Explicit bounds and sharp results for the composition operators preserving the exponential class (Q305637)

Summary: Let \(f : \Omega \subset \mathbb{R}^n \to \mathbb{R}^n\) be a quasiconformal mapping whose Jacobian is denoted by \(J_f\) and let \(\mathrm{EXP}(\Omega)\) be the space of exponentially integrable functions on \(\Omega\). We give an explicit bound for the norm of the composition operator \(T_f\): \(u \in \mathrm{EXP}(\Omega) \mapsto u \circ f^{- 1} \in \mathrm{EXP}(f(\Omega))\) and, as a related question, we study the behaviour of the norm of \(\log J_f\) in the exponential class. The \(A_{\infty}\) property of \(J_f\) is the counterpart in higher dimensions of the area distortion formula due to \textit{K. Astala} et al. [Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton, NJ: Princeton University Press (2009; Zbl 1182.30001)] in the plane and it is the key tool to prove the sharpness of our results.