Inequalities For Variation Operator

Let $f$ be a measurable function defined on $\mathbb{R}$. For each $n\in\mathbb{Z}$ define the operator $A_n$ by $$A_nf(x)=\frac{1}{2^n}\int_x^{x+2^n}f(y)\, dy.$$ Consider the variation operator $$\mathcal{V}f(x)=\left(\sum_{n=-\infty}^\infty|A_nf(x)-A_{n-1}f(x)|^s\right)^{1/s}$$ for $2\leq s<\infty$. It has been proved in \cite{jkw1} that $\mathcal{V}$ is of strong type $(p,p)$ for $1<p<\infty$ and is of weak type $(1,1)$, it maps $L^\infty$ to BMO. We first provide a completely different proofs for these known results and in addition we prove that $\mathcal{V}$ maps $H^1$ to $L^1$. Furthermore, we prove that it satisfies vector-valued weighted strong type and weak type inequalities. As a special case it follows that $\mathcal{V}$ satisfies weighted strong type and weak type inequalities.