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The purpose of this paper is to derive some Lewy-Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy-Stampacchia estimates in more classical cases as well. In particular, we can recover the known estimates for the standard Laplacian, the p-Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy-Stampacchiadoi:10.4171/rmi/750 fatcat:2w2tnlr4ebdopdmcgnqk4ezvgm