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

Summary: Hardy's inequalities for monotone functions on partially
ordered measure spaces
Nicola Arcozzi§
, Sorina Barza, Josep L. Garcia-Domingo¶
, 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 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

  

Source: Arcozzi, Nicola - Dipartimento di Matematica, Università di Bologna

 

Collections: Mathematics