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Transactions of the American Mathematical Society

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A strong type $ (2,\,2)$ estimate for a maximal operator associated to the Schrödinger equation


Authors: Carlos E. Kenig and Alberto Ruiz
Journal: Trans. Amer. Math. Soc. 280 (1983), 239-246
MSC: Primary 42A45; Secondary 35J10
DOI: https://doi.org/10.1090/S0002-9947-1983-0712258-4
MathSciNet review: 712258
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Abstract: Let $ {T^{\ast} }f(x) = \sup_{t > 0}\vert{T_t}f(x)\vert$, where $ ({T_t}f)\hat{\empty}(\xi) = {e^{it\vert\xi \vert^2}}\hat f(\xi)/\vert\xi {\vert^{1/4}}$. We show that, given any finite interval $ I$, $ \int_I {\vert{T^{\ast} }f{\vert^2}\;dx \leqslant {C_I}\int_{\mathbf{R}} {\vert f(x){\vert^2}\;dx} } $, and that the above inequality is false with $ 2$ replaced by any $ p < 2$. This maximal operator is related to solutions of the Schrödinger equation.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0712258-4
Article copyright: © Copyright 1983 American Mathematical Society