Previous |  Up |  Next

Article

Keywords:
maximum principle; plurisubharmonic function
Summary:
We prove, among other results, that $\min (u,v)$ is plurisubharmonic (psh) when $u,v$ belong to a family of functions in ${\rm psh}(D)\cap \Lambda _{\alpha }(D),$ where $\Lambda _{\alpha }(D)$ is the $\alpha $-Lipchitz functional space with $1<\alpha <2.$ Then we establish a new characterization of holomorphic functions defined on open sets of $\mathbb {C}^n.$
References:
[1] Abidi, J.: Sur le prolongement des fonctions harmoniques. Manuscripta Math. 105 (2001), 471-482. DOI 10.1007/s002290100182 | MR 1858498 | Zbl 1013.31002
[2] Abidi, J.: Analycité, principe du maximum et fonctions plurisousharmoniques (à paraitre).
[3] Carleson, L.: Selected Problems on Exceptional Sets. Van Nostrand, Princeton, N.J., 1967. (Reprint: Wadswarth, Belmont, Cal., 1983). MR 0225986 | Zbl 0505.00034
[4] Cegrell, U.: Removable singularities for plurisubharmonic functions and related problems. Proc. Lond. Math. Soc. 36 (1978), 310-336. DOI 10.1112/plms/s3-36.2.310 | MR 0484969 | Zbl 0375.32013
[5] Cegrell, U.: Removable singularity sets for analytic functions having modulus with bounded Laplace mass. Proc. Amer. Math. Soc. 88 (1983), 283-286. DOI 10.1090/S0002-9939-1983-0695259-X | MR 0695259 | Zbl 0514.32011
[6] Conway, J. B.: Functions of One Complex Variable II. Springer, Berlin (1995). MR 1344449 | Zbl 0887.30003
[7] Federer, H.: Geometric Measure Theory. Springer, Berlin (1969). MR 0257325 | Zbl 0176.00801
[8] Gunning, R. C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs (1965). MR 0180696 | Zbl 0141.08601
[9] Harvey, R.: Removable singularities for positive currents. Amer. J. Math. 96 (1974), 67-78. DOI 10.2307/2373581 | MR 0361156 | Zbl 0293.32015
[10] Harvey, R., Polking, J.: Extending analytic objects. Comm. Pure Appl. Math. 28 (1975), 701-727. DOI 10.1002/cpa.3160280603 | MR 0409890
[11] Hayman, W. K., Kennedy, P. B.: Subharmonic Functions. Academic Press (1976). Zbl 0323.32013
[12] Henkin, G. M., Leiterer, J.: Theory of Functions on Complex Manifolds. Birkhäuser, Boston, Mass. (1984). MR 0774049 | Zbl 0726.32001
[13] Hervé, M.: Les fonctions analytiques. Presses Universitaires de France (1982). MR 0696576
[14] Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton, N.J. (1966). MR 0203075
[15] Hyvönen, J., Rühentaus, J.: On the extension in the Hardy classes and in the Nevanlinna class. Bull. Soc. Math. France 112 (1984), 469-480. MR 0802536
[16] Klimek, M.: Pluripotential Theory. Clarendon Press, Oxford (1991). MR 1150978 | Zbl 0742.31001
[17] Krantz, S. G.: Function Theory of Several Complex Variables. Wiley, New York (1982). MR 0635928 | Zbl 0471.32008
[18] Krantz, S. G.: Lipschitz spaces, smoothness of functions, and approximation theory. Expo. Math. 3 (1983), 193-260. MR 0782608 | Zbl 0518.46018
[19] Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. Gordon and Breach, New York (1969). MR 0243112
[20] O'Farrell, A. G.: The 1-reduction for removable singularities, and the negative Hölder spaces. Pro. R. Ir. Acad. A 88 (1988), 133-151. MR 0986220 | Zbl 0651.46041
[21] Poletsky, E.: The minimum principle. Indiana Univ. Math. J. 51 (2003), 269-304. MR 1909290
[22] Range, R. M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, Berlin (1986). MR 0847923 | Zbl 0591.32002
[23] Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press (1995). MR 1334766 | Zbl 0828.31001
[24] Riihentaus, J.: On the extension of separately hyperharmonic functions and $H^{p}$-functions. Michigan Math. J. 31 (1984), 99-112. DOI 10.1307/mmj/1029002968 | MR 0736475
[25] Ronkin, L. I.: Introduction to the theory of entire functions of several variables. Amer. Math. Soc., Providence, RI (1974). MR 0346175 | Zbl 0286.32004
[26] Rudin, W.: Function Theory in Polydiscs. Benjamin, New York (1969). MR 0255841 | Zbl 0177.34101
[27] Rudin, W.: Function Theory in the Unit Ball of $\mathbb{C}^n$. Springer, New York (1980). MR 0601594
[28] Shiffman, B.: On the removal of singularities of analytic sets. Michigan Math. J. 15 (1968), 111-120. DOI 10.1307/mmj/1028999912 | MR 0224865 | Zbl 0165.40503
[29] Ullrich, D. C.: Removable sets for harmonic functions. Michigan Math. J. 38 (1991), 467-473. DOI 10.1307/mmj/1029004395 | MR 1116502 | Zbl 0751.31001
[30] Verdera, J.: Approximation by solutions of elliptic equations, and Calderon-Zygmund operators. Duke Math. J. 55 (1987), 157-187. DOI 10.1215/S0012-7094-87-05509-8 | MR 0883668 | Zbl 0654.35007
[31] Vladimirov, V. S.: Les fonctions de plusieurs variables complexe (et leur application à la théorie quantique des champs). Dunod, Paris (1967). MR 0218608
Partner of
EuDML logo