Variation of the dyadic maximal function

2020/06/02 by Weigt, Julian · 1 citation
#26B30 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.01853

Abstract

We prove that for the dyadic maximal operator \mathrm M and every locally integrable function f∈ L¹_loc(\mathbb R^d) with bounded variation, also \mathrm M f is locally integrable and \mathopvar\mathrm M f≤ C_d\mathopvar f for any dimension d≥1. It means that if f∈ L¹_loc(\mathbb R^d) is a function whose gradient is a finite measure then so is ∇ \mathrm M f and ‖∇ \mathrm M f‖_{L¹(\mathbb R^d)}≤ C_d‖∇ f‖_{L¹(\mathbb R^d)}. We also prove this for the local dyadic maximal operator.

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• A Vitali-type lemma for the boundary

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