$L^p$-estimates for singular integral operators along codimension one subspaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916619472601088 |
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| author | Flórez-Amatriain, Mikel |
| author_facet | Flórez-Amatriain, Mikel |
| contents | In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first $ n-1 $ variables of $ \mathbb{R}^n $. Assuming the subspace to be non degenerate in the sense that it is away from a cone around $e_n$ and the function $ f $ to be frequency supported in a cone away from $ \mathbb{R}^{n-1} $, we prove $ L^p $-bounds for these operators for $ p > 3/2 $. If we assume, additionally, that $ \widehat{f} $ is supported in a single frequency band, we are able to extend the boundedness range to $ p >1 $. The non-degeneracy assumption cannot in general be removed, even in the band-limited case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$-estimates for singular integral operators along codimension one subspaces Flórez-Amatriain, Mikel Classical Analysis and ODEs 42B20 In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first $ n-1 $ variables of $ \mathbb{R}^n $. Assuming the subspace to be non degenerate in the sense that it is away from a cone around $e_n$ and the function $ f $ to be frequency supported in a cone away from $ \mathbb{R}^{n-1} $, we prove $ L^p $-bounds for these operators for $ p > 3/2 $. If we assume, additionally, that $ \widehat{f} $ is supported in a single frequency band, we are able to extend the boundedness range to $ p >1 $. The non-degeneracy assumption cannot in general be removed, even in the band-limited case. |
| title | $L^p$-estimates for singular integral operators along codimension one subspaces |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2502.13079 |