Strichartz and local smoothing estimates for the fractional Schr\"odinger equations over fractal time

Abstract

We obtain Strichartz-type estimates for the fractional Schr\"odinger operator $f \mapsto e^{it(-\Delta)^{\gamma/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-\Delta)^{\gamma/2}} f \|_{L_t^q(\mathrm{d}\mu; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $\mu$ is a measure satisfying an $\alpha$-dimensional growth condition. In addition, we establish related inhomogeneous estimates and $L^2$ local smoothing estimates. A surprising feature of our work is that, despite dealing with rough fractal sets, we extend the known estimates for the fractional Schr\"odinger operators in a natural way, precisely consistent with the associated fractal dimensions.