Variational Inequalities For The Differences Of Averages Over Lacunary Sequences

Abstract

Let $f$ be a locally integrable function defined on $\mathbb{R}$, and let $(n_k)$ be a lacunary sequence. Define the operator $A_{n_k}$ by $$A_{n_k}f(x)=\frac{1}{n_k}{\int }_0^{n_k}f(x-t)dt.$$ We prove various types of new inequalities for the variation operator $${\mathcal{V}}_{s}f(x)={\left(\sum _{k=1}^{\infty }|A_{n_k}f(x)-A_{n_{k-1}}f(x){|}^s\right)}^{1/s}$$ when $2\le s<\infty$.