Tame algebra estimates for product and flag kernels on graded Lie groups

We prove that product kernels and flag kernels on a direct product of graded Lie groups $G_1\times \cdots \times G_{\nu }$ satisfy so-called \emph{tame algebra estimates}. Tame algebra estimates are central to the study of nonlinear partial differential equations via, for instance, the Nash-Moser inverse function theorem. In addition, the special structure of these estimates generates a new Banach-algebraic proof of an inversion theorem for product kernels and flag kernels.