An improved maximal inequality for 2D fractional order Schrödinger operators

• Autorzy: Changxing Miao , Jianwei Yang , Jiqiang Zheng
• Języki publikacji: EN

Abstrakty

EN

The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from $H^{s}(ℝ²)$ to L² for any s > 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 α = 2. The extension from α = 2 to general α > 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 L²-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.

Kategorie tematyczne

• 35Q41: Time-dependent Schr\"odinger equations, Dirac equations
• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 230
• Numer: 2
• Strony: 121-165

Daty

wydano

2015

Twórcy

autor

Changxing Miao

• Institute of Applied Physics and Computational Mathematics, 100088 Beijing, China

autor

Jianwei Yang

• Beijing International Center for Mathematical Research, Peking University, 100871 Beijing, China

autor

Jiqiang Zheng

• Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France

Identyfikatory

DOI

10.4064/sm8190-12-2015