Sobolev inequality and its extremal functions for homogeneous H\"{o}rmander vector fields

We study the Sobolev inequality and the existence of its extremal functions in the setting of homogeneous H\"{o}rmander 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 $\Omega\subset \mathbb{R}^n$. 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 H\"{o}rmander 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 H\"{o}rmander vector fields acts transitively on the maximal level set of the pointwise homogeneous dimension. This result holds for general homogeneous H\"{o}rmander vector fields in non-equiregular degenerate cases, significantly extending the analysis beyond a specific class of Grushin-type vector fields.