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

Authors: Paul Hagelstein, Teresa Luque and Ioannis Parissis
Journal: Trans. Amer. Math. Soc. 367 (2015), 7999-8032
MSC (2010): Primary 42B25; Secondary 42B35
Published electronically: April 1, 2015
MathSciNet review: 3391907
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Abstract: Let $ \mathfrak{B}$ be a homothecy invariant collection of convex sets in $ \mathbb{R}^{n}$. Given a measure $ \mu $, the associated weighted geometric maximal operator $ M_{\mathfrak{B}, \mu }$ is defined by

$\displaystyle M_{\mathfrak{B}, \mu }f(x) := \sup _{x \in B \in \mathfrak{B}}\frac {1}{\mu (B)}\int _{B}\vert f\vert d\mu .$    

It is shown that, provided $ \mu $ satisfies an appropriate doubling condition with respect to $ \mathfrak{B}$ and $ \nu $ is an arbitrary locally finite measure, the maximal operator $ M_{\mathfrak{B}, \mu }$ is bounded on $ L^{p}(\nu )$ for sufficiently large $ p$ if and only if it satisfies a Tauberian condition of the form

$\displaystyle \nu \big (\big \{x \in \mathbb{R}^{n} : M_{\mathfrak{B}, \mu }(\textbf {1}_E)(x) > \frac {1}{2} \big \}\big ) \leq c_{\mu , \nu }\nu (E).$    

As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights $ A_{\infty , \mathfrak{B}}$ for homothecy invariant Muckenhoupt bases $ \mathfrak{B}$ consisting of convex sets. Moreover, it is immediately seen that the strong maximal function $ M_{\mathfrak{R}, \mu }$, defined with respect to a product-doubling measure $ \mu $, is bounded on $ L^{p}(\nu )$ for some $ p > 1$ if and only if

$\displaystyle \nu \big (\big \{x \in \mathbb{R}^{n} : M_{\mathfrak{R}, \mu }(\textbf {1}_E)(x) > \frac {1}{2}\big \}\big ) \leq c_{\mu , \nu }\nu (E)\;$    

holds for all $ \nu $-measurable sets $ E$ in $ \mathbb{R}^{n}$. In addition, we discuss applications in differentiation theory, in particular proving that a $ \mu $-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate $ L^{\infty }(\nu )$.

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Additional Information

Paul Hagelstein
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798

Teresa Luque
Affiliation: Departamento de Analisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain

Ioannis Parissis
Affiliation: Department of Mathematics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland

Keywords: Strong maximal function, Tauberian condition, Muckenhoupt weight
Received by editor(s): August 27, 2013
Published electronically: April 1, 2015
Additional Notes: The first author was partially supported by the Simons Foundation grant 208831.
The second author was supported by the Spanish Ministry of Economy and Competitiveness grant BES-2010-030264
The third author was supported by the Academy of Finland, grant 138738.
Article copyright: © Copyright 2015 American Mathematical Society