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 $p=2n$ or $\frac{p}{p-1}=2n$, 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 (Duke Math. J. 168 (2019), no. 3, 471–504).

中文翻译：

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偶数 p 的 Riesz–Titchmarsh 变换的 ℓp 范数

长期以来的猜测是，对于$p\epsilon （1,\mathrm{无穷大}）$这${\ell }^p（\mathbb{Z}）$Riesz–Titchmarsh 离散希尔伯特变换的范数与$L^p（\mathbb{右}）$经典希尔伯特变换的范数，在以下情况下得到验证$p=2n$或者$\frac{p}{p-1}=2n$， 为了$n\epsilon \mathbb{氮}$。这个证明本质上是代数的，在很大程度上取决于对${\ell }^p（\mathbb{Z}）$该运算符的不同变体的范数适用于整个范围$p$。后一个结果最近被作者证明（Duke Math. J . 168 (2019), no. 3, 471–504）。