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

We prove a structure theorem for the differential operator in the 0-term of the $\mathcal{V}$-filtration with respect to a free divisor. Using this theorem, we give a formula for the logarithmic de Rham complex in terms of ${\cal V}_0$-modules. We also give a sufficient condition for perversity of the logarithmic de Rham complex.