Matrix-weighted estimates beyond Calderón-Zygmund theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913328524165120 |
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| author | Kakaroumpas, Spyridon Nguyen, Thu Hien Vardakis, Dimitris |
| author_facet | Kakaroumpas, Spyridon Nguyen, Thu Hien Vardakis, Dimitris |
| contents | We investigate matrix-weighted bounds for the sublinear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl. We extend their result to sublinear operators acting upon vector-valued functions. First, we dominate these operators by bilinear convex body sparse forms, adapting a recent general principle due to T. Hytönen. Then we use this domination to derive matrix-weighted bounds, adapting arguments of F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Our requirements on the weight are formulated in terms of two-exponent matrix Muckenhoupt conditions, which surprisingly exhibit a rich structure that is absent in the scalar case. Consequently, we deduce that our matrix-weighted bounds improve the ones that were recently obtained by A. Laukkarinen.
The methods we use are flexible, which allows us to complement our results with a limited range extrapolation theorem for matrix weights, extending the results of P. Auscher and J. M. Martell, as well as M. Bownik and D. Cruz-Uribe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix-weighted estimates beyond Calderón-Zygmund theory Kakaroumpas, Spyridon Nguyen, Thu Hien Vardakis, Dimitris Classical Analysis and ODEs We investigate matrix-weighted bounds for the sublinear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl. We extend their result to sublinear operators acting upon vector-valued functions. First, we dominate these operators by bilinear convex body sparse forms, adapting a recent general principle due to T. Hytönen. Then we use this domination to derive matrix-weighted bounds, adapting arguments of F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Our requirements on the weight are formulated in terms of two-exponent matrix Muckenhoupt conditions, which surprisingly exhibit a rich structure that is absent in the scalar case. Consequently, we deduce that our matrix-weighted bounds improve the ones that were recently obtained by A. Laukkarinen. The methods we use are flexible, which allows us to complement our results with a limited range extrapolation theorem for matrix weights, extending the results of P. Auscher and J. M. Martell, as well as M. Bownik and D. Cruz-Uribe. |
| title | Matrix-weighted estimates beyond Calderón-Zygmund theory |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2404.02246 |