Fourier transform and rigidity of certain distributions

TL;DR

This paper investigates the rigidity properties of certain tempered distributions supported on finite unions of affine subspaces in a finite dimensional vector space over a local field, focusing on their Fourier transforms.

Contribution

It establishes a rigidity property for distributions supported on finite unions of affine subspaces and their Fourier transforms in the setting of local fields.

Findings

Distributions supported on finite unions of affine subspaces exhibit rigidity.

The Fourier transform preserves certain support properties of these distributions.

The results extend understanding of distribution support and Fourier analysis in local field contexts.

Abstract

Let $E$ be a finite dimensional vector space over a local field, and $F$ be its dual. For a closed subset $X$ of $E$, and $Y$ of $F$, consider the space $D^{-\xi}(E;X,Y)$ of tempered distributions on $E$ whose support are contained in $X$ and support of whose Fourier transform are contained in $Y$. We show that $D^{-\xi}(E;X,Y)$ possesses a certain rigidity property, for $X$, $Y$ which are some finite unions of affine subspaces.