Global estimates of maximal operators generated by dispersive equations

Let $Tf(x,t) = e^{2\pi it\phi(D)}f$ be the solution of of the general dispersive equation with the phase function $\phi$ and initial data $f$ in the Schwartz class. In case that the phase $\phi$ has a suitable growth rate at the infinity and the origin and $f$ is a finite linear combination of radial and spherical harmonic functions, we have global $L^p$ estimates of maximal operator defined by taking the supremum w.r.t. $t$. In particular, we obtain a global estimate at the end point left open.