Maximal estimate for solutions to a class of dispersive equation with radial initial value

Abstract

Consider the general dispersive equation defined by

$$\left\{ {\begin{array}{*{20}{c}} {i{\partial _t}u + \phi \left( {\sqrt { - \Delta } } \right)u = 0,}&{\left( {x,t} \right) \in {\mathbb{R}^n} \times \mathbb{R},} \\ {u\left( {x,0} \right) = f\left( x \right),}&{f \in S\left( {{\mathbb{R}^n}} \right),} \end{array}} \right.$$

((*))

where ϕ(√−Δ) is a pseudo-differential operator with symbol ϕ(|ξ|). In this paper, for ϕ satisfying suitable growth conditions and the radial initial data f in Sobolev space, we give the local and global L ^q estimate for the maximal operator S _ϕ * defined by S _ϕ *f(x) = sup_0<t<1 |S _t,ϕ f(x)|, where S _t,ϕ f is the solution of equation (*). These estimates imply the a.e. convergence of the solution of equation (*).