Recent developments in the theory of Lorentz spaces and weighted inequalities

  1. Raposo, José A.
  2. Soria, Javier
  3. Carro, María J.
Revista:
Memoirs of the American Mathematical Society

ISSN: 0065-9266,  1947-6221

Año de publicación: 2007

Volumen: 187

Número: 877

Páginas: 0-0

Tipo: Artículo

DOI: 10.1090/MEMO/0877 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Memoirs of the American Mathematical Society

Resumen

The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator M, and thus, we consider the boundedness of M in the weighted Lorentz space Λpu(w). Two examples are historically relevant as a motivation: If w=1, this corresponds to the study of the boundedness M:Lp(u)⟶Lp(u), which was characterized by B. Muckenhoupt, giving rise to the so called Ap weights. The second case is when we take u=1. This is a more recent theory, and was completely solved by M.A. Ariño and B. Muckenhoupt in 1991. It turns out that the boundedness $M:\llo\longrightarrow\llo,$ can be seen to be equivalent to the boundedness of the Hardy operator A restricted to decreasing functions of Lp(w). The class of weights satisfying this boundedness is known as Bp. Even though the Ap and Bp classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderón--Zygmund decompositions and covering lemmas for Ap, rearrangement invariant properties and positive integral operators for Bp. It is our aim to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., u=1 and w=1), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the Hardy-Littlewood maximal operator with respect to general measures.

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