Linear continuous right inverse operator for convolution operator in spaces of holomorphic functions of polynomial growth

A. V. Abanin ; Le Hai Khoi

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2015), pp. 3-13.

Résumé

We consider a convolution operator in spaces of holomorphic functions in a convex domain of the complex plane with polynomial growth at a boundary. We proved that if this operator is surjective on the class of all bounded convex domains, then it always has a linear continuous right inverse operator.

Détail

Keywords: holomorphic function, polynomial growth, convolution operator, linear continuous right/left inverse operator.

Bibliographie

[1] Bell S., “Representation theorem in strictly pseudoconvex domains”, Ill. J. Math., 26 (1982), 19–26 | MR | Zbl

[2] Straube E. J., “Harmonic and analytic functions admitting a distribution boundary value”, Ann. Scuola Norm. Sup. Pisa, 11 (1984), 559–591 | MR | Zbl

[3] Bell S., Boas H., “Regularity of the Bergman projection and duality of holomorphic function spaces”, Math. Ann., 267:4 (1984), 473–478 | DOI | MR | Zbl

[4] Barrett D., “Duality between $A^\infty$ and $A^{-\infty}$ on domains with non-degenerate corners”, Contemp. Math. A. M. S., 185 (1995), 77–87 | DOI | MR | Zbl

[5] Abanin A. V., Le Hai Khoi, “On the duality between $A^{-\infty}(D)$ and $A^{-\infty}_D$ for convex domains”, C. R. Acad. Sci. Paris Ser. I, 347:15–16 (2009), 863–866 | DOI | MR | Zbl

[6] Abanin A. V., Le Hai Khoi, “Dual of the function algebra $A^{-\infty}(D)$ and representation of functions in Dirichlet series”, Proc. Amer. Math. Soc., 138:10 (2010), 3623–3635 | DOI | MR | Zbl

[7] Abanin A. V., Le Hai Khoi, “Pre-dual of the function algebra $A^{-\infty}(D)$ and representation of functions in Dirichlet series”, Complex Anal. Oper. Theory, 5:4 (2011), 1073–1092 | DOI | MR | Zbl

[8] Abanin A. V., Le Hai Khoi, “Cauchy–Fantappiè transformation and mutual dualities between $A^{-\infty}(\Omega)$ and $A^\infty(\widetilde\Omega)$ for lineally convex domains”, C. R. Acad. Sci. Paris Ser. I, 349:21–22 (2011), 1155–1158 | DOI | MR | Zbl

[9] Abanin A. V., Le Hai Khoi, “Mutual dualities between $A^{-\infty}(\Omega)$ and $A^\infty(\widetilde\Omega)$ for lineally convex domains”, Complex Var. Elliptic Equ., 58:11 (2013), 1615–1632 | DOI | MR | Zbl

[10] Bonet J., Domański P., “Sampling sets and sufficient sets for $A^{-\infty}$”, J. Math. Anal. Appl., 277:2 (2003), 651–669 | DOI | MR | Zbl

[11] Horowitz C. A., Korenblum B., Pinchuk B., “Sampling sequences for $A^{-\infty}$”, Michigan Math. J., 44:2 (1997), 389–398 | DOI | MR | Zbl

[12] Le Hai Khoi, Thomas P. J., “Weakly sufficient sets for $A^{-\infty}(\mathbb D)$”, Publ. Mat., 42:2 (1998), 435–448 | MR | Zbl

[13] Bruna J., Pascuas D., “Interpolation in $A^{-\infty}$”, J. London Math. Soc., 40:3 (1989), 452–466 | DOI | MR

[14] Abanin A. V., Le Hai Khoi, Nalbandyan Yu. S., “Minimal absolutely representing systems of exponentials for $A^{-\infty}(\Omega)$”, J. Approx. Theory, 163:10 (2011), 1534–1545 | DOI | MR | Zbl

[15] Abanin A. V., Ishimura R., Le Hai Khoi, “Surjectivity criteria for convolution operators in $A^{-\infty}$”, C. R. Acad. Sci. Paris Ser. I, 348:5–6 (2010), 253–256 | DOI | MR | Zbl

[16] Abanin A. V., Ishimura R., Le Hai Khoi, “Exponential-polynomial bases for null spaces of convolution operators in $A^{-\infty}$”, Contemp. Math., 547 (2011), 1–16 | DOI | MR

[17] Abanin A. V., Ishimura R., Le Hai Khoi, “Convolution operators in $A^{-\infty}$ for convex domains”, Ark. Mat., 50:1 (2012), 1–22 | DOI | MR | Zbl

[18] Abanin A. V., Ishimura R., Le Hai Khoi, “Extension of solutions of convolution equations in spaces of holomorphic functions with polynomial growth in convex domains”, Bull. Sci. Math., 136:1 (2012), 96–110 | DOI | MR | Zbl

[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., 17:3 (1970), 467–517 | MR | Zbl

[20] Ishimura R., Okada J., “Sur la condition (S) de Kawai et la propriété de croissance régulière d'une fonction sous-harmonique et d'une fonction entière”, Kyushu J. Math., 48:2 (1994), 257–263 | DOI | MR | Zbl

[21] Meise R., Vogt D., “Characterization of convolution operators on spaces of $C^\infty$-functions admitting a continuous linear right inverse”, Math. Ann., 279:1 (1987), 141–155 | DOI | MR | Zbl

[22] Momm S., “Convex univalent functions and continuous linear right inverses”, J. Funct. Anal., 103:1 (1992), 85–103 | DOI | MR | Zbl

[23] Langenbruch M., Momm S., “Complemented submodules in weighted spaces of analytic functions”, Math. Nachr., 157 (1992), 263–276 | DOI | MR | Zbl

[24] Langenbruch M., “Continuous linear right inverses for convolution operators in spaces of real analytic functions”, Studia Math., 110:1 (1994), 65–82 | MR | Zbl

[25] Meyer T., “Surjectivity of convolution operators on spaces of ultradifferentialble functions of Roumieu type”, Studia Math., 125:2 (1997), 101–129 | MR | Zbl

[26] Melikhov S. N., Momm Z., “O lineinom nepreryvnom pravom obratnom dlya operatora svertki na prostranstvakh rostkov analiticheskikh funktsii na vypuklykh kompaktakh v $\mathbb C$”, Izv. vuzov. Matem., 1997, no. 5, 38–48 | MR | Zbl

[27] Zharinov V. V., “Kompaktnye semeistva LVP i prostranstva FS i DFS”, UMN, 34:4 (1979), 97–131 | MR | Zbl

[28] Meise R., Vogt D., Introduction to functional analysis, Oxford University Press, 1997 | MR | Zbl

[29] Khermander L., Vvedenie v teoriyu funktsii nekolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[30] Varziev V. A., Melikhov S. N., “O koeffitsientakh ryadov eksponent dlya analiticheskikh funktsii polinomialnogo rosta”, Vladikavkazsk. matem. zhurn., 13:4 (2011), 18–27 | MR