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

TL;DR

This paper investigates Hardy-Sobolev inequalities within Carnot-Carathéodory spaces, establishing various forms under geometric conditions linked to subelliptic capacities and Hausdorff contents.

Contribution

It introduces new Hardy-Sobolev inequalities in Carnot-Carathéodory spaces with sharp geometric assumptions, extending classical results to subelliptic settings.

Findings

Established Hardy-Sobolev inequalities under geometric conditions

Derived trace inequalities involving weights and vector fields

Connected geometric assumptions to capacities and Hausdorff contents

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.