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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 $$S_n (f) = n^{-1} \sum^n_{i = 1} f \circ T^i$$.

##### MSC:
 60G15 Gaussian processes 60F15 Strong limit theorems
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##### References:
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