Discrete analogues of second-order Riesz transforms

Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete Analogues in Harmonic Analysis (DAHA)}. In this paper, we study the $\ell^p$ norms of discrete analogues of second-order Riesz transforms. Using probabilistic methods, we construct a new class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ on the lattice $\mathbb{Z}^d$, $d \ge 2$. We show that for $1<p<\infty$, their $\ell^p(\mathbb{Z}^d)$ norms coincide with those of the classical second-order Riesz transforms $R^{(jk)}$ on $L^p(\mathbb{R}^d)$ when $j \neq k$, and are comparable up to dimensional constants when $j = k$. The operators $\mathcal{R}^{(jk)}$ differ from the discrete analogue $R^{(jk)}_{\mathrm{dis}}$ by convolution with an $\ell^1(\mathbb{Z}^d)$ function. Applications are given to the DAHA of the Beurling--Ahlfors operator. We also show that $\mathcal{R}^{(jk)}$ arise as discrete analogues of certain Calder\'on--Zygmund operators $\mathbf{R}^{(jk)}$, which differ from $R^{(jk)}$ by convolution with an $L^1(\mathbb{R}^d)$ function. Finally, we conjecture that the $L^p$ norms of $\mathcal{R}^{(jk)}$, $R^{(jk)}_{\mathrm{dis}}$, and $\mathbf{R}^{(jk)}$ agree with those of the classical Riesz transforms, known to equal the corresponding martingale transform norms.