Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 109-133

ON THE VARIATION OF THE HARDY–LITTLEWOOD MAXIMAL FUNCTION

Ondrej Kurka

Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis
Sokolovská 83, 186 75 Prague 8, Czech Republic; kurka.ondrej 'at' seznam.cz

Abstract. We show that a function f : R → R of bounded variation satisfies

Var Mf ≤ 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 L¹(R). This answers a question of Hajlasz and Onninen in the one-dimensional case.

2010 Mathematics Subject Classification: Primary 42B25, 46E35.

Key words: Hardy–Littlewood maximal function, function of bounded variation, weak differentiability.

Reference to this article: O. Kurka: On the variation of the Hardy–Littlewood maximal function. Ann. Acad. Sci. Fenn. Math. 40 (2015), 109-133.

Full document as PDF file

doi:10.5186/aasfm.2015.4003