On the maximal directional Hilbert transform in three dimensions
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arXiv
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866929507049406464 |
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| author | Di Plinio, Francesco Parissis, Ioannis |
| author_facet | Di Plinio, Francesco Parissis, Ioannis |
| contents | We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_Ω$ along finite subsets of a finite order lacunary set of directions $Ω\subset \mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2.
The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1712_02673 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the maximal directional Hilbert transform in three dimensions Di Plinio, Francesco Parissis, Ioannis Classical Analysis and ODEs 42B20 We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_Ω$ along finite subsets of a finite order lacunary set of directions $Ω\subset \mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting. |
| title | On the maximal directional Hilbert transform in three dimensions |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/1712.02673 |