Optimal dimension-dependent $\ell^p$ and $\ell^{1,\infty}$ estimates of the second-order discrete Riesz transforms

In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{Γ\left(\frac{d+2}{2}\right)}{π^{d/2}}. \] For $j\neq k$ and