An improved maximal inequality for 2D fractional order Schrodinger operators

作者：Miao Changxing^*; Yang Jianwei; Zheng Jiqiang

来源：Studia Mathematica, 2015, 230(2): 121-165.

DOI：10.4064/sm8190-12-2015

• local maximal inequality
• multilinear restriction estimate
• induction on scales
• localization argument
• oscillatory integral operator

The local maximal operator for the Schrodinger operators of order alpha > 1 is shown to be bounded from H-s (R-2) to L-2 for any s > 3/8. This improves the previous result of Sjolin on the regularity of solutions to fractional order Schrodinger equations. Our method is inspired by Bourgain's argument in the case of alpha = 2. The extension from alpha = 2 to general alpha > 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 Schrodinger equations.

• 出版日期2015
• 单位北京应用物理与计算数学研究所; 北京大学