The ${\mathrm{\ell }}^p$ norm of the Riesz--Titchmarsh transform for even integer $p$

The long-standing conjecture that for $p\in (1,\mathrm{\infty })$ the ${\mathrm{\ell }}^p(\mathbb{Z})$ 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 ${\mathrm{\ell }}^p(\mathbb{Z})$ 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 ${\mathrm{\ell }}^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].