Bilinear pseudodifferential operators on modulation spaces
The boundedness of bilinear integral operators, in particular pseudo-differential operators with not necessarily smooth symbols, on products of modulation spaces is investigated. It is proved that if the kernel of the integral operator belongs to a modulation space \({\mathcal M}^1_{\Omega_s}({\mathbb R}^{3d})\), \(\Omega_s\) is a weight function of power type \(s\), then the operator is bounded from \({\mathcal M}^{p_1,p_2}_{\nu}({\mathbb R}^{d})\times {\mathcal M}^{q_1,q_2}_{\nu}({\mathbb R}^{d})\) into \({\mathcal M}^{r_1,r_2}_{\nu}({\mathbb R}^{d})\) if \(\frac{1}{r_1}=\frac{1}{p_1}+\frac{1}{q_1}\) and \(\nu\) is an \(s\)-moderated weight. A similar result holds for bilinear pseudodifferential operators \(T_\sigma\) with symbol \(\sigma\) belonging to a modulation space \({\mathcal M}^1_{\Omega^B_s}({\mathbb R}^{3d})\). In particular, if \(\sigma\) is an element of the Feichtinger algebra \({\mathcal M}^1({\mathbb R}^{3d})\), then \(T_\sigma\) is bounded from \(L^{p}({\mathbb R}^{d})\times L^{q}({\mathbb R}^{d})\) into \( L^{r}({\mathbb R}^{d})\) for \(2\leq p,q\leq \infty\), \(1\leq r\leq 2\) and \(\frac{1}{r}=\frac{1}{p}+\frac{1}{q}\). The key idea of the approach is the discretization of the modulation spaces by Gabor frames. In that way, the problem is reduced to the problem of infinite matrices acting on sequence spaces.
scientific article; zbMATH DE number 2133426