Estimates for unimodular multipliers on modulation hardy spaces

It is known that the unimodular fourier multipliers e i t | δ | α / 2,α > 0,are bounded on all modulation spaces m p,q s 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 m p 1,q s → m p 2,q s boundedness for the operator e i t | δ | α / 2,for the case 0 < α ≤ 2 and α ≠ 1. finally,we investigate the boundedness of the operator e i t | 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 e i t | δ | α / 2. © 2013 jiecheng chen et al.