I am working through the paper A General Theory of Equivariant CNNs on Homogeneous Spaces. The paper is primarily aimed at a computer science audience.
Here is the setting: we have a locally compact and unimodular group $G$, homogeneous space $B$ and stabilizer subgroup $H \leq G$. Furthermore, we have a vector space $V \cong \mathbb{R}^n$ carrying a group representation $\rho$ of $H$. The paper now defines the vector space $\mathcal{I_G}$ of Mackey functions as all functions $f\colon G \rightarrow V$ that satisfy the equivariance constraint $f(gh) = \rho(h^{-1})f(g)$. No further restrictions on $\mathcal{I_G}$ are stated, but, up to my understanding, continuity is assumed implicitly throughout the paper.
In formula $(5)$ of section 3 (at the very bottom of page 5 of the paper), there is the following statement: given two representations $(\rho_i, V_i)$ of $H$ and corresponding Mackey function spaces $\mathcal{I}^{i}_G$, any bounded linear operator $\mathcal{I}^{1}_G \rightarrow \mathcal{I}^{2}_G$ can be written as \begin{equation} [\kappa\cdot f](g) = \int_G\kappa(g, g') f(g')dg' \end{equation} where $\kappa(g, g')\colon G \times G \rightarrow \text{Hom}(V_1, V_2)$ is a linear operator-valued kernel (or matrix-valued kernel if we choose bases).
This looks like some version of the Schwartz kernel theorem. However, I could not find any such theorem in the context of Haar integrals. I also tried looking for similar results in the works of Mackey. However, since I am not from a mathematics background, I found those to be quite impenetrable.
Do you know a version of the Schwartz theorem that reduces to the above statement as a special case? Or is that statement only true under stricter assumptions, e.g. on the function space $\mathcal{I}_G$?
In general, I would be very happy about any advice on sources that could be helpful in this context.