Sharp embedding results and geometric inequalities for H\"{o}rmander vector fields

TL;DR

This paper establishes sharp embedding theorems and geometric inequalities for Sobolev spaces associated with Hörmander vector fields, extending classical results to a broader subelliptic setting using advanced lifting and saturation techniques.

Contribution

It introduces new sharp Sobolev inequalities and geometric inequalities in generalized Sobolev spaces defined by Hörmander vector fields, utilizing Rothschild-Stein's lifting and saturation methods.

Findings

Proved representation formula for smooth functions with compact support.

Derived sharp Sobolev inequalities with critical exponent depending on Métivier index.

Established various inequalities including isoperimetric, logarithmic Sobolev, and Moser-Trudinger inequalities.

Abstract

Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of H\"{o}rmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where $\Omega\subset\subset U$ is a general open bounded subset of $U$. By employing Rothschild-Stein's lifting technique and saturation method, we prove the representation formula for smooth functions with compact support in $\Omega$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where the critical Sobolev exponent depends on the generalized M\'{e}tivier index. As applications of these sharp Sobolev inequalities, we establish the isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and Moser-Trudinger inequality in the context of general H\"{o}rmander vector fields.