论文标题
Riesz的$ \ ell^p $规范 - 甚至整数$ p $
The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$
论文作者
论文摘要
长期以来的猜想是，$ p \ in（1，\ infty）$ $ \ ell^p（\ mathbb z）$ riesz-- titchmarsh nilbert hilbert转换与$ l^p（\ m athbb r）$相同，$ l^p（\ mathbb r）$ n $ p = 2 n $ = 2 n $ = 2 n $ \ frac} $ n \ in \ mathbb n $。本质上是代数的证明，以一种至关重要的方式取决于$ \ ell^p（\ mathbb z）$ norm for $ p $的全部范围。后者的结果最近由[Bañuelos，Kwaśnicki，在$ \ ell^p $ -Norm的$ \ ell^p $ norm，duke Math。 J. 168（3）（2019）：471-504]。
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].
