Asymptotic estimates for unimodular Fourier multipliers onmodulation spaces

Jiecheng Chen; Dashan Fan; Lijing Sun

doi:10.3934/dcds.2012.32.467

Article Contents

2012, Volume 32, Issue 2: 467-485. Doi: 10.3934/dcds.2012.32.467

This issue Previous Article On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps Next Article Pressures for asymptotically sub-additive potentials under a mistake function

Asymptotic estimates for unimodular Fourier multipliers on modulation spaces

1.
Department of Mathematics, Zhejiang Normal University, 321004 Jinhua
2.
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201

Received: July 31, 2010

Revised: May 31, 2011

Published: January 2012

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Abstract

Recently, it has been shown that the unimodular Fourier multipliers $e^{it|\Delta |^{\frac{\alpha }{2}}}$ are bounded on all modulation spaces. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multipliers $e^{it|\Delta |^{\frac{\alpha }{2}}}$ on the modulation spaces. As applications, we give the grow-up rates of the solutions for the Cauchy problems for the free Schrödinger equation, the wave equation and the Airy equation with the initial data in a modulation space. We also obtain a quantitative form about the solution to the Cauchy problem of the nonlinear dispersive equations.

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Mathematics Subject Classification: Primary: 42B15, 42B35; Secondary: 42C15.

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