Writing a distribution as a tensor product

Question

Definition about tensor product of distributions:

Let $u_j \in \mathcal{D}'(\Omega_j)$, $j=1,2$. There is only $u \in \mathcal{D}'(\Omega_1 \times \Omega_2)$ such that $$\langle u, \varphi_1 \otimes \varphi_2 \rangle=\langle u_1, \varphi_1 \rangle \langle u_2, \varphi_2 \rangle$$ $\forall \varphi_j \in C^{\infty}_{c}(\Omega_j).$ Furthermore, if $\varphi \in C^{\infty}_{c}(\Omega)$ then \begin{align} \langle u, \varphi \rangle&=\langle u_1, \langle u_2, \varphi(x_1,x_2)\rangle \rangle \\ &=\langle u_2, \langle u_1, \varphi(x_1,x_2) \rangle \rangle. \end{align}

My problem is

If $u \in \mathcal{D}'(\mathbb{R}^{n})$ and $x_nu=0$ then there is $v\in \mathcal{D}'(\mathbb{R}^{n-1})$ such that $$u(x)=v(x') \otimes \delta(x_n),$$ where $x=(x',x_n)$.

I think I should use the uniqueness of the definition about tensor product.

Thank you for your help.

Answer 1

Take $\varphi_1 \in \mathcal{D}'(\mathbb{R}^{n-1})$ and $\varphi_2 \in \mathcal{D}'(\mathbb{R}).$

If $\varphi_2(0)=0$ then there exists $\psi \in \mathcal{D}'(\mathbb{R})$ such that $\varphi_2(x) = x \, \psi(x)$ giving $$ \langle u, \varphi_1 \otimes \varphi_2 \rangle = \langle u, \varphi_1 \otimes x \psi \rangle = \langle x_n u, \varphi_1 \otimes \psi \rangle = \langle 0, \varphi_1 \otimes \psi \rangle = 0. $$

Otherwise, take $\rho \in \mathcal{D}'(\mathbb{R})$ such that $\rho(0)=1,$ and set $\hat{\varphi}_2 = \varphi_2 - \varphi_2(0) \rho.$ Then $\hat{\varphi}_2(0) = 0$ so $\langle u, \varphi_1 \otimes \hat{\varphi}_2 \rangle = 0.$ Thus $$ \langle u, \varphi_1 \otimes \varphi_2 \rangle = \langle u, \varphi_1 \otimes (\hat{\varphi}_2 + \varphi_2(0) \rho) \rangle = \langle u, \varphi_1 \otimes \hat{\varphi}_2 \rangle + \langle u, \varphi_1 \otimes \varphi_2(0) \, \rho \rangle \\ = \langle u, \varphi_1 \otimes \rho \rangle \varphi_2(0) . $$

Now, define $v_\rho \in \mathcal{D}'(\mathbb{R}^{n-1})$ by $\langle v_\rho, \varphi_1 \rangle = \langle\langle u(x',x_n), \rho(x_n)\rangle, \varphi_1(x')\rangle .$ Then $$ \langle u, \varphi_1 \otimes \rho \rangle \varphi_2(0) = \langle\langle u(x',x_n), \rho(x_n)\rangle, \varphi_1(x')\rangle \, \langle \delta, \varphi_2 \rangle = \langle v_\rho, \varphi_1\rangle \langle \delta, \varphi_2 \rangle = \langle v_\rho \otimes \delta, \varphi_1 \otimes \varphi_2 \rangle. $$

Thus, $ \langle u, \varphi_1 \otimes \varphi_2 \rangle = \langle v_\rho \otimes \delta, \varphi_1 \otimes \varphi_2 \rangle $ and by uniqueness, we must have $u = v_\rho \otimes \delta.$