Inverses of Product Kernels and Flag Kernels on Graded Lie Groups

Let $T(f)=f\ast K$, where $K$ is a product kernel or a flag kernel on a direct product of graded Lie groups $G=G_1\times \cdots \times G_{\nu }$. Suppose $T$ is invertible on $L^2(G)$. We prove that its inverse is given by $T^{-1}(g)=g\ast L$, where $L$ is a product kernel or a flag kernel accordingly.