The geometric Sobolev embedding for vector fields and the isoperimetric inequality
The authors prove an optimal embedding theorem for the Sobolev spaces associated to some general families of vector fields in \(\mathbb{R}^n\). Notable examples of systems of vector fields to which the authors' results apply are: (I) \(C^\infty\) vector fields \(X_1,\dots,X_m\) in \(\mathbb{R}^n\) with Hörmander's condition rank Lie \([X_1,\dots,X_m](x)=n\) at every \(x\in\mathbb{R}^n\); (II) Generalized Baouendi-Grushin type families \(X_j={\partial\over\partial x_j}\) \((j=1,\dots,k)\), \(X_j=(x^2_1+\cdots+x^2_k)^{\alpha/2}{\partial\over\partial x_j}\) \((j=k+1,\dots,n)\). The authors prove their basic results using a new version of the isoperimetric inequality. The main point of the paper is to show that the geometric Sobolev embedding is a consequence of two general facts: (i) The doubling condition for the balls in a metric naturally associated to the given fields; (ii) The possibility of representing a function compactly supported in a ball, in terms of a metric fractional integral which involves the (degenerate) gradient associated to the fields. The basic result is new in the context of nilpotent homogeneous Lie groups (e.g. the Heisenberg group). The authors' approach gives a new proof of the embedding theorem even for the ordinary Sobolev spaces.
scientific article; zbMATH DE number 820216