The support theorem for the complex Radon transform of distributions

The complex Radon transform $\widehat{F}$ of a rapidly decreasing distribution $F\in {\mathcal{O}}_C^{\mathrm{\prime }}({\mathbb{C}}^n)$ is considered. A compact set $K\subset {\mathbb{C}}^n$ is called linearly convex if the set $ $\mathbb{C}$^n \setminus K$isaunionofcomplexhyperplanes\mathrm{.}Let$\hat K$ denote the set of complex hyperplanes which meet $K$. The main result of the paper establishes the conditions on a linearly convex compact $K$ under which the support theorem for the complex Radon transform is true: from the relation $\hbox{supp}(\hat F)\subset\hat K$itfollowsthat$F\in\mathscr{O}^{\prime}_C($\mathbb{C}$^n)$ is compactly supported and $\text{supp}(F)\subset K$.