Micro-Local Analysis with Fourier Lebesgue Spaces. Part I

Abstract

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, $\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f)$ of $f\in \mathcal{S}'$ with respect to weighted Fourier Lebesgue space $\mathcal{F}L^{q}_{(\omega )}$. 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 that

$$\begin{array}[b]{lll}\mathrm{WF}_{\mathcal{F}L^q_{(\omega /\omega _0)}}(\operatorname {Op}(a)f)&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega )}}(f)\\[6pt]&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega/\omega _0)}}(\operatorname {Op}(a)f)\cup \operatorname {Char}(a).\end{array}$$

(*)

Here Char (a) is the set of characteristic points of a.