L1 → Lq Poincaré inequalities for 0 < q < 1 imply representation formulas

Abstract

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B0⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L1→L1 Poincaré inequality of the following form: ∫B|f−fB|dv⩽cr(B)∫Bgdμ, for all metric balls B⊂B0⊂S, implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B0, |f(x)−fB0|⩽C∫B0g(y)ρ(x,y)μ(B(x,ρ(x,y)))dμ(y)+Cr(B0)μ(B0)∫B0g(y)dμ(y). One of the main results of this paper shows that an L1 to Lq Poincaré inequality for some 0 < q < 1, i.e., (∫B|f−fB|qdv)1/q⩽cr(B)∫Bgdμ, for all metric balls B⊂B0, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, supλ>0λν({x∈B:|f(x)−fB|>λ})ν(B)⩽Cr(B)∫Bgdμ, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.