Boundedness of Maximal Operators of Schr\"odinger Type with Complex Time

TL;DR

This paper improves the understanding of the boundedness of Schr"odinger maximal operators with complex time, establishing sharp Sobolev space conditions for their boundedness and implications for pointwise convergence of solutions to dispersive PDEs.

Contribution

It determines the exact Sobolev space threshold for boundedness of Schr"odinger maximal operators with complex time, extending results to a broader class of dispersive PDE solution operators.

Findings

Sharp bounds for Schr"odinger maximal operator with complex time.

Boundedness results for a class of dispersive PDE solution operators.

Almost everywhere convergence of PDE solutions to initial data.

Abstract

Results of P. Sj\"olin and F. Soria on the Schr\"odinger maximal operator with complex-valued time are improved by determining up to the endpoint the sharp $s \geq 0$ for which boundedness from the Sobolev space $H^s(\mathbb{R})$ into $L^2(\mathbb{R})$ occurs. Bounds are established for not only the Schr\"odinger maximal operator, but further for a general class of maximal operators corresponding to solution operators for certain dispersive PDEs. As a consequence of additional bounds on these maximal operators from $H^s(\mathbb{R})$ into $L^2([-1, 1])$, sharp results on the pointwise almost everywhere convergence of the solutions of these PDEs to their initial data are determined.