On the Pointwise Convergence of Solutions to the Schr\"odinger Equation Along Certain Highly Tangential Curves

Abstract

We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schr\"odinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal estimates for expressions of the form $e^{it\Delta}f(x+\gamma(t))$ over specific $\alpha$-H\"older curves $\gamma$ and initial data $f\in H^s(\mathbb{R}^n)$. For the model family $\gamma(t)=(t^{\alpha_1},\ldots,t^{\alpha_n})$, where $\alpha=\min_j \alpha_j$, we show that the critical regularity is $s=\max\left\{\frac{1-2\alpha}{2},\frac{n}{2(n+1)}\right\}.$