Pullback theorem and rigidity for Sobolev mappings on Carnot groups

TL;DR

This paper extends the rigidity and continuity results for Sobolev mappings on Carnot groups to cases where the integrability exponent is below the homogeneous dimension, using a mollification-based pullback theorem.

Contribution

It introduces a new pullback theorem for Sobolev mappings on Carnot groups and extends classical rigidity results to lower integrability exponents, providing new insights into their regularity.

Findings

Rigidity of Sobolev mappings extended to p<ν

Continuity properties of Sobolev mappings established for lower p

Invariant horizontal gradient under higher layer vector fields

Abstract

We study the pullback theorem of Sobolev mappings on Carnot groups via mollification of mappings. With the pullback theorem we extend the classical result proved by Xiangdong Xie : Rigidity of Sobolev mappings $W^{1,p}(G_1;G_2)$ for $p>\nu$, to the case $p<\nu$, where $\nu$ is the homogeneous dimension of $G_1$. Therefore, some conclusions about continuity of Sobolev mappings on Carnot groups for $p<\nu$ are found. And also, the determine of horizontal gradient $D_Hf$ is invariant under the motion related to higher layer left-invariant vector fields. At last, we find a equivalent definition of quasiconformal mappings with lower integrability $dim(g^{[1]})<p<\nu$.