Poincaré Inequality Meets Brezis–Van Schaftingen–Yung Formula on Metric Measure Spaces

Let (X, ρ, μ) be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol C∗c(X) the space of all continuous functions f with compact support satisfying that Lip f := lim supr→0 supy∈B(·,r) | f (·) − f (y)|/r is also a continuous function with compact support and Lip f = limr→0 supy∈B(·,r) | f (·) − f (y)|/r converges uniformly. Let p ∈ [1,∞). In this article, the authors prove that, for any f ∈ C∗c(X), sup λ∈(0,∞) λ ∫ X μ ({ y ∈ X : | f (x) − f (y)| > λρ(x, y)[V(x, y)] 1 p }) dμ(x)