Superposition and propagation of singularities for extended Gevrey regularity

TL;DR

This paper extends the theory of ultradifferentiable functions, including Gevrey classes, showing their stability under superposition and inverse operations, and studies singularity propagation for PDEs with extended Gevrey regularity.

Contribution

It introduces parameter-dependent sequences defining ultradifferentiable function families, proves their closure under superposition, and extends singularity propagation results to PDEs with weaker coefficient regularity.

Findings

Families are closed under superposition

Inverse closedness of these families

Propagation of singularities under weaker regularity assumptions

Abstract

We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients satisfy the extended Gevrey regularity. To that aim we introduce appropriate wave front sets and derive a theorem on propagation of singularities. This extends related known results in the sense that weaker assumptions on the regularity of the coefficients are imposed.