Inequalities of Hardy-Sobolev type in Carnot-Carath\'eodory spaces

Abstract

We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\'eodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the H\"ormander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|\phi(x)|^{p}V(x)dx\leq C\int_{\Om}|X\phi|^{p}dx,\qquad \phi\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1<p<+\infty$. Under sharp geometric assumptions on the domain $\Om\subset \Rn$ that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.