Hardy spaces and quasiconformal maps in the Heisenberg group

Abstract

We define Hardy spaces H p , 0 < p < ∞, for quasiconformal mappings on the Korányi unit ball B in the first Heisenberg group H 1 . 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) ⊂ H 1 belongs to H p for all 0 < p < p0(K). Second, we give two equivalent conditions for the H p 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 R n to H 1 . A crucial difference between the proofs in R n and H 1 is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.