A Trace theorem for Martinet--type vector fields

Abstract

In $\mathbb{R}^3$ we consider the vector fields \[ X_1 =\frac{ \partial }{\partial x},\qquad X_2 =\frac{ \partial }{\partial y}+ |x|^\alpha \frac{ \partial }{\partial z}, \] where $\alpha\in\left[1,+\infty\right[$. Let $\mathbb{R}^3_+ =\{(x,y,z)\in\mathbb{R}^3: z\geq 0\}$ be the (closed) upper half-space and let $f\in C^1 ( \mathbb{R} ^3_+ )$ be a function such that $X_1f, X_2f \in L^ p(\mathbb{R}^3_+)$ 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-Carath\'eodory metric associated with $X_1$ and $X_2$ and the related perimeter measure.