## Arkiv för Matematik

• Ark. Mat.
• Volume 52, Number 2 (2014), 367-382.

### On the weak-type (1,1) of the uncentered Hardy–Littlewood maximal operator associated with certain measures on the plane

#### Abstract

Suppose μ is a positive measure on $\mathbb{R}^{2}$ given by μ=ν×λ, where ν and λ are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$, respectively, which do not vanish on any open interval. We prove that if for either ν or λ there exists a set of positive measure A in its domain for which the upper and lower s-densities, 0< s≤1, are positive and finite for every xA then the uncentered Hardy–Littlewood maximal operator Mμ is weak-type (1,1) if and only if ν is doubling and λ is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when n=2.

#### Article information

Source
Ark. Mat., Volume 52, Number 2 (2014), 367-382.

Dates
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802676

Digital Object Identifier
doi:10.1007/s11512-013-0183-2

Mathematical Reviews number (MathSciNet)
MR3255145

Zentralblatt MATH identifier
1316.42021

Rights
• Sjögren, P., A remark on the maximal function for measures in $\mathbb{R}^{n}$, Amer. J. Math. 105 (1983), 1231–1233.
• Vargas, A. M., On the maximal function for rotation invariant measures in $\mathbb{R}^{n}$, Studia Math. 110 (1994), 9–17.