Beyond Gevrey regularity

TL;DR

This paper introduces new classes of ultradifferentiable functions less regular than Gevrey, studies their microlocal properties, and explores how these functions behave under differential operators, expanding the understanding of ultradifferentiability.

Contribution

It defines and analyzes function classes that are less regular than Gevrey, introduces new technical tools, and investigates their microlocal properties under differential operators.

Findings

New classes of ultradifferentiable functions less regular than Gevrey.

Microlocal analysis results relating wave front sets and differential operators.

Introduction of admissibility conditions and enumeration as analytical tools.

Abstract

We define and study classes of smooth functions which are less regular than Gevrey functions. To that end we introduce two-parameter dependent sequences which do not satisfy Komatsu's condition (M.2)', which implies stability under differential operators within the spaces of ultradifferentiable functions. Our classes therefore have particular behavior under the action of differentiable operators. On a more advanced level, we study microlocal properties and prove that $${\rm WF}_{0,\infty}(P(D)u)\subseteq {\rm WF}_{0,\infty}(u)\subseteq {\rm WF}_{0,\infty}(P(D)u) \cup {\rm Char}(P),$$ where $u$ is a Schwartz distribution, $P(D)$ is a partial differential operator with constant coefficients and ${\rm WF}_{0,\infty}$ is the wave front set described in terms of new regularity conditions. For the analysis we introduce particular admissibility condition for sequences of cut-off functions, and a new technical tool called enumeration.