On smoothing estimates in modulation spaces and the nonlinear Schrodinger equation with slowly decaying initial data

We show new local L-p-smoothing estimates for the Schrodinger equation in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and L-p-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in L-p(R-d) for d >= 1 and p >= 2, which was previously known for d > 2. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrodinger equation on the line. Lastly, we show l(2)-decoupling inequalities for variable coefficient versions of elliptic and non-elliptic Schrodinger phase functions. (C) 2021 Elsevier Inc. All rights reserved.