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

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier \(\text {Op}(p)\) $\text{Op}(p)$ of order \(d>0\) $d>0$ , real-analytic near \(p^{-1}(0)\) $p^{-1}(0)$ , has a fundamental solution \(\mu _0\) ${\mu }_0$ . We give an integral representation of \(\mu _0\) ${\mu }_0$ in terms of the resolutions supplied by Hironaka’s theorem. This \(\mu _0\) ${\mu }_0$ is weakly approximated in \(H^t_{\text {loc}}({\mathbb {R}}^n)\) $H_{\text{loc}}^t({\mathbb{R}}^n)$ for \(t<d-\frac{n}{2}\) $t<d-\frac{n}{2}$ 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.