Analytic holonomicity of real C${}^{\exp }$-class distributions

Abstract

We introduce a notion of distributions on ${\mathbb{R}}^n$, called distributions of C${}^{\exp }$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C${}^{\exp }$-class functions. We prove that the framework of C${}^{\exp }$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C${}^{\exp }$-class.

Proposition 3.13 gets a new, corrected proof in the new section 4 on algebraic holonomicity. To appear in Trans. Amer. Math. Soc