A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$

Abstract

Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $\mu_0$. We give an integral representation of $\mu_0$ in terms of the resolutions supplied by Hironaka's theorem. This $\mu_0$ is weakly approximated in $H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.