Oscillation inequalities on real and ergodic H1 spaces. II

Let $(x_n)$ be a sequence and $\rho\geq 1$. For two fixed sequences $n_1<n_2<n_3<\dots$, and $M$ define the oscillation operator $$\mathcal{O}_\rho (x_n)=(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\m\in M}}\left|x_m-x_{n_k}\right|^\rho)^{1/\rho}.$$ Let $(X,\mathscr{B} ,\mu , \tau)$ be a dynamical system with $(X,\mathscr{B} ,\mu )$ a probability space and $\tau$ a measurable, invertible, measure preserving point transformation from $X$ to itself. Suppose that the sequence $(n_k)$ is a lacunary, and $M$ is any sequence of positive real numbers such that there exists an $\ell \in \mathbb{R}$ satisfying $\#\{m\in M:n_k\leq m<n_{k+1}\}\leq \ell$ for all $k\in \mathbb{N}$ to obtain the above mentioned results, where $\#$ denotes cardinality. Then the following results are proved in this article for $\rho\geq 2$. (i) Define $\phi_n(x)=\dfrac{1}{n}\chi_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that $$\|\mathcal{O}_\rho (\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$$ for all $f\in H^1(\mathbb{R})$. (ii) Let $\displaystyle A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(\tau^kx)$ be the usual ergodic averages in ergodic theory. Then $$\|\mathcal{O}_\rho (A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$$ for all $f\in H^1(X)$. (iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{O}_\rho (A_nf)$ is integrable. In the author's previously published article (S. Demir "Oscillation inequalities on real and ergodic $H^1$ spaces", Russ. Math. 67 (3), 42-52 (2023)) the above results have been obtained when both $(n_k)$ and $M$ are lacunary. Thus the results of this work extend those results to a nonlacunary sequence $M$ with a more general growth condition.