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We study maximum principles for a class of linear, degenerate elliptic differential operators of the second order. The Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypotheses on the principal part and on the degeneracy set of the operator. We prove a Poincare inequality, which then allows to define the functional setting where to study weak solutions for equations and inequalities involving this class of operators. A good example of such an operator is the Grushin operator, to which we devote...