Abstract: Gundy and Varopoulos introduced the probabilistic representation of singular integrals and Fourier multipliers, such as Hilbert transforms, and Riesz transforms, as conditional expectations of some stochastic integrals. Combined with the sharp martingale inequalities by Burkholder and Banuelos-Wang, the representations have played a crucial role in finding the sharp, or nearly sharp, $L^p$-bounds for these operators in a variety of geometric settings. Motivated by a recent breakthrough of Banuelos and Kwasnicki on the sharp $\ell^p$-norm of the discrete Hilbert transform, we construct a natural collection of discrete operators on $\mathbb{Z}^{d}$ which have $\ell^p$-norms independent of the dimension. This collection of discrete operators includes the probabilistic discrete Riesz transforms, which are the analogs of the probabilistic discrete Hilbert transform used in the paper by Banuelos-Kwasnicki. In this talk, we discuss the construction of the probabilistic discrete operators, their $\ell^p$ bounds, and related open problems. This is based on joint work with Rodrigo Banuelos and Mateusz Kwasnicki.
