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Mat. Sb., 2001, Volume 192, Number 4, Pages 37–58 (Mi msb556)  

This article is cited in 19 scientific papers (total in 19 papers)

$C^1$-approximation and extension of subharmonic functions

J. Verderaa, M. S. Mel'nikova, P. V. Paramonovb

a Universitat Autònoma de Barcelona
b M. V. Lomonosov Moscow State University

Abstract: Criteria for the uniform approximability in $\mathbb R^N$, $N\geqslant 2$, of the gradients of $C^1$-subharmonic functions by the gradients of similar functions that are harmonic in neighbourhoods of a fixed compact set are obtained. The semiadditivity of the capacity related to the problem is proved and several metric conditions for the approximation are found. An estimate of the flux of the gradient of a subharmonic function in terms of the capacity of its “sources” and a theorem on the possibility of a $C^1$-extension of a subharmonic function in a ball to a subharmonic function on the whole of $\mathbb R^N$ are established.

DOI: https://doi.org/10.4213/sm556

Full text: PDF file (337 kB)
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English version:
Sbornik: Mathematics, 2001, 192:4, 515–535

Bibliographic databases:

UDC: 517.5
MSC: Primary 31A05, 31B05; Secondary 31A15, 31B15, 30A82
Received: 15.06.2000

Citation: J. Verdera, M. S. Mel'nikov, P. V. Paramonov, “$C^1$-approximation and extension of subharmonic functions”, Mat. Sb., 192:4 (2001), 37–58; Sb. Math., 192:4 (2001), 515–535

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Proc. Steklov Inst. Math., 235 (2001), 136–149  mathnet  mathscinet  zmath
    2. Tolsa X., “On the analytic capacity $\gamma_+$”, Indiana Univ. Math. J., 51:2 (2002), 317–343  crossref  mathscinet  zmath  isi
    3. Tolsa X., “Painlevé's problem and the semiadditivity of analytic capacity”, Acta Math., 190:1 (2003), 105–149  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    4. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Tolsa X., “The semiadditivity of continuous analytic capacity and the inner boundary conjecture”, Amer. J. Math., 126:3 (2004), 523–567  crossref  mathscinet  zmath  isi
    6. Gardiner S.J., Gustafsson A., “Smooth potentials with prescribed boundary behaviour”, Publ. Mat., 48:1 (2004), 241–249  crossref  mathscinet  zmath  isi
    7. O. A. Zorina, “$C^m$-extension of subholomorphic functions from plane Jordan domains”, Izv. Math., 69:6 (2005), 1099–1111  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. P. V. Paramonov, “$C^m$-extension of subharmonic functions”, Izv. Math., 69:6 (2005), 1211–1223  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Ruiz de Villa A., Tolsa X., “Characterization and Semiadditivity of the C-1-Harmonic Capacity”, Transactions of the American Mathematical Society, 362:7 (2010), 3641–3675  crossref  mathscinet  zmath  isi  scopus  scopus
    11. M. Ya. Mazalov, “Uniform approximation problem for harmonic functions”, St. Petersburg Math. J., 23:4 (2012), 731–759  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    12. M. Ya. Mazalov, “A criterion for approximability by harmonic functions in Lipschitz spaces”, J. Math. Sci. (N. Y.), 194:6 (2013), 678–692  mathnet  crossref  mathscinet
    13. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. 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  mathnet  crossref  mathscinet  isi  elib
    15. Paramonov P.V., “On C-M-Subharmonic Extension Sets of Walsh-Type”, Complex Analysis and Potential Theory, CRM Proceedings & Lecture Notes, 55, ed. Boivin A. Mashreghi J., Amer Mathematical Soc, 2012, 201–209  crossref  mathscinet  zmath  isi
    16. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. M. Ya. Mazalov, P. V. Paramonov, “Criteria for $C^m$-approximability by bianalytic functions on planar compact sets”, Sb. Math., 206:2 (2015), 242–281  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. 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  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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