Estimates for Unimodular Multipliers on Modulation Hardy Spaces

It is known that the unimodular Fourier multipliers eit|Δ|α/2, α>0, are bounded on all modulation spaces Mp,qs for 1≤p,q≤∞. We extend such boundedness to the case of all 0<p,q≤∞ and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study the Mp1,qs→Mp2,qs boundedness for the operator eit|Δ|α/2, for the case 0<α≤2 and α≠1. Finally, we investigate the boundedness of the operator eit|Δ|α/2 for α>0 and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroup eit|Δ|α/2