Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I

Let $\omega ,\omega_0$ be appropriate weight functions and $q\in [1,\infty ]$. We introduce the wave-front set, $\WF_{\mathscr FL^q_{(\omega)}}(f)$ of $f\in \mathscr S'$ with respect to weighted Fourier Lebesgue space $\mathscr FL^q_{(\omega)}$. We prove that usual mapping properties for pseudo-differential operators $\op (a)$ with symbols $a$ in $S^{(\omega _0)}_{\rho, 0}$ hold for such wave-front sets. Especially we prove \WF_{\mathscr FL^q_{(\omega /\omega_0)}}(\op (a)f)\subseteq \WF_{\mathscr FL^q_{(\omega)}}(f) \subseteq \WF_{\mathscr FL^q_{(\omega /\omega_0)}}(\op (a)f)\ttbigcup \Char (a). %% Here $\Char (a)$ is the set of characteristic points of $a$.