A note on some variations of the maximal inequality for the fractional Schr\"odinger equation

TL;DR

This paper summarizes recent advances in the fractional Schrödinger equation, focusing on pointwise convergence along tangential curves and sets of lines, highlighting new results and optimal regularity conditions.

Contribution

It introduces new results on convergence along tangential curves and sets of lines, and discusses counterexamples indicating optimal regularity conditions.

Findings

Convergence along exponential order tangential curves.

Counterexamples showing regularity optimality.

New insights into pointwise convergence for fractional Schrödinger solutions.

Abstract

The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schr\"odinger equations; convergence along a tangential line and along a set of lines, as exhibiting some new results in each setting. For the former case, we make a simple observation on a path along a tangential curve of exponential order. We discuss counterexamples for the latter case that show some of the known smooth regularities are essentially optimal.