Wave-front sets in non-quasianalytic setting for Fourier Lebesgue and modulation spaces

We define and study wave-front sets for weighted Fourier-Lebesgue spaces when the weights are moderate with respect to the associated functions for general sequences $\{M_p\}$ which satisfy Komatsu’s conditions $(M.1)-(M.3{)}^{\mathrm{\prime }}$. In particular, when $\{M_p\}$ is the Gevrey sequence ($M_p=p{!}^s$, $s>1$) we recover some previously observed results. Furthermore, we consider wave-front sets for modulation spaces in the same setting, and prove the invariance property related to the Fourier-Lebesgue type wave-front sets.