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In this paper a Malliavin calculus for L\'evy processes based on a family oftrue derivative operators is developed. The starting point is an extension toL\'evy processes of the pioneering paper by Carlen and Pardoux [8] for thePoisson process, and our approach includes also the classical Malliavinderivative for Gaussian processes. We obtain a sufficient condition for theabsolute continuity of functionals of the L\'evy process. As an application, weanalyze the absolute continuity of the law of the solution of some stochasticdifferential equations.

Local Malliavin Calculus for L\'evy Processes and Applications