Bounded oscillation operators on BMO spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915661457915904 |
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| author | Karagulyan, Grigori A. |
| author_facet | Karagulyan, Grigori A. |
| contents | Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calderón-Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are $BO$ operators. $BO$ operators are defined on abstract measure spaces equipped with a basis of abstract balls. The abstract balls in their definition owe four basic properties of classical balls in $\mathbb{R}^n$, which are crucial in the study of singular operators on $\mathbb{R}^n$. Among various properties studied in these papers it was proved that $BO$ operators allow pointwise sparse domination, establishing the $A_2$-conjecture for those operators. In the present paper we study boundedness properties of $BO$ operators on $BMO$ spaces. In particular, we prove that general $BO$ operators boundedly map $L^\infty$ into $BMO$, and under a logarithmic localization condition those map $BMO$ into itself. We obtain these properties as corollaries of new local type bounds, involving oscillations of functions over the balls. We apply the results in the $BMO$ estimations of Calderón-Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces $H^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded oscillation operators on BMO spaces Karagulyan, Grigori A. Classical Analysis and ODEs 42B20, 42B35, 43A85 Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calderón-Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are $BO$ operators. $BO$ operators are defined on abstract measure spaces equipped with a basis of abstract balls. The abstract balls in their definition owe four basic properties of classical balls in $\mathbb{R}^n$, which are crucial in the study of singular operators on $\mathbb{R}^n$. Among various properties studied in these papers it was proved that $BO$ operators allow pointwise sparse domination, establishing the $A_2$-conjecture for those operators. In the present paper we study boundedness properties of $BO$ operators on $BMO$ spaces. In particular, we prove that general $BO$ operators boundedly map $L^\infty$ into $BMO$, and under a logarithmic localization condition those map $BMO$ into itself. We obtain these properties as corollaries of new local type bounds, involving oscillations of functions over the balls. We apply the results in the $BMO$ estimations of Calderón-Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces $H^1$. |
| title | Bounded oscillation operators on BMO spaces |
| topic | Classical Analysis and ODEs 42B20, 42B35, 43A85 |
| url | https://arxiv.org/abs/2512.07736 |