Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

The quartile operator and pointwise convergence of Walsh series

Author(s): Christoph Thiele
Journal: Trans. Amer. Math. Soc. 352 (2000), 5745-5766.
MSC (2000): Primary 42A50, 42A20, 42C10
Posted: August 3, 2000
MathSciNet review: 1695038
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Abstract | References | Similar articles | Additional information

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

, with

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.

References:

1.: J. Bergh and J. Löfström, Interpolation spaces, an introduction, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 58:2349
2.: P. Billard, Sur la convergence presque partout des series de Fourier Walsh des fonctions de l'espace , Studia Math. 28 (1967), 363-388. MR 36:599
3.: A. P. Calderon and A. Zygmund, A note on the interpolation of linear operators, Studia Math. 12 (1951), 194-204. MR 13:754e
4.: L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 33:7774
5.: C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551-571. MR 49:5676
6.: R. Hunt, Almost everywhere convergence of Walsh-Fourier series of -functions, Actes, Congrès Internat. Math., vol. 2, 1970, pp. 655-661. MR 58:23330
7.: S. Janson, On interpolation of multi-linear operators, (Proceedings Lund, 1986), Springer Lecture Notes in Math., vol. 1302, 1988. MR 89g:46114
8.: M. Lacey and C. Thiele, estimates on the bilinear Hilbert transform for $2<p<\infty$ , Ann. of Math. (2) 146 (1997), 693-724. MR 99b:42014
9.: M. Lacey and C. Thiele, Bounds for the bilinear Hilbert transform on , Proc. Nat. Acad. Sci. 94 (1997), 33-35.
10.: R.E.A.C. Paley, A remarkable series of orthogonal functions (I), Proc. London Math. Soc. 34 (1932), 241-264.
11.: P. Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series, Ark. Mat. 7 (1968), 551-570. MR 39:3222
12.: E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. MR 44:7280
13.: C. M. Thiele and L. F. Villemoes, A fast algorithm for adapted time frequency tilings, Appl. Comput. Harmon. Anal. 3 (1996), 91-99. MR 98g:65134
14.: J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 45 (1923), 5-24.

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