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

Abstract

Let $Θ\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $β$, and let $γ(x,t,θ)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(τ,θ)\in[0,1)\timesΘ}\left|e^{iτΔ}f(γ(\,\cdot\,,τ,θ))\right|\right\|_{L^2(B^n_1)}\lesssim_s\|f\|_{H^s(\mathbb{R}^n)},$$ for every $s>\tfrac{n(β+1)}{2(n+1)}$, in the range $0\leβ\le n-1$. In the range $n-1<β\leq n$ we prove the threshold $s>\min\big\lbrace\fracβ2 + \frac 1{2(n+1)},\frac n2\big\rbrace$. 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.