Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Let (X,ρ,μ) be a metric measure space of homogeneous type, and let Cc⁎(X) denote the set all Lipschitz functions f on X such that limr→0+⁡supy∈B(⋅,r)⁡|f(⋅)−f(y)|/r converges uniformly to compactly supported continuous function Lip X. p∈[1,∞) f∈Cc⁎(X) pair (f,Lipf) satisfies certain Poincaré inequality. Assume α∈R∖{0} if p∈(1,∞), α∈(0,∞) p=1. Under these assumptions, authors prove following result which extends recent formula H. Brezis, A. Seeger, J. Van Schaftingen, P.-L. Yung finite dimensional Euclidean spaces:supλ∈(0,∞)⁡λp∬Dλ[U(x,y)]α−1dμ(x)dμ(y)∼∫X[Lipf(x)]pdμ(x), whereDλ:={(x,y)∈X×X:|f(x)−f(y)|>λρ(x,y)[U(x,y)]αp} for any λ∈(0,∞), V(x,y):=μ(B(x,ρ(x,y))) U(x,y):=min⁡{V(x,y),V(y,x)} x,y∈X, positive constants equivalence are independent f. A similar remains true p∈(1,∞) lies in wider class Hajłasz–Sobolev space. As an application, establish new fractional Sobolev Gagliardo–Nirenberg inequalities These results applicable many classical examples, as spaces, with weighted Lebesgue measure, complete Riemannian manifolds, non-negative Ricci curvature.