Analytic holonomicity of real C${}^{\mathrm{e}\mathrm{x}\mathrm{p}}$-class distributions

We introduce a notion of distributions on ${\mathbb{R}}^n$, called distributions of C${}^{\mathrm{e}\mathrm{x}\mathrm{p}}$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C${}^{\mathrm{e}\mathrm{x}\mathrm{p}}$-class functions. We prove that the framework of C${}^{\mathrm{e}\mathrm{x}\mathrm{p}}$-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${}^{\mathrm{e}\mathrm{x}\mathrm{p}}$-class.