Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications

The primary objective in this paper is to give an answer to an open question posed by J. A. Barcel\'o, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on $s\geq0$ and $p\geq1$ for which the following Strichartz estimate with time-dependent weights $w$ in Morrey-Campanato type classes $\mathfrak{L}^{2s+2,p}_2$ holds: \begin{equation}\label{absset} \|e^{it\Delta}f\|_{L_{x,t}^2(w(x,t))}\leq C\|w\|_{\mathfrak{L}^{2s+2,p}_2}^{1/2}\|f\|_{\dot{H}^s}. \end{equation} Beyond the case $s\geq0$, we further ask how much regularity we can expect on this setting. But interestingly, it turns out that this estimate is false whenever $s<0$, which shows that the smoothing effect cannot occur in this time-dependent setting and the dispersion in the Schr\"odinger flow $e^{it\Delta}$ is not strong enough to have the effect. This naturally leads us to consider the possibility of having the effect at best in higher-order versions of this estimate with $e^{-it(-\Delta)^{\gamma/2}}$ ($\gamma>2$) whose dispersion is more strong. We do obtain a smoothing effect exactly for these higher-order versions. In fact, we will obtain the estimates where $\gamma\geq1$ in a unified manner and also their corresponding inhomogeneous estimates to give applications to the global well-posedness for Schr\"odinger and wave equations with time-dependent perturbations. This is our secondary objective in this paper.