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    A note on the cone restriction conjecture in the cylindrically symmetric case

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    In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The first is based on a dyadic restriction estimate, while the second follows from a strengthening version of the Hausdorff-Young inequality and the H\"older inequality in the Lorentz spaces.Comment: 9 pages, no figures. Referee's suggestions and comments incorporated; to appear the Proceedings of the AM

    On localization of the Schr\"odinger maximal operator

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    In \cite{Lee:2006:schrod-converg}, when the spatial variable xx is localized, Lee observed that the Schr\"odinger maximal operator eitΞ”f(x)e^{it\Delta}f(x) enjoys certain localization property in tt for frequency localized functions. In this note, we give an alternative proof of this observation by using the method of stationary phase, and then include two applications: the first is on is on the equivalence of the local and the global Schr\"odinger maximal inequalities; secondly the local Schr\"odinger maximal inequality holds for f∈H3/8+f\in H^{3/8+}, which implies that eitΞ”fe^{it\Delta}f converges to ff almost everywhere if f∈H3/8+f\in H^{3/8+}. These results are not new. In this note we would like to explore them from a slightly different perspective, where the analysis of the stationary phase plays an important role.Comment: 14 pages, no figure. Note
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