Vol. 40, No. 2, 1972

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Radon-Nikodým densities and Jacobians

Mitsuru Nakai

Vol. 40 (1972), No. 2, 375–396

A Dirichlet mapping between regions in Euclidean space is a homeomorphism preserving the finiteness of Dirichlet integrals of admissible functions and plays an important role in the potential theory. Two dimensional Dirichlet mappings are known to be characterized geometrically as being quasiconformal mappings. In this paper, higher dimensional Dirichlet mappings will be characterized geometrically as being quasi-isometries. In order to carry out the reasoning it is necessary to study the relation between the Radon-Nikodym density R and the Jacobian J of an arbitrary homeomorphism for which only existences of R and J almost everywhere are assured. It will be proven that R |J|, almost everywhere, which is the main result of this paper.

Mathematical Subject Classification 2000
Primary: 28A15
Secondary: 46E35, 26A57
Received: 1 September 1970
Published: 1 February 1972
Mitsuru Nakai