The return times property for the tail on logarithm-type spaces

Given a dynamical system $(Ω,Σ,μ, τ)$ with $μ$ a non-atomic probability measure and $τ$ an invertible measure preserving ergodic transformation, we prove that the maximal operator, considered by I. Assani, Z. Buczolich and R. D. Mauldin in 2005,

${N^*}f\left( x \right) = \mathop {\sup }\limits_{\alpha > 0} \alpha \# \left\{ {k \ge 1:\frac{{\left| {f\left( {{\tau ^k}x} \right)} \right|}}{k} > \alpha } \right\}$

satisfies that

${N^*}:\left[ {L \log_3 L (μ)} \right] \longrightarrow L^{1, ∞}(μ)$

is bounded where the space $\left[ {L \log_3 L (μ)} \right]$ is defined by the condition

$\Vert f\Vert_{\left[ {L \log_3 L (μ)} \right]} = ∈t_0^1 \frac{\sup\limits_{t≤q y}tf_μ^*(t)}{y} \log_3 \frac 1y dy < ∞,$

with $\log_3 x = 1+\log_+\log_+\log_+ x$ and $f^*_μ$ the decreasing rearrangement of $f$ with respect to $μ$. This space is near $L \log_3 L (μ)$, which is the optimal Orlicz space on which such boundedness can hold. As a consequence, the space $\left[ {L \log_3 L (μ)} \right]$ satisfies the Return Times Property for the Tail; that is, for every $f∈\left[ {L \log_3 L (μ)} \right]$, there exists a set $X_0$ so that $μ(X_0) = 1$ and, for all $x_0∈ X_0$, all dynamical systems $(Y,\mathcal{C},ν, S)$ and all $g∈ L^1(ν)$, the sequence

$R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$