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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Rogers, K., Villarroya, P. Sharp estimates for maximal operators associated to the wave equation. Ark Mat 46, 143–151 (2008). https://doi.org/10.1007/s11512-007-0063-8
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DOI: https://doi.org/10.1007/s11512-007-0063-8