Abstract
Let $\mathrm{M}$ be the uncentered Hardy–Littlewood maximal operator, or the dyadic maximal operator, and let $d\geq 1$. We prove that for a set $E\subset\mathbb{R}^d$ of finite perimeter, the bound $\operatorname{var}\mathrm{M} 1\_E\leq C\_d \operatorname{var} 1\_E$ holds. We also prove this for the local maximal operator.
Highlights
IntroductionB where the supremum is taken over all open balls B ⊂ Rd that contain x
The uncentered Hardy-Littlewood maximal function of a nonnegative locally integrable function f is given by Mf (x) = sup B∋x L(B) fB where the supremum is taken over all open balls B ⊂ Rd that contain x
There is the Hardy-Littlewood maximal operator, where the supremum is taken only over those balls that are centered in x, or the dyadic maximal operator which maximizes over dyadic cubes instead of balls
Summary
B where the supremum is taken over all open balls B ⊂ Rd that contain x. In one dimension for L1(R) the gradient bound has already been proven in [26] by Tanaka for the uncentered maximal function, and later in [21] by Kurka for the centered Hardy-Littlewood maximal function The latter proof turned out to be much more complicated. In this paper we prove that for M being the dyadic or the uncentered Hardy Littlewood maximal operator and E ⊂ Rd being a set with finite perimeter, we have var M1E ≤ Cd var 1E This answers the question of Hajlasz and Onninen in a special case, and is the first truly higher dimensional result for p = 1 to the best of our knowledge. Kalle Vaisala Foundation of the Finnish Academy of Science and Letters
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