Sharp estimates for maximal operators associated to the wave equation

Abstract

The wave equation, ∂_ttu=Δu, in ℝⁿ⁺¹, considered with initial data u(x,0)=f∈H^s(ℝⁿ) 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(ℝⁿ) to L^q(ℝⁿ).