I am wondering if the following integral has some known properties / name in literature. Consider the double convolution with an intermediate non-trivial product ($h$ is not a constant): \begin{equation} \mathcal{I}=g\ast(h(g\ast f) ) = \int_{-\infty}^{\infty}g_{(t_3-t_2)}h_{(t_2)}\int_{-\infty}^{\infty}g_{(t_2-t_1)}f_{(t_1)}dt_1 dt_2 \end{equation} Changing order of integration, this can be rewritten as: \begin{equation} \mathcal{I}= \int_{-\infty}^{\infty} K_{(t_1,t_3)}f_{(t_1)}dt_1 \end{equation} with the kernel function: \begin{equation} K_{(t_1,t_3)} = \int_{-\infty}^{\infty} g_{(t_3-t_2)}g_{(t_2-t_1)}h_{(t_2)}dt_2 \end{equation} Is there any literature name / useful properties for this kernel? The kernel is not a convolution power since $h$ is not constant.
Edit
Assuming $h_{(t_2)}$ admits a power expansion, then: \begin{equation} h_{(x)} = \sum_{n=0}^{\infty} h_n x^n \end{equation} Using the change of variables $t_2=t_1+\tau$ we have: \begin{equation} K_{(t_1, t_3)} = \int_{-\infty}^{\infty} g_{(t_3-t_1-\tau)}g_{(\tau)} h_{(t_1+\tau)} d\tau \end{equation} which degenerates to a convolution if $h$ is a constant.
Since: \begin{equation} h_{(t_1+\tau)} = \sum_{n=0}^{\infty} \sum_{k=0}^{n} h_n \left(\begin{matrix}n \\ k \end{matrix}\right) t_1^k \tau^{n-k} \end{equation} Then: \begin{equation} K_{(t_1, t_3)} = K'_{(t_3-t_1,t_1)} = \sum_{n=0}^{\infty} \sum_{k=0}^{n} h_n \left(\begin{matrix}n \\ k \end{matrix}\right) t_1^k \int_{-\infty}^{\infty} g_{(t_3-t_1-\tau)}g_{(\tau)} \tau^{n-k} d\tau = \sum_{n=0}^{\infty} \sum_{k=0}^{n} h_n \left(\begin{matrix}n \\ k \end{matrix}\right) t_1^k \left(g \ast (\tau^{n-k}g)\right)_{t=t_3-t_1} \end{equation} Which are related to derivative of the Fourier Transform of $g$. The problem now is that $\hat{g}$ is not differentiable in my case.
This results also means that $\mathcal{I}$ can be written as a sum of convolutions involving $t_1^kf_{(t_1)}$ in this special case, which are in turn related to the derivatives of the Fourier Transform of $f_{(t_1)}$. This integral transform appears when dealing with some spectral representations of differential equations. Are any other properties of this known?