A Note on Non-tangential Convergence for Schr\"{o}dinger Operators

TL;DR

This paper investigates non-tangential convergence of Schrödinger operators along restricted curves, relating approach region dimensions to initial data regularity, and provides bounds for the Schrödinger maximal function in certain function spaces.

Contribution

It establishes non-tangential convergence results for Schrödinger operators along restricted curves and derives bounds for the Schrödinger maximal function based on initial data regularity.

Findings

Derived an upper bound for p where the maximal function is bounded from H^s to L^p.

Established a relationship between approach region dimension and initial data regularity for convergence.

Proved non-tangential convergence results for Schrödinger operators along restricted curves.

Abstract

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for $p$ such that the Schr\"{o}dinger maximal function is bounded from $H^{s}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for any $s > \frac{n}{2(n+1)}$.