Inequalities of Hardy–Sobolev Type in Carnot–Carathéodory Spaces

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We consider various types of Hardy-Sobolev inequalities on a Carnot-Caratheodory space (Ω, d) associated to a system of smooth vector fields Χ = {Χ 1,Χ 2,…,Χm} on Rn satisfying the Hormander finite rank condition rank Lie[Χ 1,…, Ωm] ≡ n. One of our main concerns is the trace inequality \(\int\limits_\Omega ^{} {|\ell \left( x \right)} |^p V\left( x \right)dx \le C\int\limits_\Omega ^{} {|X\ell |^P dx},\ell \in C_0^\infty \left( \Omega \right),\) where V is a general weight, i.e., a nonnegative locally integrable function on Ω, and 1 < p < +∞. Under sharp geometric assumptions on the domain Ω C Rn that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.