On the solvability of convolution equations in Beurling's distributions (Q1922670)

If \({\mathcal D}_\omega'\) denotes the space of Beurling's distributions [cf. \textit{G. Björck}, Ark. Mat. 6, 351-407 (1966; Zbl 0166.36501)] and \({\mathcal E}_\omega'\) denotes the space of Beurling's distributions having compact support, \textit{S. Abdullah} [Acta Math. Hung. 52, No. 1/2, 7-20 (1988; Zbl 0661.46034)] has proved that, for \(S\in{\mathcal E}_\omega'\), the following properties are equivalent: a) there exist positive constants \(A\), \(C\) such that \[ \sup\{|S(x+\xi)|: x\in\mathbb{R}^n, |x|<A\omega(\xi)\}>C\exp(-A\omega(\xi))\quad\text{for all }\xi\in\mathbb{R}^n, \] b) \(S*{\mathcal D}_\omega'={\mathcal D}_\omega'\). In the present paper, the author proves that these two properties are equivalent with the following one: c) If \(u\in{\mathcal D}_\omega'\) and \(S*u\in{\mathcal D}_\omega\), then \(u\in{\mathcal D}_\omega\).