An inverse Sobolev lemma (Q1325215)

The inverse Sobolev lemma in this context means the following: Let \(0< p\leq n\), and suppose that \(f\) is a quasiconformal mapping of the unit ball \(B\) of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\). Then the derivative \(f'\) is in \(L^ q(B)\) for all \(0< q<p\) if and only if \(f\) is in \(L^ s(B)\) for all \(0< s<pn/(n- p)\). The proof employs the average derivative of a quasiconformal mapping [\textit{K. Astala} and \textit{W. F. Gehring}, J. Anal. Math. 46, 16-57 (1986; Zbl 0628.30026)]. This is used to relate the integrability of \(f\) to the growth of the \(f\), and thus to \(f'\).