Hardy spaces and quasiconformal maps in the Heisenberg group

Abstract

We define Hardy spaces Hp, 0<p<∞, for quasiconformal mappings on the Korányi unit ball B in the first Heisenberg group H1. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of p0(K)>0 such that every K-quasiconformal map f:B→f(B)⊂H1 belongs to Hp for all 0<p<p0(K). Second, we give two equivalent conditions for the Hp membership of a quasiconformal map f, one in terms of the radial limits of f, and one using a nontangential maximal function of f. As an application, we characterize Carleson measures on B via integral inequalities for quasiconformal mappings on B and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from Rn to H1. A crucial difference between the proofs in Rn and H1 is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.