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

Abstract

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