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An interpolation theorem related to the a.e. convergence of integral operators
Author(s):
Alexander
Kiselev
Journal:
Proc. Amer. Math. Soc.
127
(1999),
1781-1788.
MSC (1991):
Primary 42C15, 43A50;
Secondary 34L40
Posted:
February 11, 1999
MathSciNet review:
1476143
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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:
- 1.
- J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin Heidelberg 1976. MR 58:2349
- 2.
- L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 33:7774
- 3.
- A. Garsia, Topics in a.e. Convergence, Markham Pub. Company, Chicago 1970. MR 41:5869
- 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.
- E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton 1971. MR 46:4102
- 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:
10.1090/S0002-9939-99-04681-X
PII:
S 0002-9939(99)04681-X
Received by editor(s):
June 4, 1997
Received by editor(s) in revised form:
September 17, 1997
Posted:
February 11, 1999
Communicated by:
Christopher D. Sogge
Copyright of article:
Copyright
1999,
American Mathematical Society
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