Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [C] L. Carleson, Some analytical problems related to statistical mechanics, Euclidean Harmonic Analysis, Lecture Notes in Math., Vol. 779, Springer-Verlag, Berlin and New York, 1979, pp. 5-45. MR 576038 (82j:82005)
  • [DK] B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation (Proc. of Italo-American Symposium in Harmonic Analysis, University of Minnesota), Lecture Notes in Math., Vol. 908, Springer-Verlag, Berlin and New York, 1982, pp. 205-208. MR 654188 (83f:35023)
  • [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] A. Nagel, W. Rudin and J. Shapiro, Tangential boundary behavior of function in Dirichlet type spaces, preprint. MR 672838 (84a:31002)
  • [S] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1971. MR 0290095 (44:7280)
  • [Z] A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, London and New York, 1968. MR 0236587 (38:4882)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A45, 35J10

Retrieve articles in all journals with MSC: 42A45, 35J10

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society