An estimate in the spirit of Poincaré's inequality

Ponce, Augusto [UCL]

We show that if Omega subset of R-N, N greater than or equal to 2, is a bounded Lipschitz domain and (rho(n)) subset of L-1(R-N) is a sequence of nonnegative radial functions weakly converging to delta(0), then integral(Omega) |f - f(Omega)|(p) less than or equal to C integral(Omega)integral(Omega) |f(x)-f(y)|(p)/|x-y|(p) rho(n)(|x-y|)dx dy for all f is an element of L-P (Omega) and n greater than or equal to n(0), where f(Omega) denotes the average of f on Omega. The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu [2]. As n --> infinity we recover Poincare's inequality. The case N = I requires an additional assumption on (rho(n)). We also extend a compactness result of Bourgain, Brezis and Mironescu.