Artículo
Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
Autor/es | Hagelstein, Paul
Luque Martínez, Teresa Parissis, Ioannis |
Departamento | Universidad de Sevilla. Departamento de Análisis Matemático |
Fecha de publicación | 2015-11 |
Fecha de depósito | 2016-11-30 |
Publicado en |
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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 ... 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). |
Agencias financiadoras | Simons Foundation Ministerio de Economía y Competitividad (MINECO). España |
Identificador del proyecto | 208831
info:eu-repo/grantAgreement/MINECO/BES-2010-030264 138738 |
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. |
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