Variation of the dyadic maximal function

TL;DR

This paper proves that the dyadic maximal operator preserves bounded variation of functions in multiple dimensions, with a bound on the variation of the maximal function proportional to that of the original function.

Contribution

It establishes that the dyadic maximal operator maps functions of bounded variation to functions of bounded variation, with a dimension-dependent bound on the variation.

Findings

The variation of the dyadic maximal function is controlled by the variation of the original function.

The result holds for all dimensions $d \\geq 1$.

The proof extends to the local dyadic maximal operator.

Abstract

We prove that for the dyadic maximal operator $\mathrm M$ and every locally integrable function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ with bounded variation, also $\mathrm M f$ is locally integrable and $\mathop{\mathrm{var}}\mathrm M f\leq C_d\mathop{\mathrm{var}} f$ for any dimension $d\geq1$. It means that if $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \mathrm M f$ and $\|\nabla \mathrm M f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.