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Uniform estimates for families of singular integrals and double Fourier series

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Elena Prestini
Elena Prestini
Affiliation:
Instituto Matematico, Universita di Milano, Via C. Saldini, 50 20133 Milano, Italy
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Abstract

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It is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

MSC classification

Secondary: 42B05: Fourier series and coefficients
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1986

References

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