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

Abstract

Consider the general dispersive equation defined by

(∗)\{\begin{array}{l}\mathrm{i}{{\displaystyle \partial }}_{t}u+\varphi (\sqrt{-\mathrm{\Delta }})u=\mathrm{0},(x,t)\in {\mathrm{ℝ}}^n\times \mathrm{ℝ},\\ u(x,\mathrm{0})=f(x),f\in \ell ({\mathrm{ℝ}}^n),\end{array}

where φ(\sqrt{-\mathrm{\Delta }}) 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 {{\displaystyle S}}_{\varphi }^{*} defined by {{\displaystyle S}}_{\varphi }^{*}f(x) = sup_0<t<1|S_t,φf(x)|, where S_t,φfis the solution of equation (∗). These estimates imply the a.e. convergence of the solution of equation (∗).

Key words： Dispersive equation; maximal operator; local estimate; global estimate

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