# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.333125

Arkiv for Matematik, 46, (1), 143-151, 2006.10
The wave equation, ∂_{$tt$}$u$=Δ$u$, in ℝ^{$n$+1}, considered with initial data$u$($x$,0)=$f$∈$H$^{$s$}(ℝ^{$n$}) and$u$’($x$,0)=0, has a solution which we denote by $\frac{1}{2}(e^{it\sqrt{-\Delta}}f+e^{-it\sqrt{-\Delta}}f)$ . We give almost sharp conditions under which $\sup_{0<t<1}|e^{\pm it\sqrt{-\Delta}}f|$ and $\sup_{t\in\mathbb{R}}|e^{\pm it\sqrt{-\Delta}}f|$ are bounded from$H$^{$s$}(ℝ^{$n$}) to$L$^{$q$}(ℝ^{$n$}).
@inproceedings{keith2006sharp,
title={Sharp estimates for maximal operators associated to the wave equation},
author={Keith M. Rogers, and Paco Villarroya},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203622107378934},
booktitle={Arkiv for Matematik},
volume={46},
number={1},
pages={143-151},
year={2006},
}

Keith M. Rogers, and Paco Villarroya. Sharp estimates for maximal operators associated to the wave equation. 2006. Vol. 46. In Arkiv for Matematik. pp.143-151. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203622107378934.