Geodesic
A criterion for approximability by harmonic functions in Lipschitz spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 144-171 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Let $X$ be a compact subset of $\mathbb R^3$, $f$ be a function harmonic inside $X$, from Lipschitz space $C^\gamma(X)$, $0<\gamma<1$. A criterion for approximability of $f$ on $X$ in $C^\gamma(X)$ by functions harmonic on neighborhoods of $X$ is obtained in terms of Hausdorff content of order $1+\gamma$. The proof is completely constructive, and Vitushkin's scheme of singularities separation and approximation by parts is applied. 
@article{ZNSL_2012_401_a7,
     author = {M. Ya. Mazalov},
     title = {A criterion for approximability by harmonic functions in {Lipschitz} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {144--171},
     year = {2012},
     volume = {401},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a7/}
}
TY  - JOUR
AU  - M. Ya. Mazalov
TI  - A criterion for approximability by harmonic functions in Lipschitz spaces
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 144
EP  - 171
VL  - 401
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a7/
LA  - ru
ID  - ZNSL_2012_401_a7
ER  - 
%0 Journal Article
%A M. Ya. Mazalov
%T A criterion for approximability by harmonic functions in Lipschitz spaces
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 144-171
%V 401
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a7/
%G ru
%F ZNSL_2012_401_a7
M. Ya. Mazalov. A criterion for approximability by harmonic functions in Lipschitz spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 144-171. http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a7/

[1] J. Verdera, “Removability, capacity and approximation”, NATO Adv. Sci. Int. Ser. C Math. Phys. Sci., 439, Kluwer, Dordrecht, 1994, 419–473 | MR | Zbl

[2] I. M. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[3] L. Karleson, Izbrannye problemy teorii isklyuchitelnykh mnozhestv, Mir, M., 1971 | MR | Zbl

[4] Dzh. Verdera, M. S. Melnikov, P. V. Paramonov, “$C^1$-approksimatsiya i prodolzhenie subgarmonicheskikh funktsii”, Mat. sb., 192:4 (2001), 37–58 | DOI | MR | Zbl

[5] A. G. Vitushkin, “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199 | MR | Zbl

[6] P. V. Paramonov, “O garmonicheskikh priblizheniyakh v $C^1$-norme”, Mat. sb., 181:10 (1990), 1341–1365 | MR | Zbl

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

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

[9] M. Ya. Mazalov, “O zadache ravnomernogo priblizheniya garmonicheskikh funktsii”, Algeba i analiz, 23:4 (2011), 136–178 | MR

[10] P. V. Paramonov, “Nekotorye novye kriterii ravnomernoi priblizhaemosti funktsii ratsionalnymi drobyami”, Mat. sb., 186:9 (1995), 97–112 | MR | Zbl

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

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

[13] N. N. Tarkhanov, Ryad Lorana dlya reshenii ellipticheskikh sistem, Nauka, Novosibirsk, 1991 | MR | Zbl

[14] J. Verdera, “$C^m$ approximation by solutions of elliptic equations, and Calderón–Zygmund operators”, Duke Math. J., 55 (1987), 157–187 | DOI | MR | Zbl

[15] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR