IMRN International Mathematics Research Notices 1996, No. 1 A Relationship between Poincar´ e-Type Inequalities and RepresentationFormulas in Spaces of Homogeneous Type Bruno Franchi, Guozhen Lu, and Richard L. Wheeden The purpose of this note is to study the relationship between the validity of L 1 versions of Poincar´ e’s inequality and the existence of representation formulas for functions as (fractional) integral transforms of first-order vector fields. The simplest example of a representation formula of the type we have in mind is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space R N : |f(x) − f B |≤ C  B |∇f(y)| |x − y| N−1 dy, x ∈ B, where ∇f denotes the gradient of f, f B is the average |B| −1  B f(y) dy, |B| is the Lebesgue measure of B, and C is a constant which is independent of f, x, B. We are primarily interested in showing that various analogues of the formula above for more general systems of first-order vector fields Xf = (X 1 f,...,X m f) are simple corollaries of (and, in fact, often equivalent to) appropriate L 1 Poincar ´ e inequalities of the form 1 ν(B)  B |f − f B,ν | dν ≤ Cr(B) 1 µ(B)  B |Xf| dµ. (1) Here ν and µ are measures,B is a ball of radius r(B) with respect to a metric that is naturally associated with the vector fields, and f B,ν = ν(B) −1  B f dν. Recently, representation formulas in R N were derived for H ¨ ormander vector fields in [FLW] (see also [L1]) as well as for some nonsmooth vector fields of Grushin type in [FGW]. In the case of H ¨ ormander vector fields, if ρ(x, y) denotes the associated metric (see [FP], [NSW], [San]) and B(x, r) denotes the metric ball with center x and radius r, then we Received 30 August 1995. Revision received 22 September 1995. Communicated by Carlos Kenig.