Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
Let B be a homothecy invariant collection of convex sets in Rn. Given a measure μ, the associated weighted geometric maximal operator MB,μ is defined by MB,μf(x) := sup x∈B∈B 1/μ(B) B |f|dμ. It is shown that, provided μ satisfies an appropriate doubling condition with respect to B and ν is an arbitr...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/49377 |
| Acceso en línea: | http://hdl.handle.net/11441/49377 https://doi.org/10.1090/tran/6339 |
| Access Level: | acceso abierto |
| Palabra clave: | Strong maximal function Tauberian condition Muckenhoupt weight |
| Sumario: | Let B be a homothecy invariant collection of convex sets in Rn. Given a measure μ, the associated weighted geometric maximal operator MB,μ is defined by MB,μf(x) := sup x∈B∈B 1/μ(B) B |f|dμ. It is shown that, provided μ satisfies an appropriate doubling condition with respect to B and ν is an arbitrary locally finite measure, the maximal operator MB,μ is bounded on Lp(ν) for sufficiently large p if and only if it satisfies a Tauberian condition of the form ν x ∈ Rn : MB,μ(1E)(x) > 1 / 2 ≤ cμ,νν(E). As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights A∞,B for homothecy invariant Muckenhoupt bases B consisting of convex sets. Moreover, it is immediately seen that the strong maximal function MR,μ, defined with respect to a product-doubling measure μ, is bounded on Lp(ν) for some p > 1 if and only if ν x ∈ Rn : MR,μ(1E)(x) > 1 / 2 ≤ cμ,νν(E) holds for all ν-measurable sets E in Rn. In addition, we discuss applications in differentiation theory, in particular proving that a μ-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate L∞(ν). |
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