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.

**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

Email:
kiselev@math.uchicago.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04681-X

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