The ℓ p $\ell ^p$ norm of the Riesz–Titchmarsh transform for even integer p $p$

Abstract

The long-standing conjecture that for $p\in (1,\infty )$ the ${\ell }^p\left(Z\right)$ norm of the Riesz–Titchmarsh discrete Hilbert transform is the same as the $L^p\left(R\right)$ norm of the classical Hilbert transform, is verified when $p=2n$ or $\frac{p}{p-1}=2n$, for $n\in N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the ${\ell }^p\left(Z\right)$ 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).