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 \({{\left\| {{{c}_{k}}} \right\|}_{{{{\ell }^{\infty }}}}} \leqslant 1\). Define \(\mathcal{G}({{x}_{n}}) = \mathop {\sup }\limits_j \left| {\sum\limits_{k = 0}^j \,{{c}_{k}}\left( {{{x}_{{{{n}_{{k + 1}}}}}} - {{x}_{{{{n}_{k}}}}}} \right)} \right|.\) 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: 1. Define \({{\phi }_{n}}(x) = \frac{1}{n}{{\chi }_{{[0,n]}}}(x)\) on \(\mathbb{R}\). Then there exists a constant \(C > 0\) such that \({{\left\| {\mathcal{G}({{\phi }_{n}} * f)} \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}},\) for all \(f \in {{H}^{1}}(\mathbb{R})\). 2. Let \({{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{k = 0}^{n - 1} \,f({{\tau }^{k}}x),\) be the usual ergodic averages in ergodic theory. Then \({{\left\| {\mathcal{G}({{A}_{n}}f)} \right\|}_{{{{L}^{1}}(X)}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(X)}}},\) for all \(f \in {{H}^{1}}(X)\). 3. If \({{[f(x)\log (x)]}^{ + }}\) is integrable, then \(\mathcal{G}({{A}_{n}}f)\) is integrable.