BMO and the John-Nirenberg Inequality on Measure Spaces
We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions ha...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2020-01-01
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| Series: | Analysis and Geometry in Metric Spaces |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/agms-2020-0115 |
| _version_ | 1818732590836744192 |
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| author | Dafni Galia Gibara Ryan Lavigne Andrew |
| author_facet | Dafni Galia Gibara Ryan Lavigne Andrew |
| author_sort | Dafni Galia |
| collection | DOAJ |
| description | We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢. |
| first_indexed | 2024-12-17T23:36:00Z |
| format | Article |
| id | doaj.art-8489f9637d6a4a5ab9bbff6acf09b02c |
| institution | Directory Open Access Journal |
| issn | 2299-3274 |
| language | English |
| last_indexed | 2024-12-17T23:36:00Z |
| publishDate | 2020-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Analysis and Geometry in Metric Spaces |
| spelling | doaj.art-8489f9637d6a4a5ab9bbff6acf09b02c2022-12-21T21:28:33ZengDe GruyterAnalysis and Geometry in Metric Spaces2299-32742020-01-018133536210.1515/agms-2020-0115agms-2020-0115BMO and the John-Nirenberg Inequality on Measure SpacesDafni Galia0Gibara Ryan1Lavigne Andrew2Concordia University,Montréal, CanadaUniversité Laval, Département de mathématiques et de statistique, Québec, QC G1V 0A6, CanadaMcGill University,Montréal, CanadaWe study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.https://doi.org/10.1515/agms-2020-0115bounded mean oscillationjohn-nirenberg inequalitymuckenhoupt weightsdecomposable measure spacesprimary 30l15 42b35 46e30 |
| spellingShingle | Dafni Galia Gibara Ryan Lavigne Andrew BMO and the John-Nirenberg Inequality on Measure Spaces Analysis and Geometry in Metric Spaces bounded mean oscillation john-nirenberg inequality muckenhoupt weights decomposable measure spaces primary 30l15 42b35 46e30 |
| title | BMO and the John-Nirenberg Inequality on Measure Spaces |
| title_full | BMO and the John-Nirenberg Inequality on Measure Spaces |
| title_fullStr | BMO and the John-Nirenberg Inequality on Measure Spaces |
| title_full_unstemmed | BMO and the John-Nirenberg Inequality on Measure Spaces |
| title_short | BMO and the John-Nirenberg Inequality on Measure Spaces |
| title_sort | bmo and the john nirenberg inequality on measure spaces |
| topic | bounded mean oscillation john-nirenberg inequality muckenhoupt weights decomposable measure spaces primary 30l15 42b35 46e30 |
| url | https://doi.org/10.1515/agms-2020-0115 |
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