Caractère lipschitzien d'une distance associée à des champs de vecteurs engendrant une algèbre de Lie de rang maximal. Quelques conséquences. (Lipschitzian character of a distance associated to vector fields generated by a Lie algebra of maximal rank. Some consequences)
Let \(\Omega\) be a bounded domain in \({\mathbb{C}}^ N\); let the real vector fields \(Y_ 1,...,Y_ q\) be \({\mathcal C}^{\infty}\) on some neighbourhood of \({\bar \Omega}\), and such that the vectors \(Y_ i(x)\) span the space \({\mathbb{R}}^ N\) for all \(x\in {\bar \Omega}\); to each \(Y_ i\) is assigned a formal degree \(d_ i\) in such a way that \([Y_ i,Y_ j]\) is a linear combination, with coefficients \(\in {\mathcal C}^{\infty}({\bar \Omega})\), of the \(Y_ k\) for which \(d_ k\leq d_ i+d_ j\). Several equivalent distances \(\rho,...,\rho_ 4\) on \(\Omega\) were defined by \textit{A. Nagel}, \textit{E. M. Stein} and \textit{S. Wainger} [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)] as follows: \(\rho\) (x,y) is the smallest number for which exists an absolutely continuous function \(\phi\) : [0,1]\(\to {\bar \Omega}\) such that \[ (1)\quad \phi (0)=x,\quad \phi (1)=y;\quad (2)\quad \phi '(t)=\sum^{q}_{j=1}a_ j(t)Y_ j\circ \phi (t);\quad (3)\quad | a_ j(t)| \leq \rho^{d_ j}(x,y), \] for a.e. t; when the \(Y_ j\) are given vector fields \(X_ 1,...,X_ p\) (of formal degree 1) and their commutators of lengths \(\leq m\), \(\rho_ 4(x,y)\) is given by a similar definition, with the \(X_ i\) instead of the \(Y_ j\) in (2) and \(\rho_ 4\) instead of \(\rho^{d_ j}\) in (3). The paper shows: that \(\rho\) (x,) is locally Lipschitz on \(\Omega\setminus \{x\}\), but not on \(\Omega\) unless all formal degrees are 1; that in general \(\rho_ 4(x,)\) is not locally Lipschitz on \(\Omega\setminus \{x\}\); that the \(a_ j\) in (3) actually have the property \(\sup^{q}_{j=1}| a_ j(t)| =\rho^{d_ j}(x,y)\) for a.e. t. When the given vector fields \(X_ 1,...,X_ p\) generate a Lie algebra of rank N at each point of \(\Omega\), the locally Lipschitz character of \(\rho\) (x,) yields some information about the boundary points of the \(\rho\)-balls with centre x that are regular for the Dirichlet problem related to the differential operator \[ \sum^{p}_{i=1}X^ 2_ 1+\sum_{i<j}b_{i,j}[X_ i,X_ j]+\sum^{p}_{i=1}c_ iX_ i. \]
scientific article; zbMATH DE number 4169221