Localwell-posedness for the nonlinear Schrödinger equation in modulation spaces Ms p;q(Rd)

We show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation on modulation spaces Msp;q(Rd) for d 2 N, 1 p; q 1 and s > d 1 􀀀 1 q for q > 1 or s 0 for q = 1. This improves [4, Theorem 1.1] by Bényi and Okoudjou where only the case q = 1 is considered. Our result is based on the algebra property of modulation spaces with indices as above for which we give an elementary proof via a new Hölder-like inequality for modulation spaces.