2021/09/03 by Panu Lahti, Lahti, Panu, Xiaodan Zhou +1
Mathematics · #30C65 #30L10 #46E36 #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2109.01260
openalex publication_date 2021/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a homeomorphism f\colon X→ Y between Q-dimensional spaces X,Y, we show that f satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that f belongs to the Sobolev class N_\rmloc1,p(X;Y), where 1< p≤ Q, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors Q-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the classical Euclidean setting. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f∈ N_\rmloc1,Q(X;Y) without the strong assumption of the infinitesimal distortion hf belonging to L∞(X).