A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum

Abstract

In this short note, we give an easy proof of the following result: for $n\ge 2,$ $\underset{t\to 0}{\lim }{e}^{it\Delta }f\left(x+\gamma (t)\right)=f(x)$ almost everywhere whenever $\gamma$ is an $\alpha -$Hölder curve with $\frac{1}{2}\le \alpha \le 1$ and $f\in H^s({\mathbb{R}}^n)$, with $s>\frac{n}{2(n+1)}$. This is the optimal range of regularity up to the endpoint.

arXiv comment6 pages