[en] The concept of conformally equivariant quantization was introduced by Duval, Lecomte and Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure.
A natural generalization of this concept is to seek for a quantization procedure, over a manifold M, that depends on a pseudo-Riemannian metric, is natural, and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte and proved by C. Duval and V. Ovsienko for symbols of degree at most 2 and by S. Loubon Djounga for symbols of degree 3. In two recent papers, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover earlier results on the subject. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4.