AN IMPROVED POINCARE INEQUALITY

We show that a large class of domains D in R(n) including John domains satisfies the improved Poincare inequality \\u(x) - U-D\\L(q)(D) less than or equal to C\\del u(X)d(X, partial derivative D)(delta)\\L(p)(D) where p less than or equal to q less than or equal to np/n-p(1-delta), p(1-delta) < n, delta is an element of [0, 1], c = c(p, q, delta, D) < infinity, and u is in an appropriate Sobolev class.