Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^N$

S. N. Melikhov; Siegfried Momm

ArticleOriginal scientific text

Title

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^{N}$

Authors ¹, ²

Affiliations

Department of Mechanics, Mathematisches Institut and Mathematics, Rostov State University, Zorge St. 5, 344104 Rostov-na-Donu, Russia
Department of Mechanics, Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, 40225 Düsseldorf, Germany

Abstract

G \subset ℂ^{N}

is compact and convex it is known for a long time that the nonzero constant coefficients linear partial differential operators (of finite or infinite order) are surjective on the space of all analytic functions on G. We consider the question whether solutions of the inhomogeneous equation can be given in terms of a continuous linear operator. For instance we characterize those sets G for which this is always the case.

L. Ehrenpreis, Solution of some problems of division. IV, Amer. J. Math. 82 (1960), 522-588.
L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, Princeton University Press, 1967.
M. Klimek, Pluripotential Theory, Oxford University Press, 1991.
Yu. F. Korobeĭnik and S. N. Melikhov, A linear continuous right inverse for the representation operator and an application to convolution operators, Sibirsk. Mat. Zh. 34 (1) (1993), 70-84 (in Russian).
I. F. Krasičkov-Ternovski, Invariant subspaces of analytic functions. I, Math. USSR-Sb. 16 (1972), 471-500.
A. S. Krivosheev, A criterion for the solvability of nonhomogeneous convolution equations in convex domains of $ℂ^{N}$ , Math. USSR-Izv. 36 (1991), 497-517.
M. Langenbruch, Splitting of the $o v \partial$ -complex in weighted spaces of square integrable functions, Rev. Mat. Univ. Complut. Madrid 5 (1992), 201-223.
M. Langenbruch, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math. 110 (1994), 65-82.
M. Langenbruch and S. Momm, Complemented submodules in weighted spaces of analytic functions, Math. Nachr. 157 (1992), 263-276.
B. Ja. Levin, Distributions of Zeros of Entire Functions, Amer. Math. Soc., Providence, R.I., 1980.
A. Martineau, Équations différentielles d'ordre infini, Bull. Soc. Math. France 95 (1967), 109-154.
R. Meise and B. A. Taylor, Sequence space representations for (FN)-algebras of entire functions modulo closed ideals, Studia Math. 85 (1987), 203-227.
R. Meise and B. A. Taylor, Each non-zero convolution operator on the entire functions admits a continuous linear right inverse, Math. Z. 197 (1988), 139-152.
R. Meise and D. Vogt, Einführung in die Funktionalanalysis, Vieweg, 1992.
S. Momm, Convex univalent functions and continuous linear right inverses, J. Funct. Anal. 103 (1992), 85-103.
S. Momm, Convolution equations on the analytic functions on convex domains in the plane, Bull. Sci. Math. 118 (1994), 259-270.
S. Momm, Division problems in spaces of entire functions of finite order, in: Functional Analysis, K. D. Bierstedt et al. (eds.), Marcel Dekker, New York, 1993, 435-457.
S. Momm, Boundary behavior of extremal plurisubharmonic functions, Acta Math. 172 (1994), 51-75.
S. Momm, A critical growth rate for the pluricomplex Green function, Duke Math. J. 72 (1993), 487-502.
S. Momm, Extremal plurisubharmonic functions associated to convex pluricomplex Green functions with pole at infinity, preprint.
V. V. Morzhakov, On epimorphicity of a convolution operator in convex domains in $ℂ^{N}$ , Math. USSR-Sb. 60 (1988), 347-364.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993.
R. Sigurdsson, Convolution equations in domains of $ℂ^{N}$ , Ark. Mat. 29 (1991), 285-305.
B. A. Taylor, On weighted polynomial approximation of entire functions, Pacific J. Math. 36 (1971), 523-539.

Title

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in ℂN

Affiliations

Abstract

Bibliography

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^{N}$