Global estimates of maximal operators generated by dispersive equations

Abstract

Let $Tf(x, t) = e^{itφ(D)f}$ be the solution of a general dispersive equation with phase function φ and initial data $f$ in a Sobolev space. When the phase φ has a suitable growth condition and the initial data f has an angular regularity, we prove global and local L^p estimates for maximal operators generated by T. Here we do not assume the radial symmetry for the initial data. These results reveal some sufficient conditions on initial data for the boundedness of maximal operators in contrast to the negative results of [28]. We also prove a weighted L^2 maximal estimate, which is an extension of [19] to nonradial initial data.