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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.15570 |
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| _version_ | 1866912443799699456 |
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| author | la Cigoña, Fernando Benito-de Borges, Tainara D'Emilio, Francesco Pasquariello, Marcus Wagner, Nathan A. |
| author_facet | la Cigoña, Fernando Benito-de Borges, Tainara D'Emilio, Francesco Pasquariello, Marcus Wagner, Nathan A. |
| contents | We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv:2309.13943. Moreover, we identify a subclass of shifts, called $L^1$-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted $L^p$ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-Hölder inequality for scalar weights, which was used in arXiv:1710.03397 to establish $L^p$ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix Weighted $L^p$ Estimates in the Nonhomogeneous Setting la Cigoña, Fernando Benito-de Borges, Tainara D'Emilio, Francesco Pasquariello, Marcus Wagner, Nathan A. Classical Analysis and ODEs 42B20, 42B35 We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv:2309.13943. Moreover, we identify a subclass of shifts, called $L^1$-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted $L^p$ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-Hölder inequality for scalar weights, which was used in arXiv:1710.03397 to establish $L^p$ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition. |
| title | Matrix Weighted $L^p$ Estimates in the Nonhomogeneous Setting |
| topic | Classical Analysis and ODEs 42B20, 42B35 |
| url | https://arxiv.org/abs/2506.15570 |