Convolutors in spaces of holomorphic functions
Let Ω be an open convex set in ℂn and K a convex compact subset of Ωn: we provide sufficient, as well as necessary, conditions for the surjectivity of convolution operators between the spaces ℋ(Ω+K) and ℋ(Ω). Under natural hypotheses on the convolutors, we prove integral representation theorems for solutions f∈ℋ(Ω+K) of systems of homogeneous convolution equations. We apply this analysis to provide necessary and sufficient conditions for the hyperbolicity and the ellipticity of given systems of convolution equations; we also study the extension of solutions of homogeneous convolution equations to some sets which can be defined in terms of Ω, K and the convolutor.
Key wordsConvolution equations holomorphic functions difference-differential equations AMS Subject Classification 32A15 42B99 43A45
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