Doubling conformal densities (Q2757781)

Let \(\rho\) be a continuous positive function in the unit ball \(B\) of \(\mathbb{R}^n\). Define a new distance in \(B\) as \(d(x,y)= \inf\int_\gamma \rho ds\) where the infimum is taken over all rectifiable curves \(\gamma\) joining the points \(x\) and \(y\) in \(B\). If \(n=2\) and \(f:B\to\mathbb{R}^2\) is conformal, then \(\rho(z) =|f'(z)|\) defines a metric which satisfies the Harnack inequality and the volume growth condition. If a density satisfies these two conditions then it is called a conformal density. The volume growth condition means that NEWLINE\[NEWLINE\mu_\rho (B_\rho (z,r))\leq Cr^2NEWLINE\]NEWLINE for every \(z\in B\) and \(r>0\) where \(\mu_\rho\) is the measure defined as \(\mu_\rho(E)= \int_{E\cap B}\rho^2 dm\). Quasiconformal maps \(f\), instead of conformal maps, can also be used for \(n\geq 2\). Now \(|f'(z) |\) is replaced by an averaging process involving the Jacobian of \(f\), [\textit{K. Astala}, \textit{F. W. Gehring}, Mich. Math. J. 32, 99-107 (1985; Zbl 0574.30027)]. A density is called doubling if the induced measure \(\mu_\rho\) is doubling. The authors characterize up to a multiplicative constant conformal and doubling densities in the unit disk. It turns out that these densities are in bounded relation with respect to the induced density of some global plane quasiconformal mapping, restricted to \(B\). The proof is based on the metric analysis (curved cone conditions etc.) of the \(d\)-geodesics. The authors also show that no analogous characterization exists in \(\mathbb{R}^n\), \(n>2\).