Distortion of the boundary under conformal mapping (Q793180)

In this paper the following question is studied: What are the relations between the harmonic measure \(\omega^ a_{\Omega}\) of a Jordan domain \(\Omega\) at the point \(a\in \Omega\) and the Hausdorff measure \(\Lambda_ h\) with respect to a given measure function h? The authors give a new and more constructive proof of a (slight extension of a) result due to \textit{L. Carleson} [Duke Math. J. 40, 547-559 (1973; Zbl 0273.30014)]: There exist \(\beta>frac{1}{2}\) such that \(\omega^ a_{\Omega}\ll m_{\beta},\) where \(m_{\beta}=\Lambda_ h\) for \(h(t)=t^{\beta}\) (i.e. \(m_{\beta}\) is \(\beta\)-dimensional Hausdorff measure) and - in the opposite direction - that if \(h(t)=t \exp \{A(\log 1/t)^{\frac{1}{2}}\}\) for some constant \(A>0\) there exist \(\Omega\) and \(F\subset \partial \Omega\) with \(\omega^ a_{\Omega}(F)=1\) and \(\Lambda_ h(F)=0\). In fact, \(\partial \Omega\) can be chosen to be a quasi-circle (Theorem 1). On the other hand, it was known that for any simply connected set \(\Omega\) and any \(K\subset \partial \Omega\) situated on some straight line L one has \(m_ 1(K)=0\Rightarrow \omega^ a_{\Omega}(K)=0\) [the reviewer, Aspects of contemporary complex analysis, Proc. inst., Conf., Durham/Engl. 179, 469-473 (1980; Zbl 0496.30014)]. The authors extend this result to the case where L is a quasi-smooth curve (Theorem 3). A crucial ingredient in the proof is a result due to \textit{Jerison} and \textit{Kenig} [''Boundary behaviour of harmonic functions in non-tangentially accessible domains'' (preprint)] about properties of harmonic measures in domains with quasi-smooth boundaries. Other interesting results are also obtained. Quite recently \textit{N. G. Makarov} [''On the distortion of boundary sets under conformal mappings'' (preprint)] has proved a result which implies that in Theorem 1 one has in fact \(\omega^ a_{\Omega}\ll m_{\beta}\) for any \(\beta<1\). This has been an open problem for a long time.