Inequalities for the differences of averages on H1 spaces

Abstract

Let $(x_n)$ be a sequence and $\{c_k\}\in \ell^\infty (\mathbb{Z})$ such that $\|c_k\|_{\ell^\infty}\leq 1$. Define
$$\mathcal{G}(x_n)=\sup_j\left|\sum_{k=0}^j c_k(x_{n_{k+1}}-x_{n_k})\right|.$$
Let now $(X,\beta ,\mu ,\tau )$ be an ergodic, measure preserving dynamical system with $(X,\beta ,\mu )$ a totally $\sigma$-finite measure space. Suppose that the sequence $(n_k)$ is lacunary. Then we prove the following results:
(i) Define $\phi_n(x)=\dfrac{1}{n}\chi_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that
$$\|\mathcal{G}(\phi_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$$
for all $f\in H^1(\mathbb{R})$,
(ii) Let
$$A_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(\tau^kx)$$
be the usual ergodic averages in ergodic theory. Then
$$\|\mathcal{G}(A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$$
for all $f\in H^1(X)$,
(iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{G}(A_nf)$ is integrable.