Reverse Hardy-type inequalities for supremal operators with measures

Abstract

In this paper we characterize the validity of the inequalities parallel to g parallel to(p,(a, b),lambda) <= c parallel to u(x)parallel to g parallel to(infinity,(x,b),mu) parallel to(q,(a,b),nu) and parallel to g parallel to(p,(a, b),lambda) <= c parallel to u(x)parallel to g parallel to(infinity,(a,x),mu) parallel to(q,(a,b),nu) for all non-negative Borel measurable functions g on the interval (a, b) subset of R, where 0 < p <= +infinity, 0 < q <= +infinity, lambda, mu and nu are non-negative Borel measures on (a, b), and u is a weight function on (a, b)