 
Summary: Hardy's inequalities for monotone functions on partially
ordered measure spaces
Nicola Arcozzi§
, Sorina Barza, Josep L. GarciaDomingo¶
, and Javier Soria
Abstract. We characterize the weighted Hardy's inequalities for monotone functions in Rn
+.
In dimension n = 1, this recovers the classical theory of Bp weights. For n > 1, the result was
only known for the case p = 1. In fact, our main theorem is proved in the more general setting
of partially ordered measure spaces.
1 Introduction
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 onedimensional 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
