Vol. 46, No. 2, 1973

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Radial averaging transformations with various metrics

Catherine Bandle and Moshe Marcus

Vol. 46 (1973), No. 2, 337–348

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

|∇F |2g(r)r3−m dx  (dx = dx dx ⋅⋅⋅dx  ).
Ω                         1  2     m

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) = rβ β 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.

Mathematical Subject Classification 2000
Primary: 31B15
Received: 7 October 1971
Revised: 3 November 1972
Published: 1 June 1973
Catherine Bandle
Moshe Marcus