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

Let U be a connected open subset of Rⁿ, and let X=(X₁,X₂,…,Xₘ) 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 WX,0k,p(Ω), where Ω⊂⊂ 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 Ω. Combining this representation formula with weighted weak-Lᵖ estimates, we derive sharp Sobolev inequalities on WX,0k,p(Ω), 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.