# spaces of smooth functions with bounds on partial derivatives

## EDIT: As there were no takers at all... I have added below a possible approach I came up with...

I would like to ask the following elementary but tricky question about the density of spaces of smooth functions.

We are given spaces $\mathcal{J}^{u,v}, \mathcal{K}^u, \mathcal{J}^v$ such that

$\bigcap_{(u,v) \in \mathbb{R}^2} \mathcal{J}^{u,v}(\mathbb{R}^{2l+2k}) = S(\mathbb{R}^{2l + 2k})$ and

$\bigcap_{u \in \mathbb{R}} \mathcal{K}^u(\mathbb{R}^{2l}) = S(\mathbb{R}^{2l})$,

$\bigcap_{v \in \mathbb{R}} \mathcal{J}^v(\mathbb{R}^{2k}) = S(\mathbb{R}^{2k})$.

QUESTION: The tensor product $\mathcal{K}^u \otimes \mathcal{L}^v \subset \mathcal{J}^{u,v}$ see below. Is this tensor product with the induced $C^{\infty}$ Fr\'echet topology dense?

DEFINITIONS:

Here $\mathcal{J}^{u,v}(\mathbb{R}^{2l + 2k}) \subset C^{\infty}(\mathbb{R}^{2l + 2k})$, $\mathcal{K}^u(\mathbb{R}^{2l}) \subset C^{\infty}(\mathbb{R}^{2l})$ and $\mathcal{L}^v(\mathbb{R}^{2k}) \subset C^{\infty}(\mathbb{R}^{2k})$ are classes of smooth functions fulfilling some estimates on partial derivates. Here $u, v$ are arbitrary real numbers.

These estimates being:

For an $f$ in the first class: For each multi-indices $\gamma_1, \gamma_2 \in \mathbb{N}_0^l$ and $\gamma_3, \gamma_4 \in \mathbb{N}_0^k$ we can find a $C > 0$ such that $|\partial_z^{\gamma_1} \partial_w^{\gamma_2} \partial_x^{\gamma_3}$ $\partial_y^{\gamma_4} f(z, w, x, y)|$ $\leq C (1 + |z| + |w|)^{u - |\gamma_1| - |\gamma_2|} (1 + |x| + |y|)^{v - |\gamma_3| - |\gamma_4|}.$

For a $g$ in the second class: For each multi-indices $\gamma_1, \gamma_2 \in \mathbb{N}_0^l$ we find $C > 0$ such that $|\partial_z^{\gamma_1} \partial_w^{\gamma_2} g(z, w)|$ $\leq C (1 + |z| + |w|)^{u - |\gamma_1| - |\gamma_2|}$.

For $h$ in the third class: For each multi-indices $\gamma_3, \gamma_4 \in \mathbb{N}_0^k$ we find $C > 0$ such that $|\partial_x^{\gamma_3} \partial_y^{\gamma_4} g(x, y)|$ $\leq C (1 + |x| + |y|)^{v - |\gamma_3| - |\gamma_4|}$.

Define the semi-norms $p_{\gamma_1, \cdots, \gamma_4}$ as follows: $p_{\gamma_1, \cdots, \gamma_4}(f) := \sup_{w \in \mathbb{R}^{2l + 2k}} \frac{|\partial_{w_1}^{\gamma_1} \cdots \partial_{w_4}^{\gamma_4} f(w)|}{(1 + |w_1| + |w_2|)^{-u + |\gamma_1| +|\gamma_2|} (1 + |w_3| + |w_4|)^{-v + |\gamma_3| + |\gamma_4|}}$.

Hence the first space $\mathcal{J}^{u,v}$ becomes a Fr\'echet space with regard to the topology induced by these seminorms. The other two spaces have semi-norm systems defined in the analogous way and are therefore Fr\'echet as well.

Observation: I) It is interesting to note that the Schwartz space $S(\mathbb{R}^{2l}) \otimes S(\mathbb{R}^{2k})$ is dense in $S(\mathbb{R}^{2l + 2k})$. Also $S(\mathbb{R}^{2l})$ is contained in $\mathcal{K}^u(\mathbb{R}^{2l})$ and a closed subset. As well as $S(\mathbb{R}^{2k})$ contained and closed in $\mathcal{L}^v$ and $S(\mathbb{R}^{2l+2k})$ in $\mathcal{J}^{u,v}$. Because these spaces are just closed and not dense this is seems not useful in resolving the density question. But for me it gives a hint that a counterexample might lie in looking at polynomials of different orders. At least a usual Stone-Weierstraß argument fails it seems.

II) The compactly supported smooth functions $C_c^{\infty}$ are contained in the Schwartz spaces and dense in $C^{\infty}$, at least in the $C^{\infty}$-topology of convergence on compact subsets. We have the inclusions:

$C_c^{\infty}(\mathbb{R}^{2l}) \otimes C_c^{\infty}(\mathbb{R}^{2k}) \subset \mathcal{K}^u \otimes \mathcal{J}^v$.

As well as $\mathcal{J}^{u,v} \subset C^{\infty}(\mathbb{R}^{2l + 2k})$.

The idea is to use that $C_c^{\infty} \otimes C_c^{\infty}$ is dense in $C_c^{\infty}(\mathbb{R}^{2l + 2k})$ combined with the density in $C^{\infty}(\mathbb{R}^{2k + 2l})$. Is there a mistake somewhere?

EDIT: Rearranged the question. Added a further observation. Small correction.

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