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On the almost everywhere existence of the ergodic Hilbert transform


Authors: Diego Gallardo and F. J. Martín-Reyes
Journal: Proc. Amer. Math. Soc. 105 (1989), 636-643
MSC: Primary 28D05; Secondary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1989-0939964-1
MathSciNet review: 939964
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Abstract: Let $ (X,\mathfrak{M},\mu )$ be a finite measure space, $ T$ an invertible measure-preserving transformation and $ \upsilon $ a positive measurable function. For $ p = 1$, we prove that the ergodic Hubert transform $ Hf(X) = {\text{li}}{{\text{m}}_{n \to \infty }}\sum\nolimits_{i = - n}^n {'f({T^i}x)/i} $ exists a.e. for every $ f$ in $ {L^1}(\upsilon d\mu )$ if and only if $ {\text{in}}{{\text{f}}_{i \geq 0}}\upsilon ({T^i}x) > 0$ a.e. We also solve the problem for $ 1 < p \leq 2$. In this case the condition is $ {\text{su}}{{\text{p}}_{k \geq 1}}{k^{ - 1}}\sum\nolimits_{i - 0}^{k - 1} {{\upsilon ^{ - 1/(p - 1)}}} ({T^i}x) < \infty $ a.e. If the transformation $ T$ is ergodic, the characterizing conditions become that $ 1/\upsilon \in {L^\infty }$ and $ {\upsilon ^{ - 1/(p - 1)}} \in {L^1}(\mu )$, respectively. These characterizations, together with some recent results, give, for $ 1 \leq p \leq 2$, that $ Hf(x)$ exists a.e. for every $ f$ in $ {L^p}(\upsilon d\mu )$ if and only if the sequence of the Césàro-averages $ {k^{ - 1}}(f(x) + f(Tx) + \ldots f({T^{k - 1}}x))$ converge a.e. for every $ f$ in $ {L^p}(\upsilon d\mu )$. This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on $ {L^2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0939964-1
Keywords: Almost everywhere convergence, Cesàro-averages, ergodic Hilbert transform, ergodic maximal Hilbert transform, ergodic maximal operator, measure preserving transformations, weights
Article copyright: © Copyright 1989 American Mathematical Society