Let $A$ and $B$ be linear operators acting on $L^2(\mathbb{R})$ or some dense subset of that space. We assume that they are integral operators (possibly with distributional kernel) $Af(x)=\int_{\mathbb{R}}K_A(x,y) f(y) dy$, $f\in \mathcal{D}(A)\subset L^2(\mathbb{R})$ and analogically for $B$ (of course in the case of distributional kernels this integral should be replaced by distributional bracket). Formally kernel of product of operators $AB$ can be expressed as $K_{A B}(x,y)=\int_{\mathbb{R}}K_A(x,z)K_B(z,y) dz$. I'm looking for books and articles considering under what assumptions on kernels $K_A$ and $K_B$ the last formula or some distributional generalization of it has sense. I'm particularly interested in the case where $K_A,K_B\in \mathcal{S}'(\mathbb{R}^2)$ or $K_A\in \mathcal{S}'(\mathbb{R}^2)$ and $K_B\in L^2(\mathbb{R}^2)$.
