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

TL;DR

This paper studies the convergence of Schr"odinger solutions along highly tangential curves, identifying the Sobolev regularity needed for almost everywhere convergence and establishing critical regularity thresholds.

Contribution

It determines the minimal Sobolev regularity for convergence along certain tangential curves, extending understanding of Schr"odinger evolution in irregular geometric settings.

Findings

Critical regularity for convergence is s= max{(1-2α)/2, n/(2(n+1))}.

Maximal estimates are analyzed for solutions along α-H"older curves.

Results apply to model curves of the form (t^{α_1},...,t^{α_n}).

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 $s=\max\left\{\frac{1-2\alpha}{2},\frac{n}{2(n+1)}\right\}.$