Is the tensor product of distributions a continuous bilinear map with respect to the weak topology?

Question

Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is it continuous with respect to the weak-* topology?

Answer 1

I will use the convention $\mathbb{N}=\{1,2,\ldots\}$ and denote by $s(\mathbb{N})$ the space of (real) sequences $(\mu_i)_{i\ge 1}$ of rapid decay, i.e., such that for all integer $k\ge 0$, $$ \|\mu\|_k:=\sup_{i}i^k|\mu_i|\ <\infty\ . $$ We give it the locally convex topology defined by the norms $\|\cdot\|_k$, indexed by integers $k\ge 0$.

The dual can be identified with the space $s'(\mathbb{N})$ of sequences $(a_i)_{i\ge 1}$ which grow at most polynomially in $i$. We will write $$ \langle \mu,a\rangle:=\sum_i \mu_i a_i $$ for the duality pairing.

We can similarly define the two index generalizations $s(\mathbb{N}^2)$ and $s'(\mathbb{N}^2)$. The analogue of the tensor product of distributions is the map $s'(\mathbb{N})\times s'(\mathbb{N})\rightarrow s'(\mathbb{N}^2)$ given by $(a,b)\mapsto a\otimes b$ where $$ (a\otimes b)_{i,j}:=a_i b_j\ . $$ Saying that this bilinear map is continuous, where all spaces carry the weak-$\ast$ topology amounts to saying that $\forall \rho\in s(\mathbb{N}^2)$, $\exists p, q$, $\exists \mu^{(1)},\ldots,\mu^{(p)},\nu^{(1)},\ldots,\nu^{(q)}\in s(\mathbb{N})$, such that $\forall a,b\in s'(\mathbb{N})$, $$ \left|\sum_{i,j}\rho_{i,j}a_i b_j\right|\le\left(\sum_{m=1}^{p}|\langle\mu^{(m)},a\rangle|\right) \times \left(\sum_{n=1}^{q}|\langle\nu^{(n)},b\rangle|\right)\ . $$ Lemma: Let $\mu^{(1)},\ldots,\mu^{(p)}$ and $\mu$ be elements of $s(\mathbb{N})$. Suppose that for all $a\in s'(\mathbb{N})$, if $\langle\mu^{(1)},a\rangle=\cdots=\langle\mu^{(p)},a\rangle=0$ then $\langle\mu,a\rangle=0$. This hypothesis implies that $\mu$ is in the linear span of $\mu^{(1)},\ldots,\mu^{(p)}$.

For the proof, package the $\mu$'s into a $(p+1)\times\infty$ matrix and do row operations to reduce to row echelon form.

The above inequality and the lemma immediately show that the rows of the matrix $\rho_{i,j}$ must span a finite-dimensional space. So, pick say $\rho_{i,j}=e^{-i}\delta_{i,j}$ and this proves that the tensor product is not continuous for the weak-$\ast$ topology.

Now, as I said in my comment, the correct topology to use is the strong topology. If all spaces carry it then saying that $(a,b)\mapsto a\otimes b$ is continuous amounts to saying

$\forall \rho\in s(\mathbb{N}^2)$, $\exists p, q$, $\exists \mu^{(1)},\ldots,\mu^{(p)},\nu^{(1)},\ldots,\nu^{(q)}\in s(\mathbb{N})$, such that $\forall a,b\in s'(\mathbb{N})$, $$ \sum_{i,j}\left|\rho_{i,j}a_i b_j\right|\le\left(\sum_{m=1}^{p}\sum_i|\mu^{(m)}_i a_i|\right) \times \left(\sum_{n=1}^{q}\sum_j|\nu^{(n)}_j b_j|\right)\ . $$

See the difference? One has traded absolute values of sums for sums of absolute values.

The statement about the strong topology is true, with $p=q=1$, because given $\rho\in s(\mathbb{N}^2)$, $\exists \mu,\nu\in s(\mathbb{N})$, such that for all $i,j$, $$ |\rho_{i,j}|\le |\mu_i|\times|\nu_j|\ . $$

Finally, using isomorphisms with sequence spaces the above proofs also shows continuity of the tensor product operation $\mathscr{S}'(\mathbb{R})\times\mathscr{S}'(\mathbb{R})\rightarrow \mathscr{S}'(\mathbb{R}^2)$.