Sobolev inequality and its extremal functions for homogeneous Hörmander vector fields

We study the Sobolev inequality and the existence of its extremal functions in the setting of homogeneous Hörmander vector fields. A principal result establishes a mutual inclusion between the set of volume growth rates of subunit balls and the set of admissible Sobolev conjugate exponents on an arbitrary open subset $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Our analysis yields a precise characterization of the dependence of the exponents on the volume growth and determines their optimal admissible range. As a consequence, we obtain a global Sobolev inequality on ${\mathbb{R}}^n$. The second part of the paper investigates the attainability of the optimal Sobolev constant in degenerate cases. We develop a refined concentration-compactness lemma adapted to the structure of homogeneous Hörmander vector fields. We then prove that the optimal Sobolev constant is attained under a broad algebraic condition, namely, that the volume-preserving automorphism group of homogeneous Hörmander vector fields acts transitively on the maximal level set of the pointwise homogeneous dimension. This result holds for general homogeneous Hörmander vector fields in non-equiregular degenerate cases, significantly extending the analysis beyond a specific class of Grushin-type vector fields.