Micro-local analysis in Fourier Lebesgue and modulation spaces: part II

Abstract

We consider different types of (local) products f1 f2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$P(x,D)f = G(x,J_k f),$$ with appropriate polynomials P and G, where Jk denotes the k-jet of f. If the solution locally belongs to appropriate weighted Fourier Lebesgue space \({\fancyscript{F}L^q_{(\omega )} (\mathbf{R}^d)}\) and P is non-characteristic at (x0, ξ0), then we prove that \({(x_0,\xi_0)\not \in {\rm WF}_{\fancyscript{F}L^q_{(\widetilde {\omega })}} (f)}\), where \({\widetilde{\omega }}\) depends on ω, P and G.