Linear continuous right inverse to convolution operator in spaces of holomorphic functions of polynomial growth. (English. Russian original) Zbl 1319.30047

Russ. Math. 59, No. 1, 1-10 (2015); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2015, No. 1, 3-13 (2015).
Summary: We consider the convolution operator in spaces of holomorphic functions, defined in convex subdomains of the complex plane, with polynomial growth at a boundary. We prove that if this operator is surjective on the class of all bounded convex domains, then it always has a linear continuous right inverse one.


30H99 Spaces and algebras of analytic functions of one complex variable
Full Text: DOI


[1] Bell, S, Representation theorem in strictly pseudoconvex domains, Ill. J. Math., 26, 19-26, (1982) · Zbl 0475.32004
[2] Straube, E J, Harmonic and analytic functions admitting a distribution boundary value, Ann. Scuola Norm. Sup. Pisa, 11, 559-591, (1984) · Zbl 0582.31003
[3] Bell, S; Boas, H, Regularity of the Bergman projection and duality of holomorphic function spaces, Math. Ann., 267, 473-478, (1984) · Zbl 0536.32010
[4] Barrett, D, Duality between \(A\)\^{}{∞} and \(A\)\^{}{−∞} on domains with nondegenerate corners, Contemp. Math., 185, 77-87, (1995) · Zbl 0834.32002
[5] Abanin, A V; Khoi, L Hai, On the duality between \(A\)\^{}{−∞}(\(D\)) and \(A\)_{D}\^{}{∞} for convex domains, C. R. Acad. Sci. Paris, Ser. I, 347, 863-866, (2009) · Zbl 1177.46016
[6] Abanin, A V; Khoi, L Hai, Dual of the function algebra \(A\)\^{}{−∞}(\(D\)) and representation of functions in Dirichlet series, Proc. Amer. Math. Soc., 138, 3623-3635, (2010) · Zbl 1205.32004
[7] Abanin, A V; Khoi, L Hai, Pre-dual of the function algebra \(A\)\^{}{−∞}(\(D\)) and representation of functions in Dirichlet series, Complex Anal. Oper. Theory, 5, 1073-1092, (2011) · Zbl 1275.32008
[8] Abanin, A V; Khoi, L Hai, Cauchy-fantappiè transformation and mutual dualities between \(A\)\^{}{−∞}(ω) and \(A\)\^{}{∞}(ω̃) for lineally convex domains, C. R. Acad. Sci. Paris, Ser. I, 349, 1155-1158, (2011) · Zbl 1257.32004
[9] Abanin, AV; Khoi, L Hai, Mutual dualities between\(A\)\^{}{−∞}(ω) and \(A\)\^{}{∞}(ω̃) for lineally convex domains, Complex Var. Elliptic Equ., 58, 1615-1632, (2013) · Zbl 1287.46034
[10] Bonet, J; Domański, P, Sampling sets and sufficient sets for \(A\)\^{}{−∞}, J. Math. Anal. Appl., 277, 651-669, (2003) · Zbl 1019.30026
[11] Horowitz, C A; Korenblum, B; Pinchuk, B, Sampling sequences for \(A\)\^{}{−∞}, Michigan Math. J., 44, 389-398, (1997) · Zbl 0889.30034
[12] Khoi, L H; Thomas, P J, Weakly sufficient sets for \(A\)\^{}{−∞}(D), Publ. Mat., 42, 435-448, (1998) · Zbl 1140.46311
[13] Bruna, J; Pascuas, D, Interpolation in \(A\)\^{}{−∞}, J. London Math. Soc., 40, 452-466, (1989) · Zbl 0652.30026
[14] Abanin, A V; Khoi, L H; Nalbandyan, Yu S, Minimal absolutely representing systems of exponentials for \(A\)\^{}{−∞}(ω), J. Approx. Theory, 163, 1534-1545, (2011) · Zbl 1276.30008
[15] Abanin, A V; Ishimura, R; Khoi, L Hai, Surjectivity criteria for convolution operators in \(A\)\^{}{−∞}, C. R. Acad. Sci. Paris, Ser. I, 348, 253-256, (2010) · Zbl 1190.32001
[16] Abanin, A V; Ishimura, R; Khoi, L Hai, Exponential-polynomial bases for null spaces of convolution operators in \(A\)\^{}{−∞}, Contemp. Math., 547, 1-16, (2011) · Zbl 1251.30059
[17] Abanin, A V; Ishimura, R; Khoi, L Hai, Convolution operators in \(A\)\^{}{−∞} for convex domains, Ark. Mat., 50, 1-22, (2012) · Zbl 1254.32009
[18] Abanin, A V; Ishimura, R; Khoi, L Hai, Extension of solutions of convolution equations in spaces of holomorphic functions with polynomial growth in convex domains, Bull. Sci. Math., 136, 96-110, (2012) · Zbl 1239.47024
[19] Kawai, T, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 17, 467-517, (1970) · Zbl 0212.46101
[20] Ishimura, R., Okada, J. “Sur la Condition (S) de Kawai et la Propriétéde Croissance Réguliè re d’Une Fonction Sous-Harmonique et d’Une Fonction Entiè re,” Kyushu J. Math. 48, No. 2, 257-263 (1994). · Zbl 0815.32002
[21] Meise, R; Vogt, D, Characterization of convolution operators on spaces of \(C\)\^{}{∞}-functions admitting a continuous linear right inverse, Math. Ann., 279, 141-155, (1987) · Zbl 0607.42011
[22] Momm, S, Convex univalent functions and continuous linear right inverses, J. Funct. Anal., 103, 85-103, (1992) · Zbl 0771.46016
[23] Langenbruch, M; Momm, S, Complemented submodules in weighted spaces of analytic functions, Math. Nachr., 157, 263-276, (1992) · Zbl 0787.46034
[24] Langenbruch, M, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math., 110, 65-82, (1994) · Zbl 0824.35147
[25] Meyer, T, Surjectivity of convolution operators on spaces of ultradifferentialble functions of Roumieu type, Studia Math., 125, 101-129, (1997) · Zbl 0897.46023
[26] Melikhov, S N; Momm, Z, On the linear inverse from right operator for the convolution operator on the spaces of germs of analytical functions on convex compacts in C, Mathematics (Iz. VUZ), 41, 35-45, (1997) · Zbl 0906.46019
[27] Zharinov, V V, Compact families of locally convex topological vector spaces, Fréchet-Schwartz and dual Fréchet-Schwartz spaces, Russ. Math. Surv., 34, 105-143, (1979) · Zbl 0443.46002
[28] Meise, R., Vogt, D. Introduction to Functional Analysis (Oxford University Press, 1997). · Zbl 0924.46002
[29] Hörmander, L. An Introduction to Complex Analysis in Several Variables (D. van Nostrand Company, N. J.-Toronto-New York-London, 1966; Mir, Moscow, 1968). · Zbl 0138.06203
[30] Varziev, V A; Melihkov, S N, On coefficients of exponential series for analytic functions of polynomial growth, Vladikavkazsk. Matem. Zhurn., 13, 18-27, (2011) · Zbl 1326.30004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.