Endpoint maximal and smoothing estimates for Schroedinger equations

TL;DR

This paper establishes endpoint maximal and smoothing estimates for solutions to fractional Schrödinger equations, extending classical fixed-time bounds to maximal functions and sharp space-time estimates in certain Lebesgue spaces.

Contribution

It proves new endpoint $L^p$ maximal inequalities and sharp local space-time estimates for fractional Schrödinger equations, strengthening previous fixed-time results.

Findings

Proved endpoint $L^p$ maximal inequalities for $u(t)$ with initial data in Sobolev spaces.

Established sharp local space-time $L^p$ estimates for the dispersive equation.

Extended fixed-time estimates to maximal functions and space-time bounds.

Abstract

For $\alpha >1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-\Delta)^{\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.