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Harmonic measure, $$L^ 2$$ estimates and the Schwarzian derivative. (English) Zbl 0801.30024
Let $$\Phi$$ be a conformal map of the unit disk $$\mathbb{D}$$ onto the inner domain $$\Omega$$ of the Jordan curve $$\Gamma$$; many results also hold in the more general context that $$\Omega$$ is any simply connected domain. This paper can perhaps be considered as a sequel to the authors’ important paper on harmonic measure and arclength [Ann. Math., II. Ser. 132, 511-547 (1992; Zbl 0726.30019)] and contains many further interesting results.
For $$z\in \Gamma$$, $$t>0$$, the authors first study the geometric quantity $\beta(z,t)= \inf_ L t^{-1}\sup\{\text{dist}(w,L): z\in \Gamma\cap D(w,4t)\},{(*)}$ where the infimum is taken over all lines $$L$$ passing through the smaller disk $$D(z,t)$$. They consider the condition $\int^ 1_ 0 t^{-1} \beta(z,t)^ 2 dt< \infty.{(**)}$ If $$K\subset \Gamma$$ is compact and if $$(**)$$ holds uniformly for all $$z\in K$$ then $$K$$ lies on a rectifiable curve. Furthermore, except for a set of linear measure zero, $$\Gamma$$ has a tangent in $$z$$ if and only if $$(**)$$ holds. Next they show that $\iint_ D |\Phi'(z)| | S_ \Phi(z)|^ 2(1- | z|^ 2)^ 3 dx dy< \infty$ implies that $$\Phi'\in H^ p$$ for every $$p< 1/2$$. Here $$S_ \Phi$$ is the Schwarzian derivative that plays an important role in the proofs. Astala and Zinsmeister had characterized when $$\log \Phi'\in \text{BMO}$$. The present authors now give geometric characterizations. One of them (as the reviewer understands it) is: There are constants $$\delta> 0$$ and $$C$$ such that, for every $$z\in \Omega$$, there exists a domain $$G\subset \Omega$$ which is chord-arc (Lavrentiev) with constant $$C$$ such $D(z,\delta d)\subset G,\quad\text{length }\partial G\leq Cd,\quad\text{length }\partial G\cap \partial\Omega\geq \delta d,$ where $$d= \text{dist}(z,\partial\Omega)$$. Their results imply that the condition $$\log \Phi'\in \text{BMO}$$ is invariant under bi-Lipschitz homeomorphisms of the plane and that the condition holds if and only if the corresponding condition holds for the exterior conformal map (in the case of a quasidisk).
There are many details that are difficult to follow. E.g. the definition of $$\beta$$ on p. 86 is different from $$(*)$$ and does not make sense. Saying that “By adjusting the constants in the definitions one gets” some relation leaves doubt in the reader’s mind about that relation. It is stated that Lemma 3.3 was proven as Lemma 3.1 in the authors’ paper quoted above. But the status of that lemma is not quite clear, see M.R. 92c:30026 (loc. cit.). That reviewer writes “Bishop informed the reviewer that Lemma 3.1 can be salvaged by…”.

MSC:
 30C85 Capacity and harmonic measure in the complex plane
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References:
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