On the pullback on spaces of distributions with prescribed microregularity with respect to a general Banach space

Abstract

We consider the space ${\mathcal{D}}_L^{\prime E}(U)$ of all distributions on the open set $U$ whose wave front set measured with respect to a Banach space $E$ lies in a closed conic subset $L$ of $U\times ({\mathbb{R}}^n\backslash \{0\})$. The Banach space $E$ only satisfies mild technical assumptions. We show that the pullback by a diffeomorphism $f:O\to U$ is a topological isomorphism $f^{\ast }:{\mathcal{D}}_L^{\prime E}(U)\to {\mathcal{D}}_{f^{\ast }L}^{\prime E}(O)$. In the case when $E$ is the $L^p$-Sobolev space $W^{r,p}({\mathbb{R}}^n)$ of order $r\in \mathbb{R}$, we study the pullback by a smooth map $f:O\to U$ of constant rank.