dE’ which is the hyper-singular integral form of $K_{22}$.

Applying Lemma \[hadale\] we see that \[k22-a\] & [H]{}\_2((\_[22,2]{})(x,,,E))(E)\
=& ( \_[E]{}\^[E\_m]{}[[(\_[22,2]{})(x,,E’,E)]{}]{}dE’) - \_[E]{}\^[E\_m]{}[1]{}[E]{}(x,,E’,E)dE’\
& +( (\_[22,2]{})(x,,E’,E))\_[|E’=E]{}\
=& ( [H]{}\_1((\_[22,2]{})(x,,,E))(E)) - [H]{}\_1([E]{}(x,,,E))(E)\
& +( (\_[22,2]{})(x,,E’,E))\_[|E’=E]{},

Finally, we remark that the approximative $\hat {{{\mathcal{}}}K}_{22,j}$ for $j=0,1,2$ are $$(\hat  {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E)
\approx {}&
\int_{S'}\hat\sigma_j(x,E',E)\chi_{22}(E',E)
\eta_\epsilon(\omega'\cdot\omega-\mu_{22}(E,E'))\psi(x,\omega',E')d\omega' \\
=: &
\int_{S'}\tilde\sigma_j(x,\omega',\omega,E',E)\psi(x,\omega',E')d\omega'
=:(\tilde { K}_{22,j}\psi)(x,\omega,E',E).$$ Note that $\tilde K_{22,0}$ is the usual partial Schur integral operator. The approximations $\tilde {{{\mathcal{}}}K}_{22,j}$ may be useful from theoretical and practical point of view.

A. When the spatial dimension $n=2$, $$\begin{gathered}
(\hat {{{\mathcal{}}}K}_{11}\psi)(x,\omega,E',E) \\
=
\hat{\sigma}_{11}(x,E',E)\chi_{11}(E',E)\big(\psi(x,\mu_{11}(E',E)\omega+\sqrt{1-\mu_{11}(E',E)^2}\omega^\perp,E')
\\
+
\psi(x,\mu_{11}(E',E)\omega-\sqrt{1-\mu_{11}(E',E)^2}\omega^\perp,E')\big),\end{gathered}$$ where $\omega^\perp:=(-\omega_2,\omega_1)$ (the tangent vector of the unit circle $S=S_1$ at $\omega$).

B. We compute further some of the above terms. For simplicity we restrict ourselves to the case $n=2$. In this case $$\begin{gathered}
\label{n2k-1}
(\ol {{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,E',E)
=
\hat\sigma_2(x,E',E)\big(\psi(x,\mu_{22}(E',E)\omega+\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp,E')
\\
+
\psi(x,\mu_{22}(E',E)\omega-\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp,E')\big).\end{gathered}$$ Denote $$\xi_{\pm}(E',E,\omega):=\mu_{22}(E',E)\omega\pm\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp.$$ Then \[n2k-2\] ((\_[22,2]{})(x,,E’,E)) =& (\_2(x,E’,E)((x,\_+(E’,E,),E’))\
&+ (\_2(x,E’,E)((x,\_-(E’,E,),E’)). Further