In this paper we give new estimates for the solution to the Schrödinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces.

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We study the propagation of singularities for semi-linear wave equations u = F (u, Du) satisfying the null condition, and improve an earlier result of ours. Microlocal Sobolev regularity in H s (U ) ∩ H r ml propagates along null bicharacteristics, when a suitable condition is imposed on s and r. We provide here optimal conditions on s and r, and improve existing results to include values up to n/2 < s r < 3s − n when the nonlinearity F (u, Du) satisfies the null condition.

We introduce the wave front set W F'f•q by using the wave packet transform. This is another characterization of the Fourier-Lebesgue type wave front set WF.rL~• We apply this to the propagation of singularities for the wave equation.