Additional operations on generalized functions

Prove that the action of the convolution of generalized functions $F, G \in \mathcal{D}^{\prime }$ can be expressed through the direct product:

Compute the direct product of generalized functions of one variable:

Do direct products of generalized functions from $\mathcal{D}^{\prime }$ form an everywhere dense set in $\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right)$? And their linear combinations?

Find the pushforward of the function $\delta_{0}^{(k)} \times \delta_{0}^{(n)}$ under the map $\gamma : \mathbb {R}^{2} \rightarrow \mathbb {R}, \gamma (x, y) = x+y$.

Find the pullback of the $\delta$-function under the map $\gamma : \mathbb {R}^{2} \rightarrow \mathbb {R}, \gamma (x, y) = x$.

Find the pullback of the $\delta$-function under the map $\gamma : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}, \gamma (x, y) = \left(x+e^{y}, y-1\right)$.

Find the pushforward and pullback of the $\delta$-function under the linear map $\gamma : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}, \gamma (x) = B x+v$, where the matrix $B$ is nonsingular.

Prove that the pullback of the $\delta$-function under the map $\gamma : \mathbb {R}^{2} \rightarrow \mathbb {R}, \gamma (x, y) = x y$, does not exist as an element of $\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right)$, but can be defined on the subspace $\left\{ \varphi \in \mathcal{D}\left(\mathbb {R}^{2}\right): \varphi (0,0) = 0\right\}$ as follows:

Sometimes this action is written concisely by the Dirac formula