A note on the Schr\"odinger smoothing effect

Abstract

The Kato-Yajima smoothing estimate is a smoothing weighted $L^2$ estimate with a singular power weight for the Schr\"odinger propagator. The weight has been generalized relatively recently to Morrey-Campanato weights. In this paper we make this generalization more sharp in terms of the so-called Kerman-Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted $L^2$ space. Obtained results are also extended to the fractional Schr\"odinger propagator.