–Riemann equation”, Proc. Steklov Inst. Math., 236 (2002), 132–142 | MR | Zbl

[55] V. I. Danchenko, E. P. Dolzhenko, “On boundary behavior of holomorphic components of bianalytic functions”, J. Math. Sci. (N. Y.), 126:6 (2005), 1586–1592 | DOI | MR | Zbl

[56] K. M. Rasulov, “On the solution of boundary value problems of Dirichlet problem type for polyanalytic functions”, Soviet Math. Dokl., 40:3 (1990), 643–647 | MR | Zbl

[57] M. Ya. Mazalov, “The Dirichlet problem for polyanalytic functions”, Sb. Math., 200:10 (2009), 1473–1493 | DOI | DOI | MR | Zbl

[58] A. G. O'Farrell, “Annihilators of rational modules”, J. Funct. Anal., 19:4 (1975), 373–389 | DOI | MR | Zbl

[59] J. L.-M. Wang, “Approximation by rational modules on nowhere dense sets”, Pacific J. Math., 80:1 (1979), 293–295 | MR | Zbl

[60] J. L.-M. Wang, “Rational modules and higher order Cauchy transforms”, Internat. J. Math. Math. Sci., 4:4 (1981), 661–665 | DOI | MR | Zbl

[61] J. L.-M. Wang, “Approximation by rational modules on boundary sets”, Pacific J. Math., 92:1 (1981), 237–239 | MR | Zbl

[62] J. L.-M. Wang, “A localization operator for rational modules”, Rocky Mountain J. Math., 19:4 (1989), 999–1002 | DOI | MR | Zbl

[63] T. Trent, J. L.-M. Wang, “Uniform approximation by rational modules on nowhere dense sets”, Proc. Amer. Math. Soc., 81:1 (1981), 62–64 | DOI | MR | Zbl

[64] T. Trent, J. L.-M. Wang, “The uniform closure of rational modules”, Bull. London Math. Soc., 13:5 (1981), 415–420 | DOI | MR | Zbl

[65] J. J. Carmona, “A necessary and sufficient condition for uniform approximation by certain rational modules”, Proc. Amer. Math. Soc., 86:3 (1982), 487–490 | DOI | MR | Zbl

[66] J. J. Carmona, “Mergelyan's approximation theorem for rational modules”, J. Approx. Theory, 44:2 (1985), 113–126 | DOI | MR | Zbl

[67] 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

[68] 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

[69] J. Verdera, “On the uniform approximation problem for the square of the Cauchy–Riemann operator”, Pacific J. Math., 159:2 (1993), 379–396 | MR | Zbl

[70] J. L.-M. Wang, “Approximation by rational modules in $\operatorname{Lip}\alpha$ norms”, Illinois J. Math., 26:4 (1982), 632–636 | MR | Zbl

[71] J. Verdera, “Approximation by rational modules in Sobolev and Lipschitz norms”, J. Funct. Anal., 58:3 (1984), 267–290 | DOI | MR | Zbl

[72] J. J. Carmona Doménech, “The closure in $\operatorname{Lip}_\alpha$ norms of rational modules with three generators”, Illinois J. Math., 29:3 (1985), 418–431 | MR | Zbl

[73] J. L.-M. Wang, “A Mergelyan–Vitushkin approximation theorem for rational modules”, J. Approx. Theory, 63:3 (1990), 368–374 | DOI | MR | Zbl

[74] J. L.-M. Wang, “Approximation by rational modules in $L^p$ and BMO”, J. Math. Anal. Appl., 160:1 (1991), 19–23 | DOI | MR | Zbl

[75] J. L.-M. Wang, “Rational modules and Cauchy transforms. II”, Proc. Amer. Math. Soc., 115:2 (1992), 405–408 | DOI | MR | Zbl

[76] V. V. Andrievskii, V. I. Belyi, V. V. Maimeskul, “Direct and inverse theorems for approximation of functions for rational modules in domains with quasiconformal boundary”, Math. Notes, 46:1-2 (1989), 581–588 | DOI | MR | Zbl

[77] K. Yu. Fedorovskiy, “Uniform $n$-analytic polynomial approximations of functions on rectifiable contours in $\mathbb C$”, Math. Notes, 59:3-4 (1996), 435–439 | DOI | DOI | MR | Zbl

[78] J. J. Carmona, P. V. Paramonov, K. Yu. Fedorovskiy, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193:9-10 (2002), 1469–1492 | DOI | DOI | MR | Zbl

[79] J. J. Carmona, K. Yu. Fedorovskiy, “Conformal maps and uniform approximation by polyanalytic functions”, Selected topics in complex analysis, Oper. Theory Adv. Appl., 158, Birkhäuser, Basel, 2005, 109–130 | DOI | MR | Zbl

[80] A. Boivin, P. M. Gauthier, P. V. Paramonov, “On uniform approximation by $n$-analytic functions on closed sets in $\mathbb C$”, Izv. Math., 68:3 (2004), 447–459 | DOI | DOI | MR | Zbl

[81] J. J. Carmona, K. Yu. Fedorovskiy, “On the dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity”, Math. Notes, 83:1-2 (2008), 31–36 | DOI | DOI | MR | Zbl

[82] J. J. Carmona, K. Yu. Fedorovskiy, “New conditions for uniform approximation by polyanalytic polynomials”, Tr. MIAN, 279 (2012), 227–241

[83] A. B. Zaitsev, “Uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact plane sets”, Izv. Math., 68:6 (2004), 1143–1156 | DOI | DOI | MR | Zbl

[84] P. J. Davis, The Schwarz functions and its applications, Carus Math. Monogr., 17, Math. Assoc. Amer., Buffalo, NY, 1974, xi+228 pp. | MR | Zbl

[85] K. Yu. Fedorovskii, “On some properties and examples of Nevanlinna domains”, Proc. Steklov Inst. Math., 253 (2006), 186–194 | DOI | MR

[86] H. S. Shapiro, “Generalized analytic continuation” (Madras, 1967), Symposia on Theoretical Physics and Mathematics, 8, Plenum, New York, 1968, 151–163 | MR | Zbl

[87] R. G. Douglas, H. S. Shapiro, A. L. Shields, “Cyclic vectors and invariant subspaces for the backward shift operator”, Ann. Inst. Fourier (Grenoble), 20:1 (1970), 37–76 | DOI | MR | Zbl

[88] N. K. Nikol'skii, Treatise on the shift operator. Spectral function theory, Grundlehren Math. Wiss., 273, Springer-Verlag, Berlin, 1986, xii+491 pp. | MR | MR | Zbl | Zbl

[89] K. Dyakonov, D. Khavinson, “Smooth functions in star-invariant subspaces”, Recent advances in operator-related function theory, Contemp. Math., 393, Amer. Math. Soc., Providence, RI, 2006, 59–66 | DOI | MR | Zbl

[90] A. D. Baranov, K. Yu. Fedorovskiy, “Boundary regularity of Nevanlinna domains and univalent functions in model subspaces”, Sb. Math., 202:12 (2011), 1723–1740 | DOI | DOI | MR | Zbl

[91] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., 96, Academic Press, Inc., [Harcourt Brace Jovanovich, Publishers], New York–London, 1981, xvi+467 pp. | MR | MR | Zbl

[92] M. Ya. Mazalov, “An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary”, Math. Notes, 62:4 (1997), 524–526 | DOI | DOI | MR | Zbl

[93] P. R Ahern, D. N. Clark, “On functions orthogonal to invariant subspaces”, Acta Math., 124 (1970), 191–204 | DOI | MR | Zbl

[94] E. P. Dolzhenko, “On the boundary smoothness of conformal mappings of domains with nonsmooth boundaries”, Dokl. Math., 76:1 (2007), 514–518 | DOI | MR | Zbl

[95] E. P. Dolzhenko, “Bounds for the moduli of continuity for conformal mappings of domains near their accessible boundary arcs”, Sb. Math., 202:12 (2011), 1775–1823 | DOI | DOI | MR | Zbl

[96] K. Yu. Fedorovskiy, “Approximation and boundary properties of polyanalytic functions”, Proc. Steklov Inst. Math., 235 (2001), 251–260 | MR | Zbl

[97] B. Gustafsson, H. S. Shapiro, “What is a quadrature domain?”, Quadrature domains and their application, Oper. Theory Adv. Appl., 156, Birkhäuser, Basel, 2005, 1–25 | DOI | MR | Zbl

[98] H. S. Shapiro, The Schwarz function and its generalization to higher dimensions, Univ. Arkansas Lecture Notes Math. Sci., 9, John Wiley Sons, Inc., New York, 1992, xiv+108 pp. | MR | Zbl

[99] D. Aharonov, H. S. Shapiro, “Domains in which analytic functions satisfy quadrature identities”, J. Anal. Math., 30 (1976), 39–73 | DOI | MR | Zbl

[100] M. Sakai, “Regularity of boundary having a Schwarz function”, Acta Math., 166:3-4 (1991), 263–297 | DOI | MR | Zbl

[101] K. Yu. Fedorovskiy, “$C^m$-approximation by polyanalytic polynomials on compact subsets of the complex plane”, Complex Anal. Oper. Theory, 5:3 (2011), 671–681 | DOI | MR

[102] P. Mattila, “Smooth maps, null-sets for integralgeometric measure and analytic capacity”, Ann. of Math. (2), 123:2 (1986), 303–309 | DOI | MR | Zbl

[103] M. S. Mel'nikov, “Analytic capacity: discrete approach and curvature of measure”, Sb. Math., 186:6 (1995), 827–846 | DOI | MR | Zbl

[104] M. S. Melnikov, J. Verdera, “A geometric proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz graphs”, Internat. Math. Res. Notices, 1995, no. 7, 325–331 | DOI | MR | Zbl

[105] P. Mattila, M. S. Melnikov, J. Verdera, “The Cauchy integral, analytic capacity, and uniform rectifiability”, Ann. of Math. (2), 144:1 (1996), 127–136 | DOI | MR | Zbl

[106] G. David, P. Mattila, “Removable sets for Lipschitz harmonic functions in the plane”, Rev. Mat. Iberoamericana, 16:1 (2000), 137–215 | DOI | MR | Zbl

[107] G. David, “Unrectifiable $1$-sets have vanishing analytic capacity”, Rev. Mat. Iberoamericana, 14:2 (1998), 369–479 | DOI | MR | Zbl

[108] F. Nazarov, S. Treil, A. Volberg, “The $Tb$-theorem on non-homogeneous spaces”, Acta Math., 190:2 (2003), 151–239 | DOI | MR | Zbl

[109] P. Mattila, P. V. Paramonov, “On geometric properties of harmonic $\operatorname{Lip}_1$-capacity”, Pacific J. Math., 171:2 (1995), 469–491 | MR | Zbl

[110] Proc. Steklov Inst. Math., 235 (2001), 136–149 | MR | Zbl

[111] | MR | Zbl

[112] A. G. O'Farrell, “$T$-invariance”, Proc. Roy. Irish Acad. Sect. A, 92:2 (1992), 185–203 | MR | Zbl

[113] P. V. Paramonov, J. Verdera, “Approximation by solutions of elliptic equations on closed subsets of Euclidean space”, Math. Scand., 74:2 (1994), 249–259 | MR | Zbl

[114] A. Boivin, P. V. Paramonov, “Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions”, Sb. Math., 189:4 (1998), 481–502 | DOI | DOI | MR | Zbl

[115] A. Boivin, P. V. Paramonov, “On radial limit functions for entire solutions of second order elliptic equations in $\mathbb R^2$”, Publ. Mat., 42:2 (1998), 509–519 | DOI | MR | Zbl

[116] A. Boivin, P. M. Gauthier, P. V. Paramonov, “Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications”, Canad. J. Math., 54:5 (2002), 945–969 | DOI | MR | Zbl

[117] P. V. Paramonov, K. Yu. Fedorovskiy, “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 | DOI | MR | Zbl

[118] P. M. Gauthier, “Subharmonic extensions and approximations”, Canad. Math. Bull., 37:1 (1994), 46–53 | DOI | MR | Zbl

[119] M. S. Mel'nikov, P. V. Paramonov, “$C^1$-extension of subharmonic functions from closed Jordan domains in $\mathbb R^2$”, Izv. Math., 68:6 (2004), 1165–1178 | DOI | DOI | MR | Zbl

[120] P. V. Paramonov, “$C^1$-extension and $C^1$-reflection of subharmonic functions from Lyapunov–Dini domains into $\mathbb R^N$”, Sb. Math., 199:12 (2008), 1809–1846 | DOI | DOI | MR | Zbl

[121] A. Boivin, P. M. Gauthier, P. V. Paramonov, “$C^m$-subharmonic extension of Runge-type from closed to open subsets of $\mathbb R^N$”, Tr. MIAN, 279 (2012), 219–226