Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
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- by Paul Hagelstein, Teresa Luque and Ioannis Parissis PDF
- Trans. Amer. Math. Soc. 367 (2015), 7999-8032 Request permission
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 \begin{align*} M_{\mathfrak {B}, \mu }f(x) := \sup _{x \in B \in \mathfrak {B}}\frac {1}{\mu (B)}\int _{B}|f|d\mu . \end{align*} 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 \begin{align*} \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). \end{align*} 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 \begin{align*} \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)\; \end{align*} 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 $L^{\infty }(\nu )$.References
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Additional Information
- Paul Hagelstein
- Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
- MR Author ID: 683523
- ORCID: 0000-0001-5612-5214
- Email: paul_hagelstein@baylor.edu
- Teresa Luque
- Affiliation: Departamento de Analisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain
- Email: tluquem@us.es
- Ioannis Parissis
- Affiliation: Department of Mathematics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland
- MR Author ID: 827096
- ORCID: 0000-0003-3583-5553
- Email: ioannis.parissis@gmail.com
- 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. - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 7999-8032
- MSC (2010): Primary 42B25; Secondary 42B35
- DOI: https://doi.org/10.1090/S0002-9947-2015-06339-9
- MathSciNet review: 3391907