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An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators
The local maximal inequality for the Schr\"{o}dinger operators of order
\a>1 is shown to be bounded from to for any .
This improves the previous result of Sj\"{o}lin on the regularity of solutions
to fractional order Schr\"{o}dinger equations. Our method is inspired by
Bourgain's argument in case of \a=2. The extension from \a=2 to general
\a>1 confronts 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 at our disposal and
analyzing all the possibilities in using the separateness of the segments to
obtain the analogous bilinear estimates. To compensate the absence of
Galilean invariance, we resort to Taylor's expansion for the phase function.
The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to
dominate the solution of fractional order Schr\"{o}dinger equations.Comment: Pages47, 3figures. To appear in Studia Mathematic
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