The abstract lewy-Stampacchia inequality and applications

We identify submodularity as the key ingredient needed to get the Lewy–Stampacchia inequality in the potential case, by showing how it can be used in a simple and effective way to produce a very abstract and general version of such estimate.We then discuss how to reproduce more classical versions of it and, more importantly, how it can be used in conjunction with Laplacian comparison estimates to produce large class of functions with bounded Laplacian on spaces with a lower bound on the Ricci curvature.