The Variation Operator of Differences of Averages over Lacunary Sequences Maps $$H_{w}^{1}(\mathbb{R})$$ to $$L_{w}^{1}(\mathbb{R})$$

Abstract

Let \(f\) be a locally integrable function defined on \(\mathbb{R}\), and \(({{n}_{k}})\) be a lacunary sequence. Define \({{A}_{n}}f(x) = \frac{1}{n}\int_0^n f(x - t)dt,\) and let \({{\mathcal{V}}_{\rho }}f(x) = {{\left( {\sum\limits_{k = 1}^\infty {{{\left| {{{A}_{{{{n}_{k}}}}}f(x) - {{A}_{{{{n}_{{k - 1}}}}}}f(x)} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.\) Suppose that \(w \in {{A}_{p}}\), \(1 \leqslant p < \infty \), and \(\rho \geqslant 2\). Then, there exists a positive constant \(C\) such that \({{\left\| {{{\mathcal{V}}_{\rho }}f} \right\|}_{{L_{w}^{1}}}} \leqslant C{{\left\| f \right\|}_{{H_{w}^{1}}}}\) for all \(f \in H_{w}^{1}(\mathbb{R})\).