Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor

We prove a structure theorem for differential operators in the 0-th term of the V-filtration relative to a free divisor, manifold. As an application, we give a formula for the logarithmic de Rham complex with respect to a fre e divisor in terms of V-0-modules, which generalizes the classical formula for the usual de Rham complex in terms of D-modules, and the formula of Esn ault-Viehweg in the case of a normal crossing divisor. We also give a suffi cient algebraic condition for perversity of the logarithmic de Rham complex . (C) Elsevier, Paris.