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AbstractConsider the general dispersive equation defined by (*)$$\left\{ {\begin{array}{*{20}{c}} {i{\partial _t}u + \phi \left( {\sqrt { - \Delta } } \right)u = 0,}&{\left( {x,t} \right) \in {\mathbb{R}^n} \times \mathbb{R},} \\… Click to show full abstract
AbstractConsider the general dispersive equation defined by (*)$$\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.$${i∂tu+ϕ(−Δ)u=0,(x,t)∈ℝn×ℝ,u(x,0)=f(x),f∈S(ℝn), 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 Lq estimate for the maximal operator Sϕ* defined by Sϕ*f(x) = sup0
Journal Title: Frontiers of Mathematics in China
Year Published: 2017
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