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

Abstract

Let $(x_n)$ be a sequence and $\rho \ge 1$. For a fixed sequences $n_1<n_2<n_3<\dots$, and $M$ define the oscillation operators $${\mathcal{O}}_{\rho }(x_n)={\left(\sum _{k=1}^{\infty }\underset{\begin{array}{c}{n}_k\le m<n_{k+1}\\ m\in M\end{array}}{\sup }{\left|x_m-x_{n_k}\right|}^{\rho }\right)}^{1/\rho }.$$ Let $(X,\mathcal{B},\mu ,\tau )$ be a dynamical system with $(X,\mathcal{B},\mu )$ 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 $\rho \ge 2$: (i) Define ${\varphi }_n(x)=\frac{1}{n}{\chi }_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that $\Vert {\mathcal{O}}_{\rho }({\varphi }_n\ast f){\Vert }_{L^1(\mathbb{R})}\le C\Vert f{\Vert }_{H^1(\mathbb{R})}$ for all $f\in H^1(\mathbb{R})$. (ii) Let $A_{n}f(x)=\frac{1}{n}{\sum }_{k=1}^{n}f({\tau }^{k}x)$ be the usual ergodic averages in ergodic theory. Then $\Vert {\mathcal{O}}_{\rho }(A_{n}f){\Vert }_{L^1(X)}\le C\Vert f{\Vert }_{H^1(X)}$ for all $f\in H^1(X)$. (iii) If $[f(x)\log (x){]}^{+}$ is integrable, then ${\mathcal{O}}_{\rho }(A_{n}f)$ is integrable.