Geometric and analytic quasiconformality in metric measure spaces

TL;DR

This paper establishes the equivalence between geometric and analytic notions of quasiconformality in general metric measure spaces, extending classical results without requiring specific metric conditions.

Contribution

It proves the equivalence of geometric and analytic quasiconformality definitions in broad metric measure spaces, including cases with bounded geometry and Alexandrov spaces.

Findings

Equivalence of geometric and analytic quasiconformality in general metric spaces

Sharp bounds relating outer dilatations and pointwise dilatations

Extension of classical quasiconformal theory to non-smooth spaces

Abstract

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either space. When $X$ and $Y$ have locally $Q$-bounded geometry and $Y$ is contained in an Alexandrov space of curvature bounded above, the sharpness of our results implies that, as in the classical case, the modular and pointwise outer dilatations of $\map$ are related by $K_O(f)= \operatorname{esssup} H_O(x,f)$.