Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces

Emma Karoliina Kurki

doi:10.1007/s00025-023-01901-x

Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces

Emma Karoliina Kurki^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

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Abstract

The natural maximal and minimal functions commute pointwise with the logarithm on A_∞. We use this observation to characterize the spaces A₁ and RH_∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt A_p and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between A_p and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A₁ and RH_∞, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.

Original language	English
Article number	123
Pages (from-to)	1-19
Number of pages	19
Journal	Results in Mathematics
Volume	78
Issue number	4
DOIs	https://doi.org/10.1007/s00025-023-01901-x
Publication status	Published - Aug 2023
MoE publication type	A1 Journal article-refereed

Funding

Open Access funding provided by Aalto University. The author was supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.

Keywords

annular decay
Doubling metric space
Muckenhoupt weights
natural maximal function
reverse Hölder inequality

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10.1007/s00025-023-01901-xLicence: CC BY

Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure SpacesFinal published version, 411 KBLicence: CC BY

Cite this

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title = "Limiting Conditions of Muckenhoupt and Reverse H{\"o}lder Classes on Metric Measure Spaces",

abstract = "The natural maximal and minimal functions commute pointwise with the logarithm on A∞. We use this observation to characterize the spaces A1 and RH∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt Ap and reverse H{\"o}lder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between Ap and reverse H{\"o}lder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A1 and RH∞, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.",

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N2 - The natural maximal and minimal functions commute pointwise with the logarithm on A∞. We use this observation to characterize the spaces A1 and RH∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt Ap and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between Ap and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A1 and RH∞, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.

AB - The natural maximal and minimal functions commute pointwise with the logarithm on A∞. We use this observation to characterize the spaces A1 and RH∞ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt Ap and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between Ap and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A1 and RH∞, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.

KW - annular decay

KW - Doubling metric space

KW - Muckenhoupt weights

KW - natural maximal function

KW - reverse Hölder inequality

UR - https://www.scopus.com/pages/publications/85152688811

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DO - 10.1007/s00025-023-01901-x

M3 - Article

AN - SCOPUS:85152688811

SN - 1422-6383

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SP - 1

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JO - Results in Mathematics

JF - Results in Mathematics

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ER -

Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces

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