Artículo
A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz-Shimogaki theorem
Autor/es | Lerner, Andrei K.
Pérez Moreno, Carlos |
Departamento | Universidad de Sevilla. Departamento de Análisis Matemático |
Fecha de publicación | 2007 |
Fecha de depósito | 2016-06-16 |
Publicado en |
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Resumen | Given any quasi-Banach function space X over Rn it is defined an index αX that coincides with the upper Boyd index αX when the space X is rearrangement-invariant. This new index is defined by means of the local maximal ... Given any quasi-Banach function space X over Rn it is defined an index αX that coincides with the upper Boyd index αX when the space X is rearrangement-invariant. This new index is defined by means of the local maximal operator mλf . It is shown then that the Hardy-Littlewood maximal operator M is bounded on X if and only if αX < 1 providing an extension of the classical theorem of Lorentz and Shimogaki for rearrangement-invariant X. As an application it is shown a new characterization of the Muckenhoupt Ap class of weights: u ∈ Ap if and only if for any ε > 0 there is a constant c such that for any cube Q and any measurable subset E ⊂ Q, |E| |Q| logε |Q| |E| ≤ c u(E) u(Q)!1/p. The case ε = 0 is false corresponding to the class Ap,1. Other applications are given, in particular within the context of the variable Lp spaces. |
Agencias financiadoras | Ministerio de Educación y Ciencia (MEC). España |
Identificador del proyecto | MTM2006-05622 |
Cita | Lerner, A.K. y Pérez Moreno, C. (2007). A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz-Shimogaki theorem. Indiana University Mathematics Journal, 56 (6), 2697-2722. |
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