Equivalence between Q-absolute continuity and the Rado–Reichelderfer (RR) condition on Carnot groups

Determine whether every Q-absolutely continuous map f: Ω ⊂ G1 → G2 between Carnot groups G1 and G2 satisfies the Rado–Reichelderfer condition (RR). Specifically, let Q be the homogeneous dimension of G1 and define the (RR) condition by the existence of a weight w ∈ L1(Ω) such that for every Carnot–Carathéodory ball U_r(x) ⊂ Ω, osc_{U_r(x)}(f) ≤ r^{-Q} ∫_{U_r(x)} w(y) dy. Establish whether for all f in AC_Q(Ω,G2), there exists such a weight w making the inequality hold for all balls U_r(x) ⊂ Ω.

Background

The paper studies intrinsic differentiability and absolute continuity notions for maps between Carnot groups, focusing on Q-absolute continuity (AC_Q) and bounded Q-variation, as well as their relations to Sobolev regularity and area formulas.

The Rado–Reichelderfer (RR) condition provides a quantitative oscillation control via an integrable weight. In Euclidean spaces, Malý established that AC in dimension n implies the (RR) condition. The authors show in Carnot groups that (RR) implies AC_Q, but note that the converse implication remains unproven in this setting.

Resolving whether AC_Q implies (RR) in Carnot groups would align Carnot-group theory with the Euclidean case and yield a sharper characterization of absolute continuity that underpins area formulas and differentiability results.