Square function characterizations of real and ergodic $H^1$ spaces

Let $(n_k)$ be a lacunary sequence with no non-trivial common divisor and $f\in L^1(\mathbb{R})$. Define the square function
$$Sf(x)=\left(\sum_{k=1}^{\infty}\left|\frac{1}{n_{k+1}}\int_{0}^{n_{k+1}}f(x-t) dt-\frac{1}{n_k}\int_{0}^{n_k}f(x-t) dt\right|^2\right)^{1/2}.$$
We show that there exist constants $A$ and $B$ such that
$$\|f\|_{L^1(\mathbb{R})}\leq A\|Sf\|_{L^1(\mathbb{R})} \text{and} \|f\|_{H^1(\mathbb{R})}\leq B\|Sf\|_{L^1(\mathbb{R})}$$
for all $f\in L^1(\mathbb{R})$.\Let $(X,\mathscr{B} ,\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\mathscr{B} ,\mu )$ a totally $\sigma$-finite measure space. Let us consider the usual ergodic averages
$$A_nf(x)=\frac{1}{n}\sum_{i=0}^{n-1}f(\tau^ix),$$
and define the ergodic square function
$$\mathcal{S}f(x)=\left(\sum_{k=1}^{\infty}\left|A_{n_{k+1}}f(x)-A_{n_k}f(x)\right|^2\right)^{1/2}.$$
We also show that
$$\|f\|_{L^1(X)}\leq A\|\mathcal{S}f\|_{L^1(X)} \text{and} \|f\|_{H^1(X)}\leq B\|\mathcal{S}f\|_{L^1(X)}$$
for all $f\in L^1(X)$, where $H^1(X)$ denotes the ergodic Hardy space. Combining these results with the author's earlier results we also conclude that the square function $Sf$ characterizes the real Hardy space $H^1(\mathbb{R})$, and the ergodic square function $\mathcal{S}f$ characterizes the ergodic Hardy space $H^1(X)$ when the sequence $(n_k)$ is lacunary.