An isoperimetric inequality and the geometric Sobolev embedding for vector fields
Let \(X_1,\dots, X_m\) be \(C^\infty\) vector fields in \(\mathbb{R}^n\) satisfying Hörmander's condition for hypoellipticity and let \(d(x,y)\) be the Carnot-Carathéodory distance associated to \((X_1,\dots, X_m)\). It is well known that for a given \(U\subseteq \mathbb{R}^n\) there exists \(c_1>0\) and \(R_0>0\) such that for any \(x\in U\), \(0<R\leq R_0\) and \(0<t<1\) the inequality \(|B(x,R)|\leq c_1t^{-Q}|B(x,tR)|\) holds, where \(Q={{\log c_1}\over {\log 2}}\) and \(B(x,R)= \{y\in\mathbb{R}^n: d(x,y)<R\}\). The authors call the number \(Q\) the local homogeneous dimension and they prove the Gagliardo-Nirenberg type inequality for vector fields: \((-\hskip-.9em\int_{B(x,R)}|u|^k)^{1/k}\leq c_2R -\hskip-.9em\int_{B(x,R)} [\sum^m_{j=1}(X_j u)^2]^{1/2}\), where \(c_2\) is a constant, \(x\in U\), \(0<R\leq R_0\) and \(1\leq k\leq {Q\over {Q-1}}\). The exponent \({Q\over {Q-1}}\) is shown to be optimal. The above inequality is a consequence of an isoperimetric inequality proven in the paper, which involves a suitable definition of perimeter relative to the vector fields.
Hörmander's condition for hypoellipticity
Gagliardo-Nirenberg type inequality for vector fields
scientific article; zbMATH DE number 817694