Global well-posedness of critical nonlinear Schrödinger equations below $L^2$

Abstract

The global well-posedness on the Cauchy problem of nonlinearSchrödinger equations (NLS) is studied for a class of criticalnonlinearity below $L^2$ in small data setting. We considerHartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the criticalSobolev index $s_c$ is negative, it is rather difficult to analyzethe nonlinear term. To overcome the difficulty we combine weightedStrichartz estimates in polar coordinates with new Duhamel estimatesinvolving angular regularity.