The variation operator of differences of averages over lacunary sequences maps $H^1_w(\mathbb{R})$ to $L^1_w(\mathbb{R})$

Let $f$ be a locally integrable function defined on $\mathbb{R}$, and $(n_k)$ be a lacunary sequence. Define
$$A_nf(x)=\frac{1}{n}\int_0^nf(x-t) dt,$$
and let
$$\mathcal{V}_{\rho}f(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^{\rho}\right)^{1/\rho}.$$
Suppose that $w\in A_p$, $1\leq p<\infty$, and $\rho\geq 2$. Then, there exists a positive constant $C$ such that
$$\|\mathcal{V}_{\rho}f\|_{L^1_w}\leq C\|f\|_{H^1_w}$$
for all $f\in H^1_w(\mathbb{R})$.