A Trace theorem for Martinet--type vector fields

Abstract

In ${\mathbb{R}}^3$ we consider the vector fields \[ X_1 =\frac{ \partial }{\partial x},\qquad X_2 =\frac{ \partial }{\partial y}+ |x|^α\frac{ \partial }{\partial z}, \] where $\alpha \in \left[1,+\infty \right[$. Let ${\mathbb{R}}_{+}^3=\{(x,y,z)\in {\mathbb{R}}^3:z\ge 0\}$ be the (closed) upper half-space and let $f\in C^1({\mathbb{R}}_{+}^3)$ be a function such that $X_{1}f,X_{2}f\in L^p({\mathbb{R}}_{+}^3)$ for some $p>1$. In this paper, we prove that the restriction of $f$ to the plane $z=0$ belongs to a suitable Besov space that is defined using the Carnot-Carathéodory metric associated with $X_1$ and $X_2$ and the related perimeter measure.