Doubling measures and quasiconformal maps (Q2368014)

In the study of quasiconformal maps, one commonly asks, ``Which classes of maps or measures are preserved under quasiconformal maps?'', and conversely, ``When does the said preservation property imply the quasiconformality of the map''''. These questions have been previously studied by Reimann, Uchiyama, and the author with respect to the classes of BMO functions, and \(A_ \infty\)-measures. Herein the author presents results of this type regarding quasiconformal maps and doubling measures. The two main results are described below. A Borel measure \(\mu\) defined on a domain \(D\) in \(\mathbb{R}^ n\) is said to be doubling, \(D\) is in \({\mathcal D}(D,\tau)\), for \(\tau\geq 1\), if there exists a constant \(c>0\) such that \(\mu(2Q)\leq c\mu(Q)\) for all cubes \(Q\) such that \(2\tau(Q)\subset D\). Theorem: If \(f: D\to D'\) is a \(K\)-quasiconformal map, then \(\varphi: \mu\to\nu=\mu(f(\cdot))\) is a monomorphism between \({\mathcal D}(D')\) and \({\mathcal D}(D,\tau)\), \(\tau=\tau(K,n)\). Moreover, the relationship between the doubling constants for \(\mu\) and \(\nu\) is determined by \(K\) and \(n\). The second key theorem treats the converse of the above case when certain additional hypotheses on the map \(f\), such as differentiability a.e., are assumed. The proof relies on the detailed construction of a particular type of doubling measure. Assuming this measure is pulled back under the homeomorphism \(f\) to a related doubling measure, one can then show that \(f\) must be quasiconformal.