Marino, Guiseppe; Pietramala, Paolamaria; Struppa, Daniele Carlo Factorization of solutions of convolution equations. II. (English) Zbl 0809.46035 Ill. J. Math. 35, No. 3, 419-433 (1991). The main result in the paper is the following: Let \(\mu_ 1,\mu_ 2,\mu: \widehat\mu= \widehat\mu_ 1\cdot \widehat\mu_ 2\) be analytic functionals supported on convex compact sets \(K_ 1\), \(K_ 2\) and \(K\) respectively (\(\widehat\mu\) stands for the Fourier-Borel transform of \(\mu\)). \(\Omega\supset K\) is an open convex domain. Let \(W_ 1\), \(W_ 2\), \(W\) denote respectively the closed subspaces of the solutions to the convolution equations \[ \mu_ 1 * f=0,\quad \mu_ 2* f=0,\quad \mu* f= 0, \] where \(f\in H(\Omega):=\) holomorphic functions in \(\Omega\). Assume that \(\widehat\mu_ 2\) is an entire function (of exponential type, of course) of completely regular growth. Then every solution \(f\in H(\Omega)\) of the equation \(\mu* f=0\) has the representation \(f= f_ 1+ f_ 2\), where \(f_ i\) denotes a solution of the equation \(\mu_ i* f= 0\), \(i= 1,2\) if and only if the convolution operator \(f\mapsto \mu_ 2* f\) restricted to the subspace \(W_ i\) maps it onto itself.This and other theorems in the paper extend to the \(H(\Omega)\) setting previous results of a similar nature by the same authors obtained for the space \(H(C):=\) entire functions. Reviewer: D.Khavinson (Fayetteville) MSC: 46F15 Hyperfunctions, analytic functionals 46F10 Operations with distributions and generalized functions 42A85 Convolution, factorization for one variable harmonic analysis 30D15 Special classes of entire functions of one complex variable and growth estimates 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:analytic functionals supported on convex compact sets; Fourier-Borel transform; open convex domain; convolution equations; convolution operator PDFBibTeX XMLCite \textit{G. Marino} et al., Ill. J. Math. 35, No. 3, 419--433 (1991; Zbl 0809.46035)