Hardy’s inequalities for monotone functions on partially ordered measure spaces

The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimension have been considered only in very specific cases. In particular, in the diagonal case, only for p = 1 (see [5]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, contrary to the one-dimensional case where one considers intervals of the form (0, a), a > 0. It is also worth to point out that, with no monotonicity restriction, the boundedness of the Hardy operator is only known in dimension n = 2 (see [15], [12], and also [3] for an extension in the case of product weights). In this work we completely characterize the weighted Hardy’s inequalities for all values of p > 0, namely, the boundedness of the operator: