On smoothing estimates in modulation spaces and the nonlinear Schr\"odinger equation with slowly decaying initial data

We show new local $L^p$-smoothing estimates for the Schr\"odinger equation with initial data 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 $L^p(\mathbb{R}^d)$ for $d \geq 1$ and $p > 2$, which was previously known for $d \geq 2$. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schr\"odinger equation on the line. Lastly, we show $\ell^2$-decoupling inequalities for variable-coefficient versions of elliptic and non-elliptic Schr\"odinger phase functions.