Estimates for Unimodular Multipliers on Modulation Hardy Spaces

Abstract EN

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

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.