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Coupling of the GB set property for ergodic averages. (English) Zbl 0851.60037
This article pursues a connection first made by J. Bourgain [Isr. J. Math. 63, No. 1, 79-97 (1988; Zbl 0677.60042)] between dynamical systems and the notion of GB-sets within a Hilbert space, a notion of compactness motivated by the theory of Gaussian processes. More specifically, for a Hilbert space \(H\), there exists an isometry to a linear space (with covariance as inner-product) of mean-0 Gaussian random variables \((Z_a, a \in H)\), the isonormal process defined on a canonical probability space. A subset \(A \subset H\) is called a GB-set if \(E (\sup_{a \in A} Z_a) < \infty\). Bourgain’s result had said that if \(\{S_n, n \geq 1\}\) is a family of bounded operators on \(L^2 (\Omega, {\mathcal B}, \mu)\) which commute with \(T\), where \((\Omega, {\mathcal B}, \mu,T)\) is an ergodic dynamical system, and if \(p \geq 2\) is such that \(\mu (\{x \in \Omega : \sup_n |S_n (f)(x) |< \infty\}) = 1\) for all \(f \in L^p (\mu)\), then for each such \(f\), \(\{S_n (f), n \geq 1\}\) is a GB-set. This article uses techniques of Bourgain and E. Stein to extend this result, to say that if the operators \(S_n\) additionally satisfy an \(L^p\) maximal inequality for some \(p \geq 2\), and if \(A \subset L^2 (\mu)\) is nonempty with a finite metric-entropy integral, then \(\{S_n(f), f \in A, n \geq 1\}\) is again a GB-set. The main application of the result is to ergodic-average operators \(S_n\) defined by \(S_n (f) = n^{-1} \sum^n_{i = 1} f \circ T^i\).

60G15 Gaussian processes
60F15 Strong limit theorems
Full Text: DOI
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