Oscillation inequalities on real and ergodic $H^1$ spaces

Abstract

Let $(x_n)$ be a sequence and $\rho\geq 1$. For a fixed sequences $n_1<n_2<n_3<\dots$, and $M$ define the oscillation operators $$\mathcal{O}_\rho (x_n)=\left(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in M}}\left|x_m-x_{n_k}\right|^\rho\right)^{1/\rho}.$$ 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 sequences $(n_k)$ and $M$ are lacunary. Then we prove the following results for $\rho\geq 2$: (i) Define $\phi_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$ 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 $A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(\tau^kx)$ 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.