Duality theorem over the cone of monotone functions and sequences in higher dimensions

Let be a non-negative function defined on . which is monotone in each variable separately. If , and a product weight function, then equivalent expressions for are given, where the supremum is taken over all such functions . Variants of such duality results involving sequences are also given. Applications involving weight characterizations for which operators defined on such functions (sequences) are bounded in weighted Lebesgue (sequence) spaces are also pointed out.