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Choquet Integrals, Hausdorff Content and the Hardy–Littlewood Maximal Operator
Author(s) -
Orobitg Joan,
Verdera Joan
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003688
Subject(s) - mathematics , sobolev space , hausdorff space , outer measure , choquet integral , duality (order theory) , hausdorff measure , pure mathematics , combinatorics , operator (biology) , type (biology) , compact space , mathematical analysis , hausdorff dimension , fractal , fuzzy logic , biochemistry , chemistry , repressor , gene , transcription factor , ecology , linguistics , philosophy , fractal dimension , biology , minkowski–bouligand dimension
Using the BMO‐ H 1 duality (among other things), D. R. Adams proved in [ 1 ] the strong type inequality∫ M f ( x ) d H α ( x ) ⩽ C ∫ | f ( x ) |d H α ( x ) , 0 < α < n ,where C is some positive constant independent of f . Here M is the Hardy–Littlewood maximal operator in R n , H α is the α‐dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of φ ⩾ 0 with respect to a set function C is defined by ∫ φ d C = ∫ 0 ∞ C { x ∈ R n : φ ( x ) > t } d t .Precise definitions of M and H α will be given below. For an application of (1) to the Sobolev space W 1, 1 (R n ), see [ 1 , p. 114]. The purpose of this note is to provide a self‐contained, direct proof of a result more general than (1). 1991 Mathematics Subject Classification 28A12, 28A25, 42B25.