Conformally invariant quantization at order three

Let (M, g) be a pseudo-Riemannian manifold and F-lambda(M) the space of densities of degree lambda on M. Denote D-lambda,mu(k)(M) the space of differential operators from F-lambda(M) to F-mu(M) of order k and S-delta(k) with delta = mu - lambda the corresponding space of symbols. We construct (the unique) conformally invariant quantization map Q(lambda,mu)(3) : S-delta(3) --> D-lambda,mu(3). This result generalizes that of Duval and Ovsienko.