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

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