An estimate in the spirit of Poincaré's inequality

We show that if Ω ⊂ R N , N ≥ 2, is a bounded Lipschitz domain andis a sequence of nonnegative radial functions weakly converging to δ 0 , thenfor all f ∈ L p (Ω) and n ≥ n 0 , where f Ω 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 (ρ n ).We also extend a compactness result of Bourgain, Brezis and Mironescu.