A counterexample concerning integrability of derivatives of conformal mappings (Q581706)

Let \(\Omega\) be a simply connected domain in the plane which intersects the real axis and f be a conformal mapping from \(\Omega\) onto the unit disk. Write \(K={\mathbb{R}}\cap \Omega\). A theorem of Hayman and Wu (1981) asserts that \(f'\in L^ 1(K,dx)\). If \(\Omega\) is a slit disk then \(f'\not\in L^ 2(K,dx)\). The author conjectured in 1983 that \(f'\in L^ p(K)\) for every \(p\in (1,2)\). Fernández, Heinonen and Martio (1989) proved that \(f'\in L^ p(K)\) for \(p\leq 1+\epsilon\), for some absolute constant \(\epsilon >0\). In this paper I construct a counterexample to my conjecture. The domain \(\Omega\) is the complement of a certain tree. The function \(f'\) fails to be in \(L^ p(K)\) for all p sufficiently close to 2. The analogue for area integrability known s the Brennan conjecture, is the assertion that \(f'\in L^ p(\Omega,dx dy)\) for every \(p\in (2,4)\). This conjecture remains open. The best known result, due to Pommerenke (1985), is that \(f^{\exists}\in L^ p(\Omega)\) for \(2\leq p<3.399\).