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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 6, Pages 139–152 (Mi izv670)  

This article is cited in 6 scientific papers (total in 6 papers)

$C^m$-extension of subharmonic functions

P. V. Paramonov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Given $m\in(1,3)$ and any (Jordan) $B$-domain $D$ in $\mathbb R^2$, we prove that any function of class $C^m( \overline D )$ that is subharmonic in $D$ can be extended to a function of class $C^m$ that is subharmonic on the whole $\mathbb R^2$ and give an estimate of the $C^{m-1}$-norm of its gradient. The corresponding assertion for $m\in[0,1)\cup[3,+\infty)$ is false even for discs. These results also hold for balls $D$ in $\mathbb R^N$, $N\in\{3,4,…\}$. We also obtain some corollaries, including the corresponding assertions on the $\operatorname{Lip}^m$-extension of subharmonic functions.

DOI: https://doi.org/10.4213/im670

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English version:
Izvestiya: Mathematics, 2005, 69:6, 1211–1223

Bibliographic databases:

UDC: 517.5
MSC: 31A05, 41A30
Received: 23.05.2005

Citation: P. V. Paramonov, “$C^m$-extension of subharmonic functions”, Izv. RAN. Ser. Mat., 69:6 (2005), 139–152; Izv. Math., 69:6 (2005), 1211–1223

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. 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
    3. P. V. Paramonov, “On $C^m$-Extension of Subharmonic Functions from Lyapunov–Dini Domains to $\mathbb R^N$”, Math. Notes, 89:1 (2011), 160–164  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Proc. Steklov Inst. Math., 279 (2012), 207–214  mathnet  crossref  mathscinet  isi  elib
    5. 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
    6. A. Boivin, P. M. Gauthier, P. V. Paramonov, “Runge- and Walsh-type extensions of smooth subharmonic functions on open Riemann surfaces”, Sb. Math., 206:1 (2015), 3–23  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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