On weighted Hardy and Poincare-type inequalities for differences

A criterion is obtained for the Hardy-type inequality\r\n\r\n(integral(0)(a)\f(x)\(P)v(x)dx)(1/p) less than or equal to c(1){(v(a)integral(0)(a)\f(x)\(P)dx)(1/p)\r\n\r\n+(integral(0)(a) integral(0)(a)\f(x)-f(y)\(P)w(\x-y\)dxdy)(1/p)}\r\n\r\nto be valid for 0 < a less than or equal to A less than or equal to infinity and 0 < p < infinity. This weakens a criterion previously found by the first two authors and comes close to being necessary as well as sufficient. when an inequality in the criterion is reversed, a Poincare-type inequality is derived in some cases.