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Sharp estimates for maximal operators associated to the wave equation

Authors:	Keith M Rogers Paco Villarroya

Institution:	(1) Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain;(2) University of California, Los Angeles, CA 90095–1555, USA

Abstract:	The wave equation, ∂_tt u=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).

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