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

Bibliographie

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

[2] H. Whitney, “Analytic extensions of differentiable functions defined on closed sets”, Trans. Amer. Math. Soc., 36:1 (1934), 63–89 | DOI | MR | Zbl

[3] 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

[4] A. Boivin and 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 | MR | Zbl

[5] 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

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

[7] 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

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

[9] M. Ya. Mazalov, “A criterion for approximability by harmonic functions in Lipschitz spaces”, J. Math. Sci. (N. Y.), 194:6 (2013), 678–692 | DOI | MR | Zbl

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

[11] M. Ya. Mazalov and P. V. Paramonov, “Criteria for $C^m$-approximability by bianalytic functions on planar compact sets”, Sb. Math., 206:2 (2015), 242–281 | DOI | MR | Zbl

[12] 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 | 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 | MR | Zbl

[14] P. V. Paramonov and X. Tolsa, “On $C^1$-approximability of functions by solutions of second order elliptic equations on plane compact sets and $C$-analytic capacity”, Anal. Math. Phys., 9:3 (2019), 1133–1161 | DOI | MR | Zbl

[15] 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

[16] P. V. Paramonov, “Criteria for $C^1$-approximability of functions on compact sets in ${\mathbb{R}}^N$, $N \geq 3$, by solutions of second-order homogeneous elliptic equations”, Izv. Math., 85:3 (2021), 483–505 | DOI | MR | Zbl

[17] 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

[18] 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

[19] 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

[20] 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 | Zbl

[21] M. Ya. Mazalov, “Commensurability of some capacities with harmonic capacities”, Russian Math. Surveys, 78:5 (2023), 964–966 | DOI | MR | Zbl

[22] M. Ya. Mazalov, “Capacities commensurable with harmonic ones”, Sb. Math., 215:2 (2024), 250–274 | DOI | MR | Zbl

[23] 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

[24] 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

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

[26] A. P. Calderón and A. Zygmund, “Singular integrals and periodic functions”, Studia Math., 14 (1954), 249–271 | DOI | MR | Zbl

[27] X. Tolsa, The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions, 2021, 60 pp., arXiv: 2106.00680v2

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

[29] R. Harvey and J. C. Polking, “A Laurent expansion for solutions to elliptic equations”, Trans. Amer. Math. Soc., 180 (1973), 407–413 | DOI | MR | Zbl

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

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

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

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

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

[35] 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

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

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

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

[39] 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

[40] M. Ya. Mazalov, “On $\gamma_{{\mathcal L}}$-capacities of Cantor sets”, St. Petersburg Math. J., 35:5 (2024), 869–877 | DOI

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

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

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

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

[45] 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

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

[47] S. N. Mergelyan, Uniform approximations to functions of a complex variable, Amer. Math. Soc. Transl., 101, Amer. Math. Soc., Providence, RI, 1954, 99 pp. | MR | Zbl

[48] R. Narasimhan, Analysis on real and complex manifolds, Adv. Stud. Pure Math., 1, Masson Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968, x+246 pp. | MR | Zbl

[49] 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

[50] P. V. Paramonov and K. Yu. Fedorovskiy, “On $C^m$-reflection of harmonic functions and $C^m$-approximation by harmonic polynomials”, Sb. Math., 211:8 (2020), 1159–1170 | DOI | MR | Zbl

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

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

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

[54] Yu. S. Belov and K. Yu. Fedorovskiy, “Model spaces containing univalent functions”, Russian Math. Surveys, 73:1 (2018), 172–174 | DOI | MR | Zbl

[55] Yu. Belov, A. Borichev, and I. Fedorovskiy, “Nevanlinna domains with large boundaries”, J. Funct. Anal., 277:8 (2019), 2617–2643 | DOI | MR | Zbl

[56] K. Fedorovskiy, “Nevanlinna domains and uniform approximation by polyanalytic polynomial modules”, Function spaces, theory and applications, Fields Inst. Commun., 87, Springer, Cham, 2023, 207–227 | DOI | MR | Zbl

[57] 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 | MR | Zbl

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

[59] M. Ya. Mazalov, “Example of a nonrectifiable Nevanlinna contour”, St. Petersburg Math. J., 27:4 (2016), 625–630 | DOI | MR | Zbl

[60] A. D. Baranov and K. Yu. Fedorovskiy, “On $L^1$-estimates of derivatives of univalent rational functions”, J. Anal. Math., 132 (2017), 63–80 | DOI | MR | Zbl

[61] M. Ya. Mazalov, “On Nevanlinna domains with fractal boundaries”, St. Petersburg Math. J., 29:5 (2018), 777–791 | DOI | MR | Zbl

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

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

[64] 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 | MR | Zbl

[65] A. B. Zaitsev, “Uniform approximability of functions by polynomials of special classes on compact sets in $\mathbb R^2$”, Math. Notes, 71:1 (2002), 68–79 | DOI | MR | Zbl

[66] A. B. Zaitsev, “Uniform approximation of functions by polynomial solutions to second-order elliptic equations on compact sets in $\mathbb{R}^2$”, Math. Notes, 74:1 (2003), 38–48 | DOI | MR | Zbl

[67] A. Bagapsh, K. Fedorovskiy, and M. Mazalov, “On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in $\mathbb R^2$”, J. Math. Anal. Appl., 531:1 (2024), 127896, 26 pp. | DOI | MR | Zbl

[68] J. J. Carmona and K. Fedorovskiy, Carathéodory sets in the plane, Mem. Eur. Math. Soc., 14, EMS Press, Berlin, 2024, 146 pp. | DOI

[69] G. C. Verchota and A. L. Vogel, “Nonsymmetric systems on nonsmooth planar domains”, Trans. Amer. Math. Soc., 349:11 (1997), 4501–4535 | DOI | MR | Zbl