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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 3, Pages 352–370 (Mi msb1861)  

This article is cited in 7 scientific papers (total in 7 papers)

On epimorphicity of a convolution operator in convex domains in $\mathbf C^l$

V. V. Morzhakov

Abstract: Let $D$ be a convex domain and $K$ a convex compact set in $\mathbf C^l$; let $H(D)$ be the space of analytic functions in $D$, provided with the topology of uniform convergence on compact sets, and $H(K)$ the space of germs of analytic functions on $K$ with the natural inductive limit topology; and let $H'(K)$ be the space dual to $H(K)$. Each functional $T\in H'(K)$ generates a convolution operator $(\check Ty)(z)=T_\zeta(y(z+\zeta))$, $y\in H(D+K)$, $z\in D$, which acts continuously from $H(D+K)$ into $H(D)$. Further let $(\mathscr FT)(z)=T_\zeta(\exp\langle z,\zeta\rangle)$ be the Fourier–Borel transform of the functional $T\in H'(K)$.
In this paper the following theorem is proved:
Theorem. {\it Let $D$ be a bounded convex domain in $\mathbf C^l$ with boundary of class $C^1$ or $D=D_1\times…\times D_l,$ where the $D_j$ are bounded planar convex domains with boundaries of class $C^1,$ and let $T\in H'(K)$. In order that $\check T(H(D+K))=H(D)$ it is necessary and sufficient that
{\rm1)} $\mathscr L^*_{\mathscr FT}(\zeta)=h_K(\zeta)$ $\forall \zeta\in\mathbf C^l;$
{\rm2)} $(\mathscr FT)(z)$ be a function of completely regular growth in $\mathbf C^l$ in the sense of weak convergence in $D'(\mathbf C^l)$.}
Here $\mathscr L^*_{\mathscr FT}(\zeta)=\varlimsup_{z\to\zeta}  \varlimsup_{r\to\infty }\frac{\ln |(\mathscr FT)(rz)|}{r}$ is the regularized radial indicator of the entire function $(\mathscr FT)(z)$, and $h_K(\zeta)$ is the support function of the compact set $K$.
Bibliography: 29 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 60:2, 347–364

Bibliographic databases:

UDC: 517.55
MSC: 32A30, 30D99
Received: 26.11.1985

Citation: V. V. Morzhakov, “On epimorphicity of a convolution operator in convex domains in $\mathbf C^l$”, Mat. Sb. (N.S.), 132(174):3 (1987), 352–370; Math. USSR-Sb., 60:2 (1988), 347–364

Citation in format AMSBIB
\by V.~V.~Morzhakov
\paper On epimorphicity of a~convolution operator in convex domains in~$\mathbf C^l$
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 3
\pages 352--370
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 347--364

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    This publication is cited in the following articles:
    1. A. S. Krivosheev, “A criterion for the solvability of nonhomogeneous convolution equations in convex domains of the space $\mathbf C^n$”, Math. USSR-Izv., 36:3 (1991), 497–517  mathnet  crossref  mathscinet  zmath  adsnasa
    2. Ragnar Sigurdsson, “Convolution equations in domains ofC n ”, Ark Mat, 29:1-2 (1991), 285  crossref  mathscinet  zmath  isi
    3. A. S. Krivosheev, V. V. Napalkov, “Complex analysis and convolution operators”, Russian Math. Surveys, 47:6 (1992), 1–56  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Ishimura R. Okada Y., “The Existence and the Continuation of Holomorphic Solutions for Convolution Equations in Tube Domains”, Bull. Soc. Math. Fr., 122:3 (1994), 413–433  mathscinet  zmath  isi
    5. Melikhov, SN, “Analytic solutions of convolution equations on convex sets with an obstacle in the boundary”, Mathematica Scandinavica, 86:2 (2000), 293  isi
    6. Guang Wang, “Applications of the division problem in spaces of entire functions”, J Syst Sci Syst Eng, 12:3 (2003), 307  crossref
    7. A. V. Abanin, V. A. Varziev, “Sufficient sets in weighted Fréchet spaces of entire functions”, Siberian Math. J., 54:4 (2013), 575–587  mathnet  crossref  mathscinet  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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