Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• Artykuł - szczegóły

Studia Mathematica

2015 | 230 | 2 | 121-165

An improved maximal inequality for 2D fractional order Schrödinger operators

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.

121-165

wydano
2015

Twórcy

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