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We prove an abstract Nash-Moser implicit function theorem which, when appliedto control and Cauchy problems for PDEs in Sobolev class, is sharp in terms ofthe loss of regularity of the solution of the problem with respect to the data.The proof is a combination of: (i) the iteration scheme by H\"ormander (ARMA1976), based on telescoping series, and very close to the original one by Nash;(ii) a suitable way of splitting series in scales of Banach spaces, inspired bya simple, clever trick used in paradifferential calculus (for example, byM\'etivier). As an example of application, we apply our theorem to a controland a Cauchy problem for quasi-linear perturbations of KdV equations, improvingthe regularity of a previous result. With respect to other approaches tocontrol and Cauchy problems, the application of our theorem requires lighterassumptions to be verified.

A Nash-Moser-H\"ormander implicit function theorem with applications to control and Cauchy problems for PDEs