Remarks on Fourier multipliers and applications to the wave equation

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [A. Bényi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, preprint, April 2007 (available at ArXiv:0704.0833v1)]. The same arguments may apply to obtain local wellposedness for the NLKG equation.