On the Surjectivity of the Convolution Operator in Spaces of Holomorphic Functions of a Prescribed Growth

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

		Log in

Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2018.2.14713 On the Surjectivity of the Convolution Operator in Spaces of Holomorphic Functions of a Prescribed Growth Abanin, A. V. , Andreeva T. M. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2. Abstract: We consider the weighted (DFS)-spaces of holomorphic functions in a bounded convex domain \(G\) of the complex plane \(\mathbb C\) having a prescribed growth given by some sequence of weights satisfying several general and natural conditions. Under these conditions the problem of the continuity and surjectivity of a convolution operator from \(H(G+K)\) into (onto) \(H(G)\) is studied. Here \(K\) is a fixed compact subset in \(\mathbb C\). We answer the problem in terms of the Laplace transformation of the linear functional that determines the convolution operator (it is called the symbol of the convolution operator). In spaces of a general type we obtain a functional criterion for a convolution operator to be surjective from \(H(G+K)\) onto \(H(G)\). In the particular case of spaces of exponential-power growth of the maximal and normal types we establish some sufficient conditions on the symbol's behaviour for the corresponding convolution operator to be surjective. These conditions are stated in terms of some lower estimates of the symbol. In addition, we show that these conditions are necessary for the convolution operator to be surjective for all bounded convex domains \(G\) in \(\mathbb C\). In fact, we obtain a criterion for a surjective convolution operator in spaces of holomorphic functions of exponential-power growth on the class of all bounded convex domains in \(\mathbb C\). Similar previous results were available for only the particular space of holomorphic functions having the polynomial growth in bounded convex domains. Keywords: weighted spaces, holomorphic functions, convolution operator, surjectivity, spaces of exponential-power growth Language: Russian Download the full text For citation: Abanin A. V., Andreeva T. M. On the Surjectivity of the Convolution Operator in Spaces of Holomorphic Functions of a Prescribed Growth. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp.3-15. DOI 10.23671/VNC.2018.2.14713 + References 1. Korobeinik Yu. F. O razreshimosti v kompleksnoy oblasti nekotorykh klassov lineynykh operatornykh uravneniy [On the Solvability of Certain Classes of Linear Operator Equations in a Complex Domain], Rostov-na-Donu, Yuzhnyi Federalnyi Universitet, 2009, 251 p. (in Russian). 2. Melikhov S. N., Momm S. Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary, Math. Scand., 2000, vol. 86, no. 2, pp. 293-319. 3. Momm S. A Division Problem in the Space of Entire Functions of Exponential Type, Ark. Mat., 1994, vol. 32, no. 1, pp. 213-236. DOI: 10.1007/BF02559529. 4. Abanin A. V., Ishimura R., Le Hai Khoi. Surjectivity Criteria for Convolution Operators in \(A^{-\infty}\), C.R. Acad. Sci. Paris, Ser. I, 2010, vol. 348, no. 5-6, pp. 253-256. DOI: 10.1016/j.crma.2010.01.015. 5. Abanin A. V., Ishimura R., Le Hai Khoi. Convolution Operators in \(A^{-\infty}\) for Convex Domains, Ark. Mat., 2012, vol. 50, no. 1, pp. 1-22. DOI: 10.1007/s11512-011-0146-4. 6. 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., 2012, vol. 136, no. 1, pp. 96-110. DOI: 10.1016/j.bulsci.2011.06.002. 7. Epifanov O. V. On Solvability of the Nonhomogeneous Cauchy-Riemann Equation in Classes of Functions that are Bounded with Weights or Systems of Weights, Math. Notes, 1992, vol. 51, no. 1, pp. 54-60. DOI: 10.1007/BF01229435. 8. Polyakova D. A. Solvability of Inhomogeneous Cauchy-Riemann Equation in Projective Weighted Spaces, Siberian Math. J., 2017, vol. 58, no. 1, pp. 142-152. DOI: 10.1134/S0037446617010189. 9. Andreeva T. M. Duals for Weighted Spaces of Holomorphic Functions of Prescribed Growth in Bounded Convex Domains, Izvestiya vuzov. Severo-Kavkazskiy region. Estestvennye nauki [Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki], 2018, no. 1, pp. 4-9. 10. Napalkov V. V. Spaces of Analytic Functions of Prescribed Growth Near the Boundary, Math. USSR-Izv., 1988, vol. 30, no. 2, pp. 263-281. DOI: 10.1070/IM1988v030n02ABEH001008. 11. Abanin A. V., Pham Trong Tien. Continuation of Holomorphic Functions with Growth Conditions and Some of its Applications, Studia Math., 2010, vol. 200, no. 3, pp. 279-295. DOI: 10.4064/sm200-3-5. 12. Hormander L. On the Range of Convolution Operators, Ann. of Math., 1962, vol. 76, no. 1, pp. 148-170. DOI: 10.2307/1970269. 13. Abanin A. V. Thick Spaces and Analytic Multipliers, Izvestiya vuzov. Severo-Kavkazskiy region. Estestvennye nauki [Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki], 1994, no. 4, pp. 3-10 (in Russian). ← Contents of issue


	\| Home \| Editorial board \| Publication ethics \| Peer review guidelines \| Latest issue \| All issues \| Rules for authors \| Online submission system’s guidelines \| Submit manuscript \|
	© 1999-2023 Южный математический институт