Hardy spaces and quasiconformal maps in the Heisenberg group

Abstract

We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group ${\mathbb{H}}^1$. 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 $p_0(K)>0$ such that every $K$-quasiconformal map $f:B\to f(B)\subset {\mathbb{H}}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. 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 ${\mathbb{R}}^n$ to ${\mathbb{H}}^1$. A crucial difference between the proofs in ${\mathbb{R}}^n$ and ${\mathbb{H}}^1$ is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.

51 pg