Sharp convergence for sequences of Schr\"{o}dinger means and related generalizations

TL;DR

This paper establishes sharp almost everywhere convergence results for sequences of Schrödinger means, including fractional and nonelliptic cases, for functions in Sobolev spaces, advancing understanding of Schrödinger evolution behavior.

Contribution

It provides the first sharp convergence results for decreasing sequences of Schrödinger means and extends these results to fractional and nonelliptic Schrödinger operators.

Findings

Convergence results are sharp up to endpoints.

Results apply to fractional Schrödinger means.

Applicable to nonelliptic Schrödinger means.

Abstract

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schr\"{o}dinger means $e^{it_{n}\Delta}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schr\"{o}dinger means and nonelliptic Schr\"{o}dinger means.