A structure theorem for fundamental solutions of analytic multipliers in Rⁿ

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier Op(p) of order d>0, real-analytic near p-1(0), has a fundamental solution μ0. We give an integral representation of μ0 in terms of the resolutions supplied by Hironaka’s theorem. This μ0 is weakly approximated in Htloc(Rn) for t<d-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.