On the variation of the Hardy-Littlewood maximal function

By Ondvrej Kurka. 2012

101 citations recorded by the source corpus.

We show that a function $ f $ of bounded variation satisfies $$ \Var Mf \leq C \Var f $$ where $ Mf $ is the centered Hardy-Littlewood maximal function of $ f $. Consequently, the operator $ f \mapsto (Mf)' $ is bounded from $ W^{1,1}(R) $ to $ L^{1}(R) $. This answers a question of Hajlasz and Onninen in the one-dimensional case.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8cca827d45785c849511d971b10adfb28daf1d4e

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Derivative bounds for fractional maximal functions (2015, background): “It is also shown in [14] that if f ∈ W (1,1)(R), then Mf is weakly differentiable and (1.”
The variation of the maximal function of a radial function (2017, background): “5 . The author was supported by the Academy of Finland, project no. 292797. 1 2 HANNESLUIRO proved by Tanaka [Ta]. For M c the problem turns out to be very complicated also when n = 1. However, Kurka [Ku] managed to show that the answer is positive also in this case. The goal of this paper is to dev”
Endpoint Sobolev and BV continuity for maximal operators, II (2017, background): “A complete solution was reached only in dimension d = 1 in [2, 17, 29] and partial progress on the general case d > 1 was obtained by Haj lasz and Malý [11] and, more recently, by Saari [26] and Luiro [22].”