Regular polynomial field functional algebra — Examples

For real test directions $h,k$, the coefficient of $st$ in $L_g(1+sh+tk)$ is $12\int g(x)h(x)k(x)d^{4}x$, by expanding $(1+sh+tk{)}^4$. By [F3], this is exactly the pairing of $K_g$ with the product test $h\otimes k$. Thus $K_g$ represents the second derivative on product tests.

For every nonempty fixed compact $K\subseteq X\times X$, choose a compact rectangle $R$ containing its first-coordinate projection. Then $\mathrm{\mid }⟨K_g,\mathrm{\Psi }⟩\mathrm{\mid }\le 12\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{vol}(R)\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}_{\mathrm{\infty }}(\mathrm{supp}\mathrm{\Psi }\subseteq K),$ so $K_g$ is continuous on each fixed-support test space and hence on the LF test topology by [F7]. It vanishes on every test supported off the diagonal, so its support is contained there by [F6]. Choose $\rho \in \mathcal{D}({\mathbb{R}}^4)$ with $\rho (0)=1$ using [F5], and set $\mathrm{\Psi }(x,y)=g(x)\rho (x-y)$. This is a test function: its support has $x\in \mathrm{supp}g$ and $x-y\in \mathrm{supp}\rho$, so both coordinate projections are compact. Then $⟨K_g,\mathrm{\Psi }⟩=12{\int }_{{\mathbb{R}}^4}g(x{)}^2{d}^{4}x>0,$ because a nonzero continuous real function is bounded away from zero on a small ball. Thus $K_g\mathrm{\ne }0$.

Suppose $L_g$ had a regular presentation. By [F1], its second derivative at $1$ would have a smooth compactly supported kernel. Its pairings with product tests agree with those of $K_g$ by step 1.1; complex linearity extends agreement to complex product tests, and density [F2] extends it to all tests. Hence the smooth kernel would equal $K_g$ as a distribution. Its support would be contained in the diagonal by step 2.1.

The diagonal has empty interior. A continuous function supported there must vanish: any nonzero value would persist on an open neighborhood not contained in the diagonal. This contradicts $K_g\mathrm{\ne }0$ from step 2.1. Therefore $L_g$ has no regular presentation.

By step 4.1, $L_g\mathrm{\notin }{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$. Fact [F8] defines ${\star }_W$ only on pairs of regular polynomial functionals, and [F9] defines each $T_n^H$ on regular polynomial inputs. Neither operation therefore applies to $L_g$. Any extension across the diagonal is a separate construction and is not proved here.

Substituting the derivatives in [F1] into [F3], terms with $k\ge 3$ vanish, so the only contributions are $k=0,1,2$.

The $k=0$ term is ${\mathrm{\ell }}_f^2{\mathrm{\ell }}_g^2$. The $k=1$ term is $\eta ⟨W,(2{\mathrm{\ell }}_{f}f)\otimes (2{\mathrm{\ell }}_{g}g)⟩=4\eta W(f,g){\mathrm{\ell }}_f{\mathrm{\ell }}_g$.

The $k=2$ term is $\frac{{\eta }^2}{2}⟨W^{\otimes 2},(2f\otimes f)\otimes (2g\otimes g)⟩=2{\eta }^{2}W(f,g{)}^2$, because the two independent pairings each equal $W(f,g)$. Adding the three contributions proves the displayed formula.