Given a dynamical system
satisfies that
is bounded where the space
$\Vert f\Vert_{\left[ {L \log_3 L (μ)} \right]} = ∈t_0^1 \frac{\sup\limits_{t≤q y}tf_μ^*(t)}{y} \log_3 \frac 1y dy < ∞,$
with
$R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$
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