${{\mathcal{D}}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M))\subset 
{\mathrm{Hom}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M)).$

The relation between the Schwarzian derivative (\[clas\]) and the cohomology group above is as follows: recall that in the one dimensional case the space ${{\mathcal{S}}}^k_\d (S^1)$ is nothing but ${{\mathcal{F}}}_{\d-k}.$ In this case, the problem of deformation with respect to the Lie algebra ${\mathrm{sl}}_2({\mathbb{R}})$ is related to the cohomology group $$\label{ser}
{\mathrm H}^1({\mathrm{Vect}}(S^1), {\mathrm{sl}}_2 ({\mathbb{R}}); {{\mathcal{D}}}({{\mathcal{F}}}_{\d-k}, {{\mathcal{F}}}_{\d-l})),$$ where $k,l=0,1,2.$ The cohomology group (\[ser\]) was calculated in [@bo1], it is one dimension for $k=2,\, l=0,$ and zero otherwise. The (unique) non-trivial class, for $k=2$ and$l=0,$ can be integrated to the group of diffeomorphisms ${\mathrm{Diff}}(S^1);$ it is a zero-order operator given as a multiplication by the Schwarzian derivative (\[clas\]) (see [@bo1] for more details).

In the multi-dimensional case and for $\d=0,$ the first group of differential cohomology of ${\mathrm{Vect}}(M)$, with coefficients in the space ${{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M))$ of linear differential operators from ${\cal S}^k(M)$ to ${\cal S}^m(M)$ was calculated in [@lo2]. For $n\geq2$ the result is as follows $$\label{cal}
\mathrm H^1({\mathrm{Vect}}(M),  {{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M)))=
\left\{
\begin{array}{ll}
{\mathbb{R}}\oplus \mathrm{H}_{\mathrm{DR}}^1(M), & \mbox{if}\quad  k-m=0,\\
{\mathbb{R}},& \mbox{if} \quad k-m=1,m\not=0,\\
{\mathbb{R}},& \mbox{if} \quad k-m=2,\\
0,&\hbox{otherwise.}
\end{array}
\right.$$ We believe, by analogy for the one-dimensional case, that the “infinitesimal” multi-dimensional 