Spaces of differential forms and mappings with controlled distortion (Q1760954)

In this paper necessary and sufficient conditions for an approximately differentiable map \(f:\mathbb M\to\mathbb M'\) between Riemannian manifolds are studied. For these maps there exists a bounded transfer operator of differential forms with norm in Lebesgue spaces. As a corollary, it is shown that every homeomorphism \(f:\mathbb M\to\mathbb M'\) of class \(\text{ACL}(\mathbb M)\) whose transfer operator of differential forms with norm in \(\mathcal L_p\) is an isomorphism must necessarily be either quasi-conformal or quasi-isometric. Some applications of these results to the study of the functorial cohomology in Lebesgue spaces are also given.