BMO and Teichmüller space. (English) Zbl 0746.30021

Hamilton, David H.

Ann. Acad. Sci. Fenn., Ser. A I, Math. 14, No. 2, 213-224 (1989).

The author has as his main theme the dependence of solutions to the Beltrami equation (*) \(F_{\bar z}=\mu F_ z\) and connections to Teichmüller theory and harmonic analysis. He considers the function \(\mu(z)\) with \(\|\mu\|_ \infty<1\), sometimes with compact support, and lets \(F^ \mu\) be a “normalized” solution of (*). Theorem 1 shows that the map \(\mu\to\log(F_ z^ \mu)\) is a complex holomorphic map of BMO\((\mathbb{R}^ 2)\). This is an extension of a well-known result of H. M. Reimann [Comment. Math. Helv. 49, 260-276 (1974; Zbl 0289.30027)] but uses careful analysis of the argument of \((F^ \mu)_ z\), via approximation.
An open set \(\Omega\ni\infty\) has a univalence criterion if there is an \(a=a(\Omega)\) such that if \(g\) is analytic in \(\Omega\) with \(| zg(z)|=o(1)\) at \(\infty\) and \(| g'(z)|\text{dist}(z,\partial\Omega)<\infty\) then \(g\) is one-to-one. Theorem 2 asserts that this is equivalent to several things, one being that \(g\) has a representation as a Hilbert transform of a function \(h\in L^ \infty(\Omega^ c)\), and makes contact with the improved Thurston- Sullivan \(\lambda\)-lemma [cf. L. Bers and H. L. Royden, Acta Math. 157, 259-286 (1986; Zbl 0619.30027)]. These results require no regularity of \(\partial\Omega\).
Analogues of some results are given for VMO. The paper is compactly written, and has a few inessential typographical errors.

MSC:

30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30D55	\(H^p\)-classes (MSC2000)
30F60	Teichmüller theory for Riemann surfaces

Keywords:

Beltrami equation; Teichmüller theory; VMO

Citations:

Zbl 0289.30027; Zbl 0619.30027