Outer measures and weak type estimates of Hardy-Littlewood maximal operators

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We will introduce the k times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for k > 0. We will prove that the k times modified centered Hardy-Littlewood maximal operator is weak type (1,1) bounded with constant 1 when k ≥ 2 if the Radon measure of the space has “continuity” in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures. Copyright © 2006 Yutaka Terasawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Hardy-Littlewood maximal operators were first introduced by Hardy and Littlewood ([6]) in one dimensional case for the purpose of the application to Complex Analysis. Then Wiener ([14]) introduced this operator in higher dimensional Eucledian spaces for the purpose of the application to Ergodic Theory. Later, Coifman and Weiss ([4]) defined Hardy-Littlewood maximal operators on quasi-metric measure spaces satisfying doubling conditions (which we call homogeneous spaces). More recently, Nazarov et al. ([9]) defined modified Hardy-Littlewood maximal operators on quasi-metric measure spaces possesing a Radon measure that does not satisfy a doubling condition (which we call nonhomogeneous spaces), which are used in harmonic analysis on nonhomogeneous spaces. In this paper, we will treat weak type (1,1) inequalities satisfied by several types of Hardy-Littlewood maximal operators. As is well known, weak type (1,1) inequalities satisfied by Hardy-Littlewood maximal operators are keys to prove their strong type (p, p) boundedness via Marcinkiewicz’s interpolation theorem. To prove their weak type (1,1) inequalities, the unification of our approach is the use of outer measures. The advantage of the use of outer measures over usual measures is that they could measure any subsets of a total space, even when they are nonmeasurable. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 15063, Pages 1–13 DOI 10.1155/JIA/2006/15063

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Outer measures and maximal functions

Let (X,μ) be a metric space possesing a nondegenerate Radon measure such that μ(B(x,r)) is continuous with respect to the variable r > 0 when the variable x ∈ X is fixed, where B(x,r) denotes a ball centered at x and of radius r. We will define the k times modified centered Hardy-Littlewood maximal operator as follows: 1  Mk f (x) = sup  r>0 μ B(x,kr)

 B(x,r)

   f (y)dμ(y).

(1.1)

We will prove that the k times modified centered Hardy-Littlewood maximal operator Mk is weak-(1,1) bounded when k is larger than or equal to 2, and that their weak-(1,1) constant (which is the infimum (consequently the minimum) of the constant appearing in the weak type (1,1) inequality) is less than or equal to 1. We will sta