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On the epimorphicity of the convolution operator in convex domains of \(C^{\ell}\). (Russian) Zbl 0632.46034

Let D be a convex domain and K a convex compact set in \(C^{\ell}\), \(\ell >1\), H(D) the space of analytic functions in D with the compact convergence topology and H(K) the space of analytic functions of K with the natural inductive limit topology, H’(D) and H’(K) are the dual spaces. For any \(T\in H'(K)\) define the convolution operator Ť by \[ (\check Ty)(z)=<y(z+\xi),T_{\xi}>,\quad y\in H(D+K),\quad z\in D. \] Then Ť: H(D\(+K)\to H(D)\) is continuous. The author gives sufficient conditions on \(T\in H'(K)\) which guarantee that the equation Ťy\(=f\) has a solution \(y\in H(D+K)\) for arbitrary \(f\in H(D)\). There are given also necessary conditions on T for the solvability of the equation Ťy\(=f\) in the case of rather general domains D. The solvability of Ťy\(=f\) has been considered by several authors in special cases. The present work is closely connected with another work of the author.
Reviewer: L.Simon

MSC:

46F10 Operations with distributions and generalized functions
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