Pressures for asymptotically sub-additive potentials under a mistake function
On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps
February 2012, 32(2): 467-485. doi: 10.3934/dcds.2012.32.467
Asymptotic estimates for unimodular Fourier multipliers on modulation spaces
Jiecheng Chen 1, , Dashan Fan 2, and Lijing Sun 2,
Department of Mathematics, Zhejiang Normal University, 321004 Jinhua, China
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, United States, United States
Received August 2010 Revised June 2011 Published September 2011
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.
Keywords: Airy equation., Unimodular multipliers, Schrödinger equation, modulation spaces, wave equation.
Mathematics Subject Classification: Primary: 42B15, 42B35; Secondary: 42C1.
Citation: Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467
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Jiecheng Chen Dashan Fan Lijing Sun