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Some connections between ergodic theory and harmonic analysis. (English) Zbl 0752.28006

Almost everywhere convergence II, Proc. 2nd Int. Conf., Evanston/IL (USA) 1989, 17-40 (1991).
[For the entire collection see Zbl 0741.00045.]
Recall that the rotated ergodic Hilbert transform, or, more briefly, the helical transform (of an integrable function \(f\) on a measure space \(X\) with respect to a measure-preserving transformation \(T\)) is defined as \[ H_ \theta f(x)=\lim_{n\to\infty}\sum^ n_{k=- n}{e^{ik\theta}f(T^ k x)\over k}. \] From the authors’ Introduction: “We discuss the relationships among strong \((p,p)\) inequalities for the maximal helical transform and its variants and the supremum of the partial sums of Fourier series. For the case \(p=2\), the Carleson-Hunt estimate is equivalent to the \(L^ 2\) boundedness of the maximal helical transform. For \(p\neq 2\), many implications among these maximal inequalities still hold, but certain ones fail. Weak (1,1) also fails for the double maximal helical transform, as does the Wiener-Wintner property. The \(L^ 2\) boundedness of the double maximal helical transform extends to measure-preserving flows as well as to higher- dimensional actions. ... We give a proof of the partial Fourier coefficients lemma in restricted form by means of kernel estimates and Fourier transforms in the style of Gaposhkin and Bourgain. Then Kolmogorov’s theorem follows quickly, by the same path that leads from the full partial Fourier coefficients lemma to the Carleson-Hunt theorem; we give this proof in detail. Finally, we show how these same easy estimates allow one to prove a quadratic-variation version of the Hardy- Littlewood maximal lemma on \(l^ 2\) similar to Bourgain’s estimate along the sequence of squares and consequently a quadratic-variation strengthening of Birkhoff’s pointwise ergodic theorem ...”.

MSC:

28D05 Measure-preserving transformations
43A50 Convergence of Fourier series and of inverse transforms
42B25 Maximal functions, Littlewood-Paley theory
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