The failure of \(L^ p\) estimates for harmonic measure in chord-arc domains
Suppose that D is a bounded plane domain whose boundary is a rectifiable Jordan curve. D is said to be a chord-arc domain if there is a constant C such that for any w,w'\(\in \partial D\) we have \(\sigma(w,w')\leq C| w- w'|,\) where \(\sigma\) denotes the length of the shorter arc of \(\partial D\) connecting w and w'. These domains were introduced by Lavrent'ev in 1936, who proved that conformal mappings from the unit disk onto chord-arc domains satisfy \(1/f'\in H^ p\) for some \(p>0,\) where \(H^ p\) denotes the Hardy class. Chord-arc domains have attracted considerable attention recently because of their connections with functions of bounded mean oscillation, the theory of \(A_ p\) weights, and the theory of Cauchy integrals along curves. If \(\partial D\) is Lipschitz then \(1/f'\in H^ 1.\) The question arose whether this would be true for every chord-arc domain. In this paper the author decisively disproves this by showing that given any \(p>0\) there exists a chord-arc domain D for which \(1/f'\not\in H^ p.\) The counterexamples are obtained by clever construction starting from the non-Smirnov domain of Keldysh and Lavrent'ev. The same result has been proved non-constructively and independently by \textit{P. W. Jones} and \textit{M. Zinsmeister} [C. R. Acad. Sci., Paris Sér. I 295, 563-566 (1982; Zbl 0536.30008)].
scientific article; zbMATH DE number 3851446