(E',E))\psi(x,\omega',E')d\omega'
=
\int_{0}^{2\pi}
\psi(x,\gamma_{\mu_{11}(E',E)}(s),E')ds,$$ as desired, since $\gamma_{\mu_{11}(E',E)}(s)=\gamma(s)$ ($=\gamma_{11}(E',E,\omega)(s)$).

It thus follows that (K\_[11]{})(x,,E) =&\_[I’]{} (\_[11]{} )(x,,E’,E)dE\
=& \_[I’]{}\_[11]{}(E’,E)\_[11]{}(x,E’,E) \_[0]{}\^[2]{}(x,(s),E’)ds.

Let us approximate $\delta$-distribution with a smooth function, say $\eta_\epsilon$ as above. Then $$(\hat {{{\mathcal{}}}K}_{11}\psi)(x,\omega,E',E)
\approx
\hat{\sigma}_{11}(x,E',E)\chi_{11}(E',E)\int_{S'}
\eta_\epsilon(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega'$$ and so &(K\_[11]{})(x,,E’,E) =\_I (\_[11]{} )(x,,E’,E)dE\
& \_[I’]{}\_[S’]{}\_[11]{}(x,E’,E)\_[11]{}(E’,E)\_(’-\_[11]{}(E’,E))(x,’,E’)d’ dE’\
=:& \_[I’]{}\_[S’]{}\_[11]{}(x,’,,E’,E)(x,’,E’)d’ dE’\
=:& (K\_[11,]{})(x,,E), where $K_{11,\epsilon}$ is a partial Schur integral operator satisfying the below assumptions (\[ass2\]).

In the case where $\psi$ has the first order derivatives (in the weak sense) with respect to $\omega$ that is, $\psi$ belongs to the mixed-norm Sobolev-Slobodevskij space $H^{2,(0,1,0)}(G\times S\times I^\circ)$. We conjecture that [K\_[11,]{}-K\_[11]{}]{}\_[L\^2(GSI)]{}C[\_-]{}\_[H\^[-1]{}()]{}\_[H\^[2,(0,1,0)]{}(GSI\^)]{} and so the approximation of $K_{11}$ with $K_{11,\epsilon}$ would be under control.

\[re:S\_Radon:example\] We make the following remark regarding the fact, as explained in Remark \[re:S\_Radon\], that the unit sphere $S$ could have been equipped with more general Radon measure $\rho$, instead of its typical (Lebesgue-measure induced) measure $\mu_S$. Letting $\rho=H^1$ be the 1-dimensional Hausdorff measure on $S$ (see [@falconer86 pp. 7–10]), and writing $$\underline{\sigma}_{11}(x,\omega',\omega,E',E)
=\chi_{11}(E',E)\frac{\sigma_{11}(x,E',E)}{\sqrt{1-\mu_{11}(E',E)^2}}\chi_{{\mathcal{M}}}(\omega',\omega,E',E),$$ where $\chi_{{\mathcal{M}}}$ is the characteristic function of the set $${\mathcal{M}}:=\{(\omega