On the Almost Everywhere Existence of the Ergodic Hilbert Transform

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 Césà ro-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 ${L^2}$.