Variation of the dyadic maximal function

Abstract

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

updated notation in section 3, fixed typos, improved formulations