Beyond Gevrey regularity

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.