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

TL;DR

This paper uses resolution of singularities to represent and analyze fundamental solutions of elliptic multipliers in real-analytic settings, providing explicit formulas in symmetric cases and approximations in Sobolev spaces.

Contribution

It introduces a novel integral representation of fundamental solutions for elliptic multipliers using Hironaka's resolution, with explicit formulas in symmetric cases and Sobolev space approximations.

Findings

Integral representation of fundamental solutions derived

Explicit formulas obtained in symmetric cases

Fundamental solutions approximated in Sobolev spaces

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.