Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields
Volume 139 / 2000
Studia Mathematica 139 (2000), 213-244
DOI: 10.4064/sm-139-3-213-244
Abstract
We study the notion of fractional $L^p$-differentiability of order $s∈(0,1)$ along vector fields satisfying the Hörmander condition on $ℝ^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p} ⊂ W^{s,q}$, where q is a suitable exponent greater than p.
