Solvability of convolution equations in spaces of generalized distributions with restricted growth
\textit{L. Ehrenpreis} [Am. J. Math. 82, 522-588 (1960; Zbl 0098.084)]\ established necessary and sufficient conditions on the Fourier transform of a distribution \(S\in {\mathcal E} (\mathbb{R}^n)\) in order that \(S*{\mathcal D}' (\mathbb{R}^n) \supset {\mathcal D}' (\mathbb{R}^n)\). This holds if and only if there are positive constants \(A_1\), \(A_2\) and \(A_3\) such that for every \(\xi\in \mathbb{R}^n\) there exists an \(\eta\in \mathbb{R}^n\) satisfying the condition \[ |\xi- \eta|\leq A_1\log (2+|\xi|) \qquad \text{and} \qquad |\widehat {S} (\eta)|\geq (A_2+ |\xi |)^{-A_3}. \] In this case \(S\) is called invertible. Later on \textit{S. Abdullah} [Hokkaido Math. J. 17, No. 2, 197-209 (1988; Zbl 0661.46033)], \textit{Ch.-Ch. Chou} [`La transformation de Fourier complexe et l'equation de convolution', Lect. Notes Math. 325 (1973; Zbl 0257.46037)]\ used other versions of the invertibility conditions to prove the existence of solutions of convolution equations in various spaces of distributions and generalized distributions. In this paper the authors have studied convolution equations in the spaces of generalized distributions of \textit{G. Björck} with restricted growth [Ark. Mat. 6, 351-407 (1966; Zbl 0166.365)]. They have characterized the convolution operators \(S\) in \({\mathcal K}_{M,\omega}'\) for which \(S*{\mathcal K}_{M, \omega}' \supset {\mathcal K}_{M, \omega}'\). Incidentally, Abdullah's (referred above) result comes out as a particular case of the result obtained in this paper. The Section 3 of the paper deals with the solvability of convolution equations in \({\mathcal K}_{M, \omega}'\).
scientific article; zbMATH DE number 78118