  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.

Classical Analysis and ODEs