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The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear operators and we show that our previous results can be appropriately generalized to this case.

A general form of the weak maximum principle and some applications