Superposition in Modulation Spaces with Ultradifferentiable Weights

In the theory of nonlinear partial differential equations we need to explain superposition operators. For modulation spaces equipped with particular ultradifferentiable weights this was done in \cite{rrs}. In this paper we introduce a class of general ultradifferentiable weights for modulation spaces $\mathcal{M}^{w_*}_{p,q}(\mathbb{R}^n)$ which have at most subexponential growth. We establish analytic as well as non-analytic superposition results in the spaces $\mathcal{M}^{w_*}_{p,q}(\mathbb{R}^n)$.