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

TL;DR

This paper advances the understanding of generalized Hörmander spaces of distributions, focusing on multilinear maps, wavefront set conditions, and their applications in algebraic quantum field theory, including nuclearity and Poisson structures.

Contribution

It introduces hypocontinuity results for tensor multiplication and compositions, and applies these to generalize microcausal functionals and construct Poisson algebra structures.

Findings

Established hypocontinuity and failure of continuity in tensor multiplication

Proved nuclearity and completeness of generalized spaces

Constructed retarded products with field-dependent propagators

Abstract

We continue our study and applications of generalized H\"ormander spaces of distributions $\mathcal{D}'_{\gamma,\Lambda}$ with $C^\infty$ wavefront set included in a cone $\Lambda$ and the union of $H^s$-wave front sets in a second cone $\gamma\subset \Lambda$. 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 $\mathcal{D}'_{\gamma,\Lambda}$ or some $\epsilon$-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.