On epimorphicity of a convolution operator in convex domains in $\mathbf C^l$

Let $D$ be a convex domain and $K$ a convex compact set in $\mathbf C^l$; let $H(D)$ be the space of analytic functions in $D$, provided with the topology of uniform convergence on compact sets, and $H(K)$ the space of germs of analytic functions on $K$ with the natural inductive limit topology; and let $H'(K)$ be the space dual to $H(K)$. Each functional $T\in H'(K)$ generates a convolution operator $(\check Ty)(z)=T_\zeta(y(z+\zeta))$, $y\in H(D+K)$, $z\in D$, which acts continuously from $H(D+K)$ into $H(D)$. Further let $(\mathscr FT)(z)=T_\zeta(\exp\langle z,\zeta\rangle)$ be the Fourier–Borel transform of the functional $T\in H'(K)$.
In this paper the following theorem is proved:
Theorem. {\it Let $D$ be a bounded convex domain in $\mathbf C^l$ with boundary of class $C^1$ or $D=D_1\times\dots\times D_l,$ where the $D_j$ are bounded planar convex domains with boundaries of class $C^1,$ and let $T\in H'(K)$. In order that $\check T(H(D+K))=H(D)$ it is necessary and sufficient that
{\rm1)} $\mathscr L^*_{\mathscr FT}(\zeta)=h_K(\zeta)$ $\forall\,\zeta\in\mathbf C^l;$
{\rm2)} $(\mathscr FT)(z)$ be a function of completely regular growth in $\mathbf C^l$ in the sense of weak convergence in $D'(\mathbf C^l)$.}
Here $\mathscr L^*_{\mathscr FT}(\zeta)=\varlimsup_{z\to\zeta}\, \varlimsup_{r\to\infty }\frac{\ln |(\mathscr FT)(rz)|}{r}$ is the regularized radial indicator of the entire function $(\mathscr FT)(z)$, and $h_K(\zeta)$ is the support function of the compact set $K$.
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