The Beltrami coefficient and the maximal dilatation

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 2 §1, printed pp. 49–51. For an orientation-preserving nonsingular real-linear map, Bishop obtains the complex dilatation $\mu =f_{\bar{z}}\mathrm{/}{f}_z$, $\mathrm{\mid }\mu \mathrm{\mid }<1$, and the dilatation $D=(1+\mathrm{\mid }\mu \mathrm{\mid })\mathrm{/}(1-\mathrm{\mid }\mu \mathrm{\mid })$, equivalently $\mathrm{\mid }\mu \mathrm{\mid }=(D-1)\mathrm{/}(D+1)$.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 2 §§11.1.2–11.3, printed pp. 177–181. Section 11.1.2 defines the pointwise Beltrami coefficient; equation (11.3) on p. 178 gives $\mathrm{Dil}=(1+\mathrm{\mid }\mu \mathrm{\mid })\mathrm{/}(1-\mathrm{\mid }\mu \mathrm{\mid })$; §11.3 adds the ACL/distributional regularity and bounded-dilatation requirements. The last K-conversion on p. 181 prints denominator $k-1$; this is a sign typo, so the correct conversion from Bishop and the library definition is used here.

Assume the Axiom of Choice. Let $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ be a homeomorphism of complex domains whose components lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(\mathrm{\Omega })$, with weak Wirtinger derivative classes as in The ACL and Sobolev analytic definition of quasiconformality, The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions, and Weak derivative of a locally integrable function.

Choose finite Borel representatives of the real and imaginary components of these derivative classes. Such representatives exist: extend each component by zero outside $\mathrm{\Omega }$, use that Lebesgue measure is the completion of its Borel restriction and that every completion-measurable function equals a Borel function almost everywhere (Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$, $\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets, A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra), then restrict the resulting Borel representatives to $\mathrm{\Omega }$ using the Borel trace identity (The Borel sigma-algebra of a topological space, The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra). Replacing infinite values on the resulting Borel null sets by zero gives finite representatives.

The Beltrami coefficient of $f$ is the Borel function ${\mu }_f(z):=\{\begin{array}{ll}{\mathrm{\partial }}_{\bar{z}}f(z)\mathrm{/}{\mathrm{\partial }}_{z}f(z),& {\mathrm{\partial }}_{z}f(z)\mathrm{\ne }0,\\ 0,& {\mathrm{\partial }}_{z}f(z)=0.\end{array}$ It is a finite-valued measurable function modulo equality almost everywhere. Changing the chosen Borel representatives changes ${\mu }_f$ only on a Lebesgue-null set, so this class is independent of those choices. For a general $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}$ homeomorphism, ${\mu }_f$ need not be essentially bounded. If $f$ is $K$-analytically quasiconformal and $k=(K-1)\mathrm{/}(K+1)$, then $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le k<1$ almost everywhere and it defines a complex $L^{\mathrm{\infty }}(\mathrm{\Omega })$ class (Complex Lp classes and Euclidean test-function conventions). The value zero on $\{{\mathrm{\partial }}_{z}f=0\}$ is a fixed convention; analytic quasiconformality gives ${\mathrm{\partial }}_{\bar{z}}f=0$ almost everywhere on that set.

Let $m_f=\mathrm{\parallel }{\mu }_f{\mathrm{\parallel }}_{\mathrm{\infty }}$ be the essential supremum of $\mathrm{\mid }{\mu }_f\mathrm{\mid }$ (The essential supremum of a measurable function with respect to a measure). Define the maximal dilatation by $K_f:=\{\begin{array}{ll}{\displaystyle \frac{1+m_f}{1-m_f}},& m_f<1\text{ and }{\mathrm{\partial }}_{\bar{z}}f=0\text{ a.e. on }\{{\mathrm{\partial }}_{z}f=0\},\\ +\mathrm{\infty },& \text{otherwise.}\end{array}$ Then $K_f\in [1,+\mathrm{\infty }]$, and for every finite $K\ge 1$, $f$ is analytically $K$-quasiconformal exactly when $K_f\le K$. Indeed, off $\{{\mathrm{\partial }}_{z}f=0\}$ the analytic inequality is equivalent to $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le (K-1)\mathrm{/}(K+1)$, and on that set the separate derivative condition in the definition of $K_f$ is exactly what makes the inequality hold. For finite $m_f$, intersecting the almost-everywhere bounds $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le m_f+1\mathrm{/}n$ gives $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le m_f$ almost everywhere. For an analytically quasiconformal map, the separate derivative condition holds and $m_f<1$, so $K_f=(1+m_f)\mathrm{/}(1-m_f)$ is the least admissible constant. In this class, $K_f=1$ exactly when ${\mu }_f=0$ almost everywhere, equivalently when ${\mathrm{\partial }}_{\bar{z}}f=0$ as an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ class. The separate one-quasiconformal theorem on this page supplies the holomorphic conclusion in that case; it is not an input to this definition. The nullity of $\{{\mathrm{\partial }}_{z}f=0\}$ will follow from the inverse theorem's area formula and null-set properties; it is not an assumption here.