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 \( n\ge 2, \) \(\underset{t\rightarrow 0}{\lim }\ \,e^{it\Delta }f\left( x+\gamma (t)\right) = f(x) \) 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.