An improved maximal inequality for 2D fractional order Schrödinger operators
Volume 230 / 2015
Abstract
The local maximal operator for the Schrödinger operators of order $\alpha \gt 1$ is shown to be bounded from $H^s(\mathbb {R}^2)$ to $L^2$ for any $s \gt 3/8$. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of $\alpha =2$. The extension from $\alpha =2$ to general $\alpha \gt 1$ faces 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 and analyzing all the possibilities in using the separation of the segments to obtain the analogous bilinear $L^2$-estimates. To compensate for the absence of Galilean invariance, we resort to Taylor’s expansion for the phase function. The Bourgain–Guth inequality (2011) is also generalized to dominate the solution of fractional order Schrödinger equations.
