On the surjectivity of the convolution operator in spaces of holomorphic functions of a prescribed growth

A. V. Abanin   ; T. M. Andreeva

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 3-15

Résumé

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.

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