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Area distortion of quasiconformal mappings. (English) Zbl 0815.30015

B. V. Bojarskij [Mat. Sb., N. Ser. 43(85), 451-503 (1958; Zbl 0084.304)] proved that the derivative of a plane quasiconformal mapping \(f\) is locally \(L^ p\)-integrable for some \(p> 2\). This means that the Jacobian determinant \(J_ f\) of \(f\) belongs to \(L^ p_{\text{loc}}\), \(p> 1\). Let \(p(K)= \sup\{p: J_ f\in L^ p_{\text{loc}}(\Omega)\) for each \(K\)-quasiconformal \(f: \Omega\in \mathbb{R}^ 2\}\). Since the work of Bojarskij has been a long standing open problem to determine the value of \(p(K)\); the conjecture, based e.g. on Hölder continuity reasons, has been \(p(K)= K/(K- 1)\). In this paper it is shown that this, indeed, is true. The problem is related to the area distortion of plane quasiconformal mappings. The author gives the optimal solution \(| f(E)|\leq M| E|^{1/K}\) for a \(K\)-quasiconformal mapping \(f: B\to B\), \(f(0)= 0\) and for each Borel set \(E\) in the unit disk \(B\).
The solution is based on holomorphic deformations of Cantor sets. This leads to the thermodynamic formalism [R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms (1975; Zbl 0308.28010)]. In practice, the area distortion problem goes down to a precise estimate of the measures of the images of disjoint disks under a quasiconformal mapping \(f: B\to B\), \(f(0)= 0\), such that \(f\) is conformal in each small disk.
As a byproduct several other related problems receive their answers: The change of the Hausdorff dimension under a plane quasiconformal mapping and the removability problem for bounded \(K\)-quasiregular mappings of the plane.
Additional remarks: For quasiconformal mappings in \(\mathbb{R}^ n\) [F. Gehring, Acta Math. 130, 265-277 (1973; Zbl 0258.30021)] has shown that \(p(K)> 1\) but the exact value is unknown for \(n\geq 3\). The \(L^ p\)- integrability problem of the derivative of a plane quasiconformal mapping is intimately connected to the norm of the two-dimensional Hilbert transform in \(L^ p\), \(p>2\) [O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane (1973; Zbl 0267.30016)]. However, the solution of the former problem does not solve the latter.

MSC:

30C62 Quasiconformal mappings in the complex plane
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