La métrique attachée de façon naturelle à des champs de vecteurs $C^{\infty }$ est susceptible de plusieurs définitions voisines ; on montre que, suivant la définition adoptée, elle peut avoir, ou ne pas avoir, un caractère localement lipschitzien qui a pour conséquence l’existence de points $L$-réguliers, pour certains opérateurs différentiels $L$, sur les frontières des boules pour la métrique.

We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.

Let the spaces ${𝐑}^m$ and ${𝐑}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of ${𝐑}^m$, and let $f:A⟶{𝐑}^n$ be an order-preserving function. Suppose that $P$ is generating in ${𝐑}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on ${𝐑}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f\left(A\right)$ is connected.