A priori estimates for Schrödinger type multipliers

Source abstract

We present an elementary proof of two a priori estimates for Schrödinger type multipliers on the circle. The first is an $L^4 - L^2$ inequality of Bourgain, while the second is a new $L^6 - L^{3/2}$ inequality. Estimates of this type are useful for the study of the Cauchy problem for Schr\"odinger type equations. The proofs are based on a counting argument and standard real and harmonic analysis techniques.