A trace theorem for Martinet-type vector fields

Journal article, 2021

In R3ℝ3 we consider the vector fields

X1=∂∂x,X2=∂∂y+|x|α∂∂z,X1=∂∂x,X2=∂∂y+|x|α∂∂z,
where α∈[1,+∞[α∈1,+∞. Let R3+={(x,y,z)∈R3:z≥0}ℝ+3={(x,y,z)∈ℝ3:z≥0} be the (closed) upper half-space and let f∈C1(R3+)f∈C1(ℝ+3) be a function such that X1f,X2f∈Lp(R3+)X1f,X2f∈Lp(ℝ+3) for some p>1p>1. In this paper, we prove that the restriction of ff to the plane z=0z=0 belongs to a suitable Besov space that is defined using the Carnot–Carathéodory metric associated with X1X1 and X2X2 and the related perimeter measure.