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

Abstract

Bounded and compact Carleson measures in the unit ball Bof Rn , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self-map Ta taking a ∈ B to the origin. The same proof applies in the unit ball of Cn . It is also proved that the powers of the Jacobian of Ta 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. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)