Local polynomial functionals are microcausal

Expand the finite polynomial in the definition of $F$ after substituting $\varphi +{\sum }_j{t}_j{h}_j$ and take the coefficient of $t_1\cdots t_r$. Each resulting term is $\int a(x){\prod }_{j=1}^r{\mathrm{\partial }}^{{\alpha }_j}{h}_j(x)d^{4}x$, with $a\in C_c^{\mathrm{\infty }}(\mathbb{M})$. Write $X=(x_1,\dots ,x_r)$, $i(x)=(x,\dots ,x)$ and $A={\sum }_j\mathrm{\mid }{\alpha }_j\mathrm{\mid }$. Its kernel is $u=(-1{)}^A{\mathrm{\partial }}_{x_1}^{{\alpha }_1}\cdots {\mathrm{\partial }}_{x_r}^{{\alpha }_r}[a(x_r){\prod }_{j<r}\delta (x_j-x_r)]$: pairing it with a product test and using distributional differentiation gives the displayed integral. More generally its value on $v(X)\mathrm{\mid }dX\mathrm{\mid }$ is $\int a(x)({\mathrm{\partial }}_{x_1}^{{\alpha }_1}\cdots {\mathrm{\partial }}_{x_r}^{{\alpha }_r}v)(i(x))d^{4}x$. Thus the complete kernel is a finite sum of finite derivatives of smooth compact diagonal distributions, supported on ${\mathrm{\Delta }}_r$. This includes $r=1$, with the empty delta product equal to $1$, and gives the zero kernel above the finite polynomial degree. Derivatives of the relative deltas are retained; they cannot in general all be transferred to $a$.

Fix an arbitrary smooth ambient chart $\psi :O\to V\subseteq {\mathbb{R}}^{4r}$, a point $q=i(x_0)\in O$, and a nonzero chart covector ${\eta }_0$ with $D(\psi \circ i)(x_0{)}^T{\eta }_0\mathrm{\ne }0$. Put $f=\psi \circ i$ on $i^{-1}(O)$. By continuity, there are a relatively compact neighborhood $B$ of $x_0$, an open cone $\mathcal{C}$ about ${\eta }_0$, and $c>0$ such that $\mathrm{\mid }Df(x{)}^T\eta \mathrm{\mid }\ge c\mathrm{\mid }\eta \mathrm{\mid }$ on $\bar{B}\times \mathcal{C}$. Indeed the left side divided by $\mathrm{\mid }\eta \mathrm{\mid }$ is continuous in $x$ and $\eta \mathrm{/}\mathrm{\mid }\eta \mathrm{\mid }$ and positive at $(x_0,{\eta }_0\mathrm{/}\mathrm{\mid }{\eta }_0\mathrm{\mid })$; restrict to a small closed ball and a small closed spherical cap contained in that positive neighborhood. Choose $\chi \in C_c^{\mathrm{\infty }}(V)$ equal to one near $\psi (q)$, with $i^{-1}({\psi }^{-1}(\mathrm{supp}\chi ))\subset B$. Such a localization is obtained by taking a sufficiently small chart ball inside $\psi (O\cap \{X:x_r\in B\})$. Smooth plateaus in nested balls follow explicitly from $H(t)=0$ for $t\le 0$, $H(t)=e^{-1\mathrm{/}t}$ for $t>0$: for radii $0<b<R$, the ratio $H(R^2-\mathrm{\mid }y-y_0{\mathrm{\mid }}^2)\mathrm{/}(H(R^2-\mathrm{\mid }y-y_0{\mathrm{\mid }}^2)+H(\mathrm{\mid }y-y_0{\mathrm{\mid }}^2-b^2))$ is one on the smaller ball and zero outside the larger, and is smooth because all derivatives of $H$ at zero vanish. No family of charts or localizations is selected. In the test-density convention underlying [F1]–[F2], put $J_{\psi }(X)=\mathrm{\mid }\det D\psi (X)\mathrm{\mid }$. For a summand $u$ from step 1.1 its chart Fourier pairing is $⟨u_{\psi },\chi (y)e^{-2\pi iy\cdot \eta }\mathrm{\mid }dy\mathrm{\mid }⟩=\int a(x){{\mathrm{\partial }}_X^{\alpha }\text{\hspace{0.17em}⁣}[J_{\psi }(X)\chi (\psi (X))e^{-2\pi i\eta \cdot \psi (X)}]\mid }_{X=i(x)}{d}^{4}x\mathrm{.}$ Here ${\mathrm{\partial }}_X^{\alpha }$ denotes the finite ambient derivative from step 1.1. The absolute Jacobian is smooth since the chart determinant never vanishes. Repeated product and chain rules make this a finite sum $\int A_{\beta }(x,\eta )e^{-2\pi i\eta \cdot f(x)}{d}^{4}x$, where each amplitude is a polynomial in $\eta$ of degree at most $m:=\mathrm{\mid }\alpha \mathrm{\mid }$, with smooth coefficients supported in one compact $K\subset B$. The coefficients include every derivative of the Jacobian, localization and chart map. Consequently, for every integer $k\ge 0$, their derivatives satisfy ${\max }_{\mathrm{\mid }\gamma \mathrm{\mid }\le k}{\sup }_x\mathrm{\mid }{\mathrm{\partial }}_x^{\gamma }{A}_{\beta }(x,\eta )\mathrm{\mid }\le C_k(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^m$. This follows by bounding the finitely many coefficient derivatives on $K$; no bound on a varying test function is asserted.

For $\eta \in \mathcal{C}$, $\mathrm{\mid }\eta \mathrm{\mid }\ge 1$, set ${\mathrm{\Phi }}_{\eta }(x)=\eta \cdot f(x)$, $b_{\eta }=\mathrm{\nabla }{\mathrm{\Phi }}_{\eta }\mathrm{/}\mathrm{\mid }\mathrm{\nabla }{\mathrm{\Phi }}_{\eta }{\mathrm{\mid }}^2$, and $L_{\eta }=-(2\pi i{)}^{-1}{b}_{\eta }\cdot {\mathrm{\nabla }}_x$. Then $L_{\eta }{e}^{-2\pi i{\mathrm{\Phi }}_{\eta }}=e^{-2\pi i{\mathrm{\Phi }}_{\eta }}$. Every derivative of $\mathrm{\nabla }{\mathrm{\Phi }}_{\eta }$ is bounded by a constant times $\mathrm{\mid }\eta \mathrm{\mid }$ on $\bar{B}$, while its squared norm is at least $c^2\mathrm{\mid }\eta {\mathrm{\mid }}^2$. Differentiating the quotient therefore gives ${\sup }_{\bar{B}}\mathrm{\mid }{\mathrm{\partial }}_x^{\gamma }{b}_{\eta }\mathrm{\mid }\le C_{\gamma }\mathrm{\mid }\eta {\mathrm{\mid }}^{-1}$: equivalently write $b_{\eta }=\mathrm{\mid }\eta {\mathrm{\mid }}^{-1}{b}_{\eta \mathrm{/}\mathrm{\mid }\eta \mathrm{\mid }}$, and differentiate the smooth rational expression on the compact base ball and closed spherical cap where its denominator is bounded below. The transpose for the bilinear integral is $L_{\eta }^{t}A=(2\pi i{)}^{-1}\mathrm{div}(b_{\eta }A)$. Compact support of each amplitude eliminates boundary terms, so $N$ integrations by parts give $\int A_{\beta }{e}^{-2\pi i{\mathrm{\Phi }}_{\eta }}dx=\int (L_{\eta }^t{)}^N{A}_{\beta }{e}^{-2\pi i{\mathrm{\Phi }}_{\eta }}dx,\mid \int A_{\beta }{e}^{-2\pi i{\mathrm{\Phi }}_{\eta }}dx\mid \le C_N\mathrm{\mid }\eta {\mathrm{\mid }}^{m-N}\mathrm{.}$ For completeness, induction on $N$ expands $(L_{\eta }^t{)}^N{A}_{\beta }$ into finitely many terms, each a derivative of $A_{\beta }$ of order at most $N$ times $N$ factors that are derivatives of components of $b_{\eta }$ of orders at most $N$. The bounds of step 2.1 and the preceding quotient bounds give the estimate, and integration over a fixed rectangle containing $K$ supplies its finite volume factor. Integration by parts itself follows componentwise from the ordinary product rule and the fundamental theorem on that rectangle, with zero boundary products. For $\mathrm{\mid }\eta \mathrm{\mid }\le 1$ the original finite polynomial amplitudes give a uniform integral bound directly. Given any requested decay power $p$, choose a finite integer $N\ge p+\max \mathrm{\mid }\alpha \mathrm{\mid }$ over the finitely many kernel terms; the finite sum of all kernel terms then has $(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^p$ times its localized Fourier pairing bounded throughout $\mathcal{C}$.

If $q\mathrm{\notin }{\mathrm{\Delta }}_r$, a localization disjoint from the closed diagonal makes the kernel zero by step 1.1. At a diagonal point, steps 2.1–3.1 exclude every chart direction for which $Df(x_0{)}^T\eta \mathrm{\ne }0$. The intrinsic covector represented by $\eta$ is $\xi =D{\psi }_q^T\eta$; the ordinary chain rule gives $Df(x_0{)}^T\eta =D{i}_{x_0}^T\xi ={\sum }_j{\xi }_j$. Thus, in every arbitrary ambient chart, the only possible singular directions annihilate the tangent to the diagonal. Taking the all-chart union in [F4]'s wavefront convention therefore proves precisely $\mathrm{WF}(F^{(r)}(\varphi ))\subseteq N^{\ast }{\mathrm{\Delta }}_r\setminus 0$. No chart-invariance theorem is invoked. For $r=1$, $i$ is the identity and $D\psi$ is invertible, so every nonzero direction has been excluded and the wavefront set is empty.

At a diagonal point, if ${\xi }_1,\dots ,{\xi }_r$ are all future causal, their metric-dual time components have the same weak sign, and each nonzero causal covector has a strictly nonzero time component. Hence ${\sum }_j{\xi }_j=0$ forces every ${\xi }_j=0$; the same holds for the past cone. Step 4.1 therefore excludes every nonzero tuple in the forbidden cones of [F3]. The finite Taylor expansion of the original polynomial at $\varphi =0$ gives its symmetric compact coefficient kernels $f_m=F^{(m)}(0)\mathrm{/}m!$ (and its constant term), so it is also a polynomial functional in the kernel sense of [F3]. Its derivative kernels have the conormal bound just proved for every smooth $\varphi$ and every $r\ge 1$. This proves the asserted membership in ${\mathcal{F}}_{\mu c}^{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}(\mathbb{M})$. All localizations, finite expansions and integrations above were constructed for a fixed chart and direction; this proof adds no choice principle.