.$

Let us prove that the operator (\[MultiSchwar1\]) vanishes on the conformal group ${\mathrm{O}}(p+1,q+1)$ in the case when $M:={\mathbb{R}}^n$ is endowed with the flat metric $\g_0:=\mathrm{diag}(1,\ldots,1,-1,\ldots,-1)$ whose trace is $p-q.$ Any $f\in {\mathrm{O}}(p+1,q+1)$ satisfies ${{f^*}}^{-1} {\g_0}=F\cdot{\g_0},$ where $F$ is a non-zero positive function. This relation implies $$\begin{aligned}
2 \ell(f)^k_{ij} +\mathrm{Sym}_{i,j}\,\ell(f)^s_{it}\,\g_0^{tk}\,
{\g_0}_{sj}&=&\frac{1}{F}\mathrm{Sym}_{i,j}\, \partial_i F\, \delta^k_j, 
\nonumber \\ 
\mathrm{Sym}_{i,j}\, \ell(f)^s_{li} \, {\g_0}_{sj}&=&\frac{\partial_l F}{F}
{\g_0}_{ij},\nonumber \\
\ell(f)_t&=& \frac{n}{2}\frac{\partial_t F}{F}\cdot  
\nonumber\end{aligned}$$ Sibstitute these formul[æ]{} into (\[MultiSchwar1\]). Then we obtain by straightforward computation that ${\cal A}(f)\equiv 0.$\

------------------------------------------------------------------------

Suppose now that ${\rm dim }(M)>2.$

Our second main definition is the linear differential operator ${\cal B}(f)$ acting from ${\cal S}^2_{\d}(M)$ to ${\cal S}^0_\d(M)$ defined by $$\begin{aligned}
{\cal B}(f)_{ij}&=&
{f^*}^{-1} \left(\g^{st}\, \g_{ij}\nabla_s\nabla_t \right ) -\g^{st}\,\g_{ij} 
\nabla_s \nabla_t
+c_1\,\left( \ell(f)_{ij}^s -\frac{1}{n}\, 
\mathrm{Sym}_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right)\nabla_s \nonumber \\
&&+\,c_2 \, \ell(f)_i\,\ell(f)_j  +c_3 \n_{s} 
\left ( \ell(f)_{ij}^s-\frac{1}{n}
\,\mathrm{Sym}_{i,j}\, \d_{i}^s \ell_{j}(f)\right) +c_4\, \ell(f)_{ij}^s\ell(f)_s 
\nonumber  \\
&& +\,c_5\, \ell(f)_{si}^u\ell(f)_{uj}^s + c_6\left( 
 {f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ),
\label{MultiSchwar2}\end{aligned}$$ where $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ and the constants $c_1,\ldots,c_6,$ are given by $$\begin{array}{ll}
c_1=2+n(1-2\delta ), & \displaystyle c_2=\frac{(2+n(1-2\delta ))(\d- 1)}{n}, 
\nonumber \\[3mm]
c_3=\displaystyle \frac{(2+n(1-2\delta ))(\d n-2)}{n-2} , &c_4=
\displaystyle \frac{(2+n(1-2\delta ))(2\d-2)}{n-2}, \\[3mm]
c_5= \displaystyle \frac{(2+n(1-2\delta ))(1-\d)n}{n-2},& c_6=
\displaystyle \frac{n(\d 