Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions

Abstract

Abstract.A characterization is given of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded. This is done by determining the weights for which Hardy's inequality holds for nonincreasing functions. An alternate characterization, valid for nondecreasing weights, is also derived.

On weighted iterated Hardy-type inequalities

“…The first results on the problem Λ p (v) ֒→ Γ p (v), 1 < p < ∞, which is equivalent to inequality (1.1) restricted to the cones of non-increasing functions, were obtained by Boyd [5] and in an explicit form by Ariño and Muckenhoupt [3]. The problem with w v and p q, 1 < p, q < ∞ was first successfully solved by Sawyer [40].…”

“…Let 0 < p < ∞, 0 < q ≤ ∞ and 1 < s < ∞. Assume that u, w ∈ W(0, ∞) and v ∈ W(0, ∞) be such that (3.13) Then inequality (1.7) with the best constant c 3 …”

Mixed Norm and Multidimensional Lorentz Spaces

“…This condition is that the Hardy-Littlewood maximal operator is bounded on Λ p (v). The weights for which this holds were first characterized by M. A. Ariño and B. Muckenhoupt [1], and it is known as the B p condition: there exists C > 0 such that, for all r > 0,…”

Embeddings of concave functions and duals of Lorentz spaces

“…Since [6] and [9] in the early 50's, techniques involving properties of monotone functions have been used effectively to address a wide variety of questions in weighted norm inequalities, interpolation theory, and function space theory. For a few of the many see [1], [3], [7], [8], [13], [14], [15], [16], [17]. The study of the collection of concave functions has also had its successes.…”