Pullback theorem and rigidity for Sobolev mappings on Carnot groups

Abstract

This paper establishes a pullback theorem via mollification to extend the rigidity theory of Sobolev mappings between Carnot groups to the low-integrability regime where the Sobolev exponent $p$ is less than the homogeneous dimension $\hat{I}{½}_1$ of the source group. The core technical achievement is the rigorous analysis of the convergence of mollified approximations $f_{\epsilon }$ for a mapping $f\in W^{1,p}$, demonstrating that the pullbacks of left-invariant differential forms converge appropriately. This allows for the recovery of more properties of Pansu differentiable mappings. The main results are: (1) A generalization of the rigidity theorem to the range $p<\hat{I}{½}_1$ for $f\in W^{1,p}$. (2) When $p>Q$, such mappings are shown to be locally Hölder continuous with exponent $1-\frac{Q}{p}$, where $p>Q$ and $Q=\max \left\{\text{homogeneous dim of}{G}_i\right\}$. (3) with the stratified structure of contact Sobolev mappings, we generalize the non-embedding theorem to contact Sobolev mappings.