Discrete analogues of second-order Riesz transforms

Discrete analogues of many classical operators in harmonic analysis have been widely studied for many years with interesting connections to other areas in mathematics, including ergodic theory and analytic number theory. This paper concerns the problem of identifying the $\ell^p$-norms of discrete analogues of second-order Riesz transforms. Using probabilistic techniques, a class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ is constructed on the lattice $\mathbb{Z}^d$, $d\geq 2$. It is shown that their $\ell^p(\mathbb{Z}^d)$ norms, $1<p<\infty$, are the same as the norms of the classical second-order Riesz transforms $R^{(jk)}$ in $L^p(\mathbb{R}^d)$. The operators $\mathcal{R}^{(jk)}$ differ from the harmonic analysis analogues of the discrete Riesz transforms, $R^{(jk)}_{\mathrm{dis}}$, by convolution with a function in $\ell^1(\mathbb{Z}^d)$. Applications are given to the discrete Beurling-Ahlfors operator. It is shown that the operators $\mathcal{R}^{(jk)}$ arise by discretization of a class of Calder\'on-Zygmund singular integrals $T^{(jk)}$ which differ from the classical Riesz transforms $R^{(jk)}$ by convolution with a function in $L^1(\mathbb{R}^d)$.