The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913238089728000 |
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| author | Bañuelos, Rodrigo Kwaśnicki, Mateusz |
| author_facet | Bañuelos, Rodrigo Kwaśnicki, Mateusz |
| contents | The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the $\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full range of $p$. The latter result was recently proved by the authors in [Bañuelos, Kwaśnicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00027 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$ Bañuelos, Rodrigo Kwaśnicki, Mateusz Classical Analysis and ODEs Functional Analysis Primary: 42A50, 42A05. Secondary: 39A12 The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the $\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full range of $p$. The latter result was recently proved by the authors in [Bañuelos, Kwaśnicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504]. |
| title | The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$ |
| topic | Classical Analysis and ODEs Functional Analysis Primary: 42A50, 42A05. Secondary: 39A12 |
| url | https://arxiv.org/abs/2210.00027 |