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 $(M.1) - (M.3)'$. 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.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1806.0440