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Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

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Autor: Hagelstein, Paul
Luque Martínez, Teresa
Parissis, Ioannis
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2015-11
Publicado en: Transactions of the American Mathematical Society, 367 (11), 7999-8032.
Tipo de documento: Artículo
Resumen: 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...
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Cita: Hagelstein, P., Luque Martínez, T.E. y Parissis, I. (2015). Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Transactions of the American Mathematical Society, 367 (11), 7999-8032.
Tamaño: 865.6Kb
Formato: PDF

URI: http://hdl.handle.net/11441/49377

DOI: 10.1090/tran/6339

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