Compactly supported polynomial microcausal functionals

Let $M$ be an oriented, time-oriented, globally hyperbolic smooth Lorentzian $n$-manifold without boundary, $n\ge 2$, with signature $(-,+,\dots ,+)$. A cotangent vector is future or past causal when its metric dual belongs to the corresponding closed causal tangent cone. For $r\ge 1$, let ${\bar{V}}_{\pm }^r\subset T^{\ast }{M}^r\setminus 0$ consist of tuples $(x_1,\dots ,x_r;{\xi }_1,\dots ,{\xi }_r)$ such that every ${\xi }_j$ is either zero or future causal for $+$, respectively past causal for $-$, and the total tuple is nonzero. The wavefront set used below is the set of nonzero cotangent tuples where compactly localized Fourier transforms fail rapid decay in every conic neighborhood.

A compactly supported polynomial functional is a finite expression $F(\varphi )={\sum }_{m=0}^N⟨f_m,{\varphi }^{\otimes m}{\mu }_g^{\otimes m}⟩,$ where $\varphi \in C^{\mathrm{\infty }}(M;\mathbb{R})$, $f_m\in {\mathcal{E}}^{\mathrm{\prime }}(M^m)$ is symmetric under permutations of its $m$ variables, ${\mu }_g$ is the metric density fixed by the Lorentzian metric, and the pairing is defined by inserting a compactly supported cutoff equal to one near $\mathrm{supp}{f}_m$ in the smooth factor. Its $r$th functional derivative $F^{(r)}(\varphi )\in {\mathcal{E}}^{\mathrm{\prime }}(M^r)$ is specified, for a test density $\omega$ on $M^r$, by $⟨F^{(r)}(\varphi ),\omega ⟩={\sum }_{m=r}^N\frac{m!}{(m-r)!}⟨f_m,\omega \otimes {\varphi }^{\otimes (m-r)}{\mu }_g^{\otimes (m-r)}⟩,$ where a compactly supported cutoff equal to one near $\mathrm{supp}{f}_m$ is inserted in the smooth $\varphi$ factors. Expanding the finite polynomial shows that this kernel represents the directional derivative in smooth compactly supported real directions. The functional is microcausal when $\mathrm{WF}(F^{(r)}(\varphi ))\cap ({\bar{V}}_{+}^r\cup {\bar{V}}_{-}^r)=\mathrm{\varnothing }$ for every $r\ge 1$ and every $\varphi \in C^{\mathrm{\infty }}(M;\mathbb{R})$. Write ${\mathcal{F}}_{\mu c}^{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}(M)$ for this class, including constants.

Scalar distributions on smooth manifolds

Wavefront and positive-frequency Hadamard data

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)