An estimate in the spirit of poincaré's inequality

Abstract

We show that if Ω ⊂ ℝN, N ≥ 2, is a bounded Lipschitz domain and (ρn) ⊂ L1 (ℝN) is a sequence of nonnegative radial functions weakly converging to δ0, then ∫Ω |f-fΩ|p ≤ C ∫Ω ∫Ω |f(x)-f(y)|p/|x-y|p ρn(|x-y|) dx dy for all f ∈ Lp(Ω) and n ≥ n0, where fΩ denotes the average of f on Ω. The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu [2]. As n → ∞ we recover Poincaré's inequality. The case N = 1 requires an additional assumption on (ρn). We also extend a compactness result of Bourgain, Brezis and Mironescu.