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CHOQUET INTEGRALS, HAUSDORFF CONTENT AND THE HARDY–LITTLEWOOD MAXIMAL OPERATOR

Published online by Cambridge University Press:  01 March 1998

JOAN OROBITG  and
JOAN VERDERA
JOAN OROBITG
Affiliation:
Departament de Matematiques, Universitat Antònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
JOAN VERDERA
Affiliation:
Departament de Matematiques, Universitat Antònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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Abstract

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality

∫Mf(x)dHα(x) [les ]C∫[mid ]f(x)[mid ]dHα(x), 0<α<n, (1)

where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in ℝn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of ϕ[ges ]0 with respect to a set function C is defined by

formula here

Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (ℝn), see [1, p. 114].

The purpose of this note is to provide a self-contained, direct proof of a result more general than (1).

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 2 , March 1998 , pp. 145 - 150
Copyright
© The London Mathematical Society 1998

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