Hokkaido Mathematical Journal

Cho Yonggeun, Shim Yongsun,

Global estimates of maximal operators generated by dispersive equations.

Hokkaido Mathematical Journal, 37 (2008) pp.773-794

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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.

MSC(Primary)42B25
MSC(Secondary)42A45
Uncontrolled Keywordsdispersive equation, maximal operator, phase function, angular regularity.