Functional properties of Generalized H\"ormander spaces of distributions II : Multilinear maps and applications to spaces of functionals with wave front set conditions

We continue our study and applications of generalized H"ormander spaces of distributions ${\mathcal{D}}_{\gamma ,\mathrm{\Lambda }}^{\mathrm{\prime }}$ with $C^{\mathrm{\infty }}$ wavefront set included in a cone $\mathrm{\Lambda }$ and the union of $H^s$-wave front sets in a second cone $\gamma \subset \mathrm{\Lambda }$. We give hypocontinuity results and failure of continuity of tensor multiplication maps between these spaces and deduce hypocontinuity results for various compositions on spaces of multilinear maps. We apply this study to a generalization of microcausal functionals from algebraic quantum field theory with derivatives controlled by spaces either of the form ${\mathcal{D}}_{\gamma ,\mathrm{\Lambda }}^{\mathrm{\prime }}$ or some $ϵ$-tensor product of them. We prove nuclearity and completeness results and give general results to build Poisson algebra structures (with at least hypocontinuous bilinear products). We also apply our general framework to build retarded products with field dependent propagators.