  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.
