Critical Sobolev inequalities and global extremals for homogeneous H\"{o}rmander vector fields on $\mathbb{R}^n$

Abstract

We study critical Sobolev inequalities and Sobolev extremal functions for homogeneous H\"{o}rmander vector fields on $\mathbb R^n$. The focus is the non-equiregular case. In this setting, the sub-Riemannian flag may have different growth at different points, the volume of subunit balls is not governed by a single local dimension, and the translation structure available on Carnot groups is absent in general. These features make both the range of admissible Sobolev exponents and the attainment of the optimal Sobolev constant substantially more delicate, especially on unbounded domains. We first give a domain-dependent description of the admissible Sobolev exponents in terms of the volume growth rates of subunit balls. This description separates the roles of the pointwise homogeneous dimension, the non-isotropic dimension of the domain, and the singular strata that may be approached at infinity. As a consequence, we prove a global Sobolev inequality for all homogeneous H\"{o}rmander vector fields on $\mathbb R^n$, without assuming any underlying group law. We then prove that the optimal Sobolev constant is attained. The main new geometric ingredient is a smooth family of maps $T(w,x)$ along the maximal level set $H$ of the pointwise homogeneous dimension. These maps preserve Lebesgue measure and the horizontal gradient, and they play the role of left translations in a setting where no group multiplication is available. In this sense, $T(w,x)$ compensates for the lack of an intrinsic translation structure and allows one to recenter minimizing sequences in the critical concentration-compactness argument. The compactness theorem obtained in this way applies to all homogeneous H\"{o}rmander vector fields. We also prove that, on every open set meeting this maximal level set, the optimal Sobolev constant is independent of the domain.