Doubling measures, monotonicity, and quasiconformality (Q2384747)

The authors construct quasiconformal mappings in \(n\)-dimensional Euclidean space \({\mathbb R}^n\) by integration of a discontinuous kernel against doubling measures with suitable decay. For example, the authors show that if a doubling measure \(\mu\) in \({\mathbb R}^n\) satisfies the decay condition \[ \int_{| z| >1} | z| ^{-1}\,d\mu(z)<\infty, \] then the mapping \(f_\mu\), defined by \[ f_\mu(x) = {1\over 2}\int_{{\mathbb R}^n}\bigg({{x-z}\over{| x-z| }}+{z\over{| z| }}\bigg)\,d\mu(z), \] is \(\eta\)-quasisymmetric with \(\eta\) depending only on the doubling constant of \(\mu\). Note that every sense-preserving quasisymmetric mapping \(f: {\mathbb R}^n\to {\mathbb R}^n\), \(n\geq 2\) is quasiconformal and vice versa [\textit{P. Tukia} and \textit{J. Väisälä}, Ann. Acad. Sci. Fenn., Ser. A I 5, 97--114 (1980; Zbl 0403.54005)]. A Radon measure \(\mu\) on \({\mathbb R}^n\) is said to be isotropic doubling if there there exists a constant \(A\geq 1\) such that \[ A^{-1} \leq {{\mu(R_1)}\over{\mu(R_2)}} \leq A \] whenever \(R_1\) and \(R_2\) are congruent rectangular boxes with nonempty intersection. A mapping \(F\) from a convex domain \(\Omega\subset \mathbb R^n\) into \({\mathbb R}^n\) is called \(\delta\)-monotone for \(\delta \in (0,1]\) if for all \(x,y\in \Omega\) \[ \langle F(x)-F(y),x-y\rangle \geq \delta | F(x)-F(y)| | x-y| . \] A nonconstant \(\delta\)-monotone mapping \(f\:\Omega \to {\mathbb R}^n\), \(n\geq 2\) is \(\eta\)-quasisymmetric on a closed ball \(B\) if \(2B\subset \Omega\) with \(\eta\) depending only on \(\delta\) [J. Lond. Math. Soc., II. Ser. 75, No. 2, 391--408 (2007; Zbl 1134.47038)]. The authors prove that for a nonconstant \(\delta\)-monotone mapping \(f:{\mathbb R}^n\to {\mathbb R}^n\), \(n\geq 2\), the weight \(\| Df\| \) is isotropic doubling. Finally, a construction is given for an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.