\(H^p\)-theory for quasiconformal mappings (Q652911)

A quasiconfomal mapping \(f\) of the unit ball \(B^n\) into \(\mathbb{R}^n\), \(n \geq 2\), belongs to the Hardy space \(H^p\) if \(\sup_{0<r<1}\int_{S^{n-1}} |f(r \omega)|^p \,d\omega < \infty\). A theorem of \textit{D. Jerison} and \textit{A. Weitsman} [``On the means of quasiregular and quasiconformal mappings'', Proc. Am. Math. Soc. 83, 304--306 (1981; Zbl 0471.30008)] says that there exists \(p_o = p_o(n,K) > 0\) such that every \(K\)-quasiconformal mapping belongs to \(H^p\) for \(p < p_o\). The authors give a new proof that yields the sharp estimate \(p_o(2, K) = 1/(2K)\) in the plane. For conformal mappings, i.e., \(K = 1\), the result was proved by Prawitz in 1927. The authors also characterize quasiconformal mappings \(f \in H^p\) in terms of the integral condition \(\int^1_0(1-r)^{n-2}M(r, f)^p \, dr < \infty\) where \(M(r,f)\) denotes the maximum modulus of \(f\) on \(S^{n-1}(r)\). They also give a new proof of a theorem by \textit{M. Zinsmeister} [``A distortion theorem for quasiconformal mappings'', Bull. Soc. Math. Fr. 114, 123--133 (1986; Zbl 0602.30027)], which characterizes quasiconformal mappings in \(H^p\) in terms of the nontangential maximal function. The proof makes direct use of quasiconformality instead of the connection between Carleson measures and quasiconformality as in the original proof. The paper contains many other characterizations for quasiconformal mappings in \(H^p\)-classes like \(\int_{B^n} |f(x)|^{p-1} |f'(x)| \,dx < \infty\). Coordinate functions of quasiconformal mappings and their role in \(H^p\)-theory is thoroughly explained as well as various BMO and VMO conditions for quasiconformal mappings.