DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

AVERAGING DOMAINS: FROM EUCLIDEAN SPACES
TO HOMOGENEOUS SPACES
SUSAN G. STAPLES
We explore the role of averaging domains from their earliest definition in Euclidean
spaces and subsequent weighted forms of the definition to most recent results in homogeneous
spaces. Various examples and applications of these domains, including Poincaré
inequalities, are also discussed.
Copyright © 2006 Susan G. Staples. 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. Averaging domains in Euclidean spaces
Averaging domains were first introduced by Staples [21] in order to answer a basic question
on the definition of BMO functions in R n . Let us frame some of the relevant background
and set notations. By Ω we mean a proper subdomain (i.e., open and connected)
of R n . Recall that the BMO norm of a function u ∈ L 1 loc (Ω)isdefinedas
∫
1
‖u‖ ∗ = sup ∣ ∣
u − u B dx, (1.1)
|B|
B⊂Ω
where B is any ball in Ω with Lebesgue measure |B|, u B = (1/|B|) ∫ B udx,anddx represents
Lebesgue measure.
It is a well-established fact that if the class of balls in Ω is replaced by the class of cubes
in Ω, then the corresponding supremum defines an equivalent norm.
A natural question arises: what types of domains D canreplaceballsandproducean
equivalent norm? Here the supremum would be taken over all domains D ′ ⊂ Ω with D ′
similar to D. The pursuit of the solution to this problem led in turn to the definition
of L p -averaging domains. The case p = 1 of bounded L 1 -averaging domains provided
precisely the class of domains which produce equivalent BMO norms.
Definition 1.1. Let D be a domain in R n ,with|D| < ∞, andletp ≥ 1. It is said that D is
an L p -averaging domain, if for some τ>1 the following holds:
( ∫ 1
|D|
D
B
) ∣ ∣ 1/p ( ∫
)
u(x) − u D p 1
dx ≤ C sup ∣ ∣ 1/p
u(x) − u B p dx . (1.2)
|B|
τB⊂D
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1041–1048
B