An improved maximal inequality for 2D fractional order Schrödinger operators

Changxing Miao¹ ; Jianwei Yang² ; Jiqiang Zheng³

¹Institute of Applied Physics and Computational Mathematics 100088 Beijing, China
²Beijing International Center for Mathematical Research Peking University 100871 Beijing, China
³Université Nice Sophia-Antipolis 06108 Nice Cedex 02, France

Studia Mathematica, Tome 230 (2015) no. 2, pp. 121-165

Résumé

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.

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DOI : 10.4064/sm8190-12-2015

Keywords: local maximal operator schr dinger operators order alpha shown bounded mathbb improves previous result lin regularity solutions fractional order schr dinger equations method inspired bourgain argument alpha extension alpha general alpha faces three essential obstacles lack lee reduction lemma absence algebraic structure symbol inapplicable galilean transformation deduction main theorem get around these difficulties establishing reduction lemma analyzing possibilities using separation segments obtain analogous bilinear estimates compensate absence galilean invariance resort taylor expansion phase function bourgain guth inequality generalized dominate solution fractional order schr dinger equations