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

Melikhov, S.; Momm, Siegfried

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

ℂ^{N}

Melikhov, S. ; Momm, Siegfried

Studia Mathematica, Tome 113 (1995), p. 79-99 / Harvested from The Polish Digital Mathematics Library

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Résumé

$G \subset ℂ^{N}$ 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.

Publié le : 1995-01-01

Zbl 0842.46051

EUDML-ID : urn:eudml:doc:216243

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     title = {Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $$\mathbb{C}$^N$
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     journal = {Studia Mathematica},
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     year = {1995},
     pages = {79-99},
     zbl = {0842.46051},
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Melikhov, S.; Momm, Siegfried. Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^N$
            . Studia Mathematica, Tome 113 (1995) pp. 79-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv117i1p79bwm/

Bibliographie

[00000] [1] L. Ehrenpreis, Solution of some problems of division. IV, Amer. J. Math. 82 (1960), 522-588. | Zbl 0098.08401

[00001] [2] L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170. | Zbl 0109.08501

[00002] [3] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Princeton University Press, 1967. | Zbl 0138.06203

[00003] [4] M. Klimek, Pluripotential Theory, Oxford University Press, 1991.

[00004] [5] 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).

[00005] [6] I. F. Krasičkov-Ternovski, Invariant subspaces of analytic functions. I, Math. USSR-Sb. 16 (1972), 471-500.

[00006] [7] 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. | Zbl 0723.45005

[00007] [8] M. Langenbruch, Splitting of the $\partial$ -complex in weighted spaces of square integrable functions, Rev. Mat. Univ. Complut. Madrid 5 (1992), 201-223. | Zbl 0772.32017

[00008] [9] M. Langenbruch, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math. 110 (1994), 65-82. | Zbl 0824.35147

[00009] [10] M. Langenbruch and S. Momm, Complemented submodules in weighted spaces of analytic functions, Math. Nachr. 157 (1992), 263-276. | Zbl 0787.46034

[00010] [11] B. Ja. Levin, Distributions of Zeros of Entire Functions, Amer. Math. Soc., Providence, R.I., 1980.

[00011] [12] A. Martineau, Équations différentielles d'ordre infini, Bull. Soc. Math. France 95 (1967), 109-154. | Zbl 0167.44202

[00012] [13] R. Meise and B. A. Taylor, Sequence space representations for (FN)-algebras of entire functions modulo closed ideals, Studia Math. 85 (1987), 203-227. | Zbl 0669.46002

[00013] [14] 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. | Zbl 0618.32014

[00014] [15] R. Meise and D. Vogt, Einführung in die Funktionalanalysis, Vieweg, 1992.

[00015] [16] S. Momm, Convex univalent functions and continuous linear right inverses, J. Funct. Anal. 103 (1992), 85-103. | Zbl 0771.46016

[00016] [17] S. Momm, Convolution equations on the analytic functions on convex domains in the plane, Bull. Sci. Math. 118 (1994), 259-270. | Zbl 0819.46039

[00017] [18] 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. | Zbl 0803.46025

[00018] [19] S. Momm, Boundary behavior of extremal plurisubharmonic functions, Acta Math. 172 (1994), 51-75. | Zbl 0802.32024

[00019] [20] S. Momm, A critical growth rate for the pluricomplex Green function, Duke Math. J. 72 (1993), 487-502. | Zbl 0830.31005

[00020] [21] S. Momm, Extremal plurisubharmonic functions associated to convex pluricomplex Green functions with pole at infinity, preprint. | Zbl 0848.31008

[00021] [22] V. V. Morzhakov, On epimorphicity of a convolution operator in convex domains in $ℂ^{N}$ , Math. USSR-Sb. 60 (1988), 347-364. | Zbl 0678.46032

[00022] [23] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993. | Zbl 0798.52001

[00023] [24] R. Sigurdsson, Convolution equations in domains of $ℂ^{N}$ , Ark. Mat. 29 (1991), 285-305. | Zbl 0794.32004

[00024] [25] B. A. Taylor, On weighted polynomial approximation of entire functions, Pacific J. Math. 36 (1971), 523-539. | Zbl 0211.14904