Carleson measures and conformal self-mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R-n, n >= 2, are characterized by means of global Dirichlet integrals of the conformal self-map T-a taking a is an element of B to the origin. The same proof applies in the unit ball of C-n. It is also proved that the powers of the Jacobian of T-a. satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim