Choquet Integrals, Hausdorff Content and the Hardy–Littlewood Maximal Operator

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.