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.

The index groups for which weighted Young inequalities hold in both the continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, and we also obtain the weak

Article Hausdorff operator on real Hardy spaces was published on November 1, 2014 in the journal Analysis (volume 34, issue 4).