Sharp embedding results and geometric inequalities for Hörmander vector fields

Abstract

Let $U$ be a connected open subset of ${\mathbb{R}}^n$, and let $X=(X_1,X_2,\dots ,X_m)$ be a system of Hörmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space ${\mathcal{W}}_{X,0}^{k,p}(\hat{I}\text{c◯})$, where $\hat{I}\text{c◯}\subset \subset U$ 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 $\hat{I}\text{c◯}$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on ${\mathcal{W}}_{X,0}^{k,p}(\hat{I}\text{c◯})$, where the critical Sobolev exponent depends on the generalized Métivier 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 Hörmander vector fields.

43 pages