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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
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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  • [C] Lennart Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 5–45. MR 576038
  • [DK] Björn E. J. Dahlberg and Carlos E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 205–209. MR 654188
  • [KS] A. N. Kolmogorov and G. Seliverstov, Sur la convergence de séries de Fourier, C. R. Acad. Sci. Paris Sér. I Math. 178 (1925), 303-305.
  • [NRS] Alexander Nagel, Walter Rudin, and Joel H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math. (2) 116 (1982), no. 2, 331–360. MR 672838,
  • [S] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [Z] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587

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Article copyright: © Copyright 1983 American Mathematical Society