Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An interpolation theorem related to
the a.e. convergence of integral operators

Author: Alexander Kiselev
Journal: Proc. Amer. Math. Soc. 127 (1999), 1781-1788
MSC (1991): Primary 42C15, 43A50; Secondary 34L40
Published electronically: February 11, 1999
MathSciNet review: 1476143
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm estimates.

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

  • 1. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • 2. Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 0199631
  • 3. Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, vol. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
  • 4. A. Kiselev, Stability of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators, to appear in Duke Math. J.
  • 5. A. Kiselev, Stability of the absolutely continuous spectrum of Jacobi matrices under slowly decaying perturbations, in preparation.
  • 6. R.E.A.C. Paley, Some theorems on orthonormal functions, Studia Math. 3 (1931), 226-245.
  • 7. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • 8. A. Zygmund, A remark on Fourier transforms, Proc. Camb. Phil. Soc. 32 (1936), 321-327.

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Additional Information

Alexander Kiselev
Affiliation: Mathematical Sciences Research Institute, 5 1000 Centennial Drive, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637-1546

Received by editor(s): June 4, 1997
Received by editor(s) in revised form: September 17, 1997
Published electronically: February 11, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society