The support theorem for the complex Radon transform of distributions

Abstract

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

8 pages