Generalized functions of several variables

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

Prove that a solution of the equation $L u = v$ ($v \in \mathcal{D}^{\prime }(\mathbb {R}^{n})$ a given function) can be found in the form $u = E * v$, where $E$ is a fundamental solution of the operator $L$ (if this convolution exists).

Prove that the fundamental solution for the operator $L y = y^{(n)}+a_{n-1} y^{(n-1)}+\ldots +a_{0} y$ in $\mathcal{D}^{\prime }(\mathbb {R})$, where $a_{k} \in \mathbb {C}$, has the form $\theta (x) y_{0}(x)$, where $y_{0}$ is the solution of the equation $L y = 0$ with initial conditions $y_{0}(0) = y_{0}^{\prime }(0) = \ldots = y_{0}^{(n-2)}(0) = 0, y_{0}^{(n-1)}(0) = 1$.

Find the fundamental solution of the operator $L$:

Let $l(y) = y^{(n)}+a_{n-1} y^{(n-1)}+\ldots +a_{0} y$ be a differential expression with constant coefficients $a_{k} \in \mathbb {C}$. Define an operator $A \in \mathcal{B}(C[0,1]), A: x \mapsto y$, as the solution of the differential equation $l(y) = x$ with initial conditions $y(0) = y^{\prime }(0) = \ldots = y^{(n-1)}(0) = 0$. Prove that $A$ is an integral operator, i.e., $(A x)(t) = \int_{0}^{1} K(t, s) x(s) d s$. Find the function $K(t, s), \operatorname {Ker} A$ and $\operatorname {Im} A$.

Find the fundamental solution for the Cauchy-Riemann operator $\frac{\partial }{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial }{\partial x}+i \frac{\partial }{\partial y}\right)$.

Find the fundamental solution for the Laplace operator $\Delta = \frac{\partial^{2}}{\partial x_{1}^{2}}+\ldots +\frac{\partial^{2}}{\partial x_{n}^{2}}, n \geqslant 2$.

Prove that the fundamental solution of a differential operator with constant coefficients cannot have compact support.

Prove that the solution of the Cauchy problem $L[u(t, x)] = 0, u(0, x) = \frac{\partial u}{\partial t}(0, x) = \ldots = \frac{\partial^{m-2} u}{\partial t^{m-2}}(0, x) = 0, \frac{\partial^{m-1} u}{\partial t^{m-1}}(0, x) = v(x) \quad$ ($v \in \mathcal{E}^{\prime }(\mathbb {R}^{n})$ a given function) can be found in the form $u = E(t, x) * v(x)$, where $E$ is the fundamental solution of the Cauchy problem for the operator $L$ (the convolution is taken with respect to the variables $x$).

Find the fundamental solution of the Cauchy problem for the operators

Find the solution $E(t, x)$ of the Cauchy problem $L(E) = 0, E(0, x) = \delta (x)$, for the operator $L = \frac{\partial^{2}}{\partial t^{2}}+\frac{\partial^{2}}{\partial x^{2}}$, satisfying the condition $E(t, \cdot ) \rightarrow 0$ as $t \rightarrow \infty$.