Riesz transforms for bounded Laplacians on graphs

We study several problems related to the ℓ^p boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of the gradient of a function, we prove for p∈ (1 , 2] an ℓ^p estimate for the gradient of the continuous time heat semigroup, an ℓ^p interpolation inequality as well as the ℓ^p boundedness of the modified Littlewood–Paley–Stein function for a graph with bounded Laplacian. This yields an analogue to Dungey’s results in [21] while removing some additional assumptions. Coming back to the classical notion of the gradient, we give a counterexample to the interpolation inequality and hence to the boundedness of Riesz transforms for bounded Laplacians for 1 < p< 2. Finally, we prove the boundedness of the Riesz transform for 1 < p< ∞ under the assumption of positive spectral gap.