In [5] one of the authors
introduced the notion of a radial averaging transformation of domains in the plane,
which was based on the metric given by the line element ds2= (1∕r2)(dx12+ dx22)
where (x1,x2) are the cartesian and (r,𝜃) are the polar coordinates. This
transformation is useful in obtaining estimates for conformal capacity of condensers
and for conformal radius of domains. In this paper we discuss averaging
transformations in m-dimensional spaces (m ≧ 2), based on various metrics of the
form ds2= g2(r)∑i = 1m(dxi)2, where g(r) is a positive, continuous function of
r(0 < r < ∞).
With the help of these transformations we are able to obtain estimates for energy
integrals of the form
These estimates can be used to compare capacities of different condensers filled with
nonhomogeneous dielectric [cf. Kühnau [8] and the literature cited there]. As a
further application we derive inequalities for conformal capacity and conformal radius
in the plane and similar results in higher dimensional spaces. In this direction
we have results for the case where g(r) = rββ ≧ m − 3. They include the
symmetrization results obtained by Szegö in [7]. The method presented seems to be
quite general, and we believe that it might be employed also with other classes of
metrics g.
