标题： 分数阶薛定谔方程在分形时间上的Strichartz和局部平滑估计 显示英文标题

标题： Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time

摘要： 我们得到了分数薛定谔算子$f \mapsto e^{it(-\Delta)^{\gamma/2}} f$在具有分形维数的时间集$E$上的斯特里哈茨型估计。为了获得捕捉$E$分形性质的估计，我们采用了类似于阿苏阿德维数的概念，如有界阿苏阿德特征和阿苏阿德谱。我们还证明了估计$$ \| e^{it(-\Delta)^{\gamma/2}} f \|_{L_t^q(\mathrm{d}\mu; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$，其中$\mu$是满足$\alpha$维增长条件的测度。此外，我们建立了相关的非齐次估计和$L^2$局部平滑估计。我们工作的显著特点是，尽管处理的是粗糙的分形集，但我们以自然的方式扩展了已知的分数薛定谔算子的估计，精确地与相关的分形维数一致。 显示英文摘要

摘要： 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.