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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The quartile operator and pointwise convergence of Walsh series


Author: Christoph Thiele
Journal: Trans. Amer. Math. Soc. 352 (2000), 5745-5766
MSC (2000): Primary 42A50, 42A20, 42C10
Published electronically: August 3, 2000
MathSciNet review: 1695038
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Abstract:

The bilinear Hilbert transform is given by

\begin{displaymath}H(f,g)(x):= p.v. \int f(x-t)g(x+t)\frac{dt}{t}. \end{displaymath}

It satisfies estimates of the type

\begin{displaymath}\Vert H(f,g)\Vert _r\le C(s,t)\Vert f\Vert _s \Vert g\Vert _t.\end{displaymath}

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in $L^p[0,1]$, with $1<p$ converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.


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Additional Information

Christoph Thiele
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095-1555
Email: thiele@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02577-0
PII: S 0002-9947(00)02577-0
Keywords: Bilinear Hilbert transform, convergence of Fourier series, Walsh functions
Received by editor(s): September 25, 1997
Published electronically: August 3, 2000
Article copyright: © Copyright 2000 American Mathematical Society