An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators

TL;DR

This paper establishes an improved maximal inequality for 2D fractional Schr"odinger operators, extending regularity results to a broader class of fractional orders by overcoming key analytical obstacles.

Contribution

It introduces a new reduction lemma and bilinear estimates to extend Bourgain's approach from the classical case to fractional orders greater than one.

Findings

Boundedness from $H^s(\mathbb{R}^2)$ to $L^2$ for $s>3/8$

Extension of maximal inequality to fractional order Schr"odinger operators

Reconstruction of Bourgain-Guth inequality for fractional cases

Abstract

The local maximal inequality for the Schr\"{o}dinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to fractional order Schr\"{o}dinger equations. Our method is inspired by Bourgain's argument in case of $\a=2$. The extension from $\a=2$ to general $\a>1$ confronts three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear $L^2-$estimates. To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to dominate the solution of fractional order Schr\"{o}dinger equations.