# Inequalities For Variation Operator

25 Jan 2020 Demir Sakin

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 $2\leq s<\infty$... (read more)

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