The support theorem for the complex Radon transform of distributions

The complex Radon transform $\hat{F}$ of a rapidly decreasing distribution $F\in\mathcal{O}'_C(\mathbb{C}^n)$ is considered. A compact set $K\subset\mathbb{C}^n $ is called linearly convex if the set $\mathbb{C}^n\backslash K$ is a union of complex hyperplanes. 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 supp $(\hat{F})\subset\hat{K}$ it follows that $F\in\mathcal{O}'_C(\mathbb{C}^n)$ is compactly supported and supp $(F)\subset K$.