Tensor product distributions and iterated pairings

Fix compact $K\subseteq U\times V$ and let $K_U,K_V$ be its compact projections. F2 gives bounds $C_u,p_{m_u}$ for $u$ on $K_U$ and $C_v,p_{m_v}$ for $v$ on $K_V$. For $\mathrm{\Phi }\in {\mathcal{D}}_K$, the inner paired function has support in $K_U$ by F1, and F3 gives its derivatives by pairing $x$ derivatives of $\mathrm{\Phi }$. Applying the two bounds yields $\mathrm{\mid }(u\otimes v)(\mathrm{\Phi })\mathrm{\mid }\le C_u{C}_v{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m_u,\mathrm{\mid }\beta \mathrm{\mid }\le m_v}{\sup }_{K_U\times K_V}\mathrm{\mid }{\mathrm{\partial }}_x^{\alpha }{\mathrm{\partial }}_y^{\beta }\mathrm{\Phi }\mathrm{\mid }\le C_u{C}_v{p}_{m_u+m_v}(\mathrm{\Phi })$. F2 proves it is a distribution. The same argument applies with the pairing order reversed.

The two orders have the same values on all products by F1. Their difference is a continuous functional zero on all finite product sums, so F4 makes it zero on every test. The same reasoning shows uniqueness of any distribution with the product values.

If $x\mathrm{\notin }\mathrm{supp}u$, choose an open neighborhood $A$ of $x$ on which $u=0$. On tests supported in $A\times V$, the outer test in F1 is supported compactly in $A$, so the tensor vanishes. If $y\mathrm{\notin }\mathrm{supp}v$, the reversed pairing of step 2.1 gives the corresponding vanishing near $U\times \{y\}$. F5 proves support containment in the product. Conversely, if $(x,y)$ lies in that product, every open neighborhood contains $A\times B$ with $x\in A,y\in B$. F5 implies there exist tests $\phi \in \mathcal{D}(A)$, $\psi \in \mathcal{D}(B)$ with nonzero pairings; otherwise one factor would vanish on that neighborhood. F1 makes their product pairing nonzero. Thus the tensor does not vanish on any neighborhood of $(x,y)$, proving equality of supports. Only two local witnesses were used.

Applying the distribution construction to two blocks at a time gives distributions $(u\otimes v)\otimes w$ and $u\otimes (v\otimes w)$ on a triple product, each taking value $u(\phi )v(\psi )w(\eta )$ on pure triple tests. To prove equality, fix $\eta$: their difference on $H(x,y)\eta (z)$ is a continuous functional in $H$, since multiplying by this fixed test preserves compact support and bounds derivatives by fixed constants. F4 in $U\times V$ makes it zero for every $H$. F4 again, now for $(U\times V)\times W$, makes the original difference zero on every triple test. This proves associativity and agreement of the nested orders.

For the $x$ derivative, F6 applied to the outer distribution and F3 applied to the inner test give $(({\mathrm{\partial }}^{\alpha }u)\otimes v)(\mathrm{\Phi })=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}(u\otimes v)({\mathrm{\partial }}_x^{\alpha }\mathrm{\Phi })$, the asserted derivative. For the $y$ derivative apply F6 directly to the inner pairing. If a factor is zero, all formulas give zero and the support product is empty; empty domains behave the same way. Order-zero derivatives are identities. All estimates and density passages were choice-free. $\mathrm{\square }$