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 \(\text {Op}(p)\) 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 \(H^t_{\text {loc}}({\mathbb {R}}^n)\) 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.