A criterion for the solvability of nonhomogeneous convolution equations in convex domains of the space $\mathbf C^n$

A criterion is given, in terms of the Laplace transform of an analytic functional , for the convolution operator $M_u\colon H(\mathscr D+k)\to H(\mathscr D)$, corresponding to this functional, to be an epimorphism, in the case when $\mathscr D$ is an arbitrary convex domain and $K$ an arbitrary convex compact set in $\mathbf C^n$.