  We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on
the Kor\'{a}nyi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our
definition is stated in terms of the Heisenberg polar coordinates introduced by
Kor\'{a}nyi 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 Kor\'{a}nyi unit sphere with its two
characteristic points.
