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).