Variation of the dyadic maximal function

Abstract

We prove that for the dyadic maximal operator M and every locally integrable functionWe also prove this for the local dyadic maximal operator.

Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\mathbb{R}^d)$

“…He originally proved it for the uncentered operator M α , but he observed shortly after that almost the same proof also works for the centered operator M α , see [24,Remark 1.3]. The proof in [24] is based on the corresponding bound for the dyadic maximal operator in the non-fractional case α = 0 in [25]. Other interesting related results in the context of fractional maximal functions have recently been proven in [4,9,12,21,22].…”