Proves embeddings, convergence, nonlinear smoothing, and spatial decay for the generalized Ostrovsky equation using high-low frequency techniques, maximal functions, and Strichartz estimates via Stein interpolation.
On localization of the Schr\"odinger maximal operator
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abstract
In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schr\"odinger maximal operator $e^{it\Delta}f(x)$ enjoys certain localization property in $t$ 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 $f\in H^{3/8+}$, which implies that $e^{it\Delta}f$ converges to $f$ almost everywhere if $f\in H^{3/8+}$. 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.
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2026 1verdicts
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Spatial decay and nonlinear smoothing of the generalized Ostrovsky equation
Proves embeddings, convergence, nonlinear smoothing, and spatial decay for the generalized Ostrovsky equation using high-low frequency techniques, maximal functions, and Strichartz estimates via Stein interpolation.