A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves

Let $\mathrm{\Theta }\subset {\mathbb{R}}^n$, $n\ge 1$ be compact with upper Minkowski dimension $\beta$, and let $\gamma (x,t,\theta )$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $${\parallel \underset{(\tau ,\theta )\in [0,1)\times \mathrm{\Theta }}{\sup }\mid e^{i\tau \mathrm{\Delta }}f(\gamma (\cdot ,\tau ,\theta ))\mid \parallel }_{L^2(B_1^n)}{\lesssim }_s\mathrm{\parallel }f{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)},$$ for every $s>\frac{n(\beta +1)}{2(n+1)}$, in the range $0\le \beta \le n-1$. In the range $n-1<\beta \le n$ we prove the threshold $s>\min \{\frac{\beta }{2}+\frac{1}{2(n+1)},\frac{n}{2}\}$. The proof relies on a fractal $L^2$ estimate of Du–Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.