Smoothing of commutators for a H\"ormander class of bilinear pseudodifferential operators

Commutators of bilinear pseudodifferential operators with symbols in the H\"ormander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calder\'on-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.Comment: 16 page

Local well-posedness of nonlinear dispersive equations on modulation spaces

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces $M{p, 1}_{0,s}$.Comment: 11 page

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schr\"odinger equation. As an example, we also show that the energy-critical cubic nonlinear Schr\"odinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.Comment: 20 pages. Introduction modified. References updated. To appear in Excursions in Harmonic Analysi

Anisotropic Classes of Inhomogeneous Pseudodifferential Symbols

We introduce a class of pseudodifferential operators in the anisotropic setting induced by an expansive dilation A which generalizes the classical isotropic class Smγ,δ of inhomogeneous symbols. We extend a well-known L 2-boundedness result to the anisotropic class S0δ,δ(A), 0 ≤ δ \u3c 1. As a consequence, we deduce that operators with symbols in the anisotropic class S01,0(A) are bounded on L p spaces, 1 \u3c p \u3c ∞