A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz-Shimogaki theorem
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 b...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/42373 |
| Acceso en línea: | http://hdl.handle.net/11441/42373 https://doi.org/10.1512/iumj.2007.56.3112 |
| Access Level: | acceso abierto |
| Palabra clave: | maximal operators rearrangement-invariant spaces Muckenhoupt weights |
| Sumario: | 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. |
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