Discrete Besov spaces via semigroups associated to the discrete Laplacian and regularity of non-local operators

In this paper we prove characterizations of the discrete Besov spaces in terms of the heat and Poisson semigroups associated with the discrete Laplacian that will allow us to prove regularity results for the fractional powers of the discrete Laplacian and the discrete Bessel potentials. Moreover, we provide new estimates for the derivatives of the discrete heat kernel and semigroup which are of independent interest.