Space of linear differential operators on the real line as a module over the Lie algebra of vector fields

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One of the basic structures on the space of linear differential operators is a natural family of module structures over the group of diffeomorphisms Diff(M) (and of the Lie algebra of vector fields Vect(M)). These Diff(M)(and Vect(M))-module structures are defined if one considers the arguments of differential operators as tensor-densities of degree λ on M. In this paper we consider the space of differential operators on R.1 Denote by D the space of kth-order linear differential operators