A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

For given measurable functions $g,h:\mathbb{R}^d \rightarrow \mathbb{R}$ and a (weighted) $L^2$-function $v$ on $\mathbb{R}$ we study existence and uniqueness of a solution $w$ to the integral equation $v(x) = \int_{\mathbb{R}^d} g(s) w(h(s)x)ds$. Such integral equations arise in the study of infinitely divisible moving average random fields. As a consequence of our solution theory to the aforementioned equation, we can thus derive non-parametric estimators for the L\'{e}vy density of the underlying random measure.