Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces

Details Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces

2008-04-17

arxiv.org/abs/0804.2833v1

Mathematics

Analysis of PDEs

31 pages

Scientific paper

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

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