Finding distributions on $\mathbb{R^2}$ constrained by 2 equations.

Question

The question asks us to classify all distributions in $\mathbb{R}^2$, say $u$, such that $$(xy)u(x,y) = (x^2-y^2)u(x,y)= 0.$$

I am genuinely stuck, any help is extremely appreciated.

Answer 1

Solving $xy \, u(x,y)=0$:

We know that $x\,v(x)=0$ has solutions $v(x)=C\,\delta(x),$ where $C$ is a constant. Generalizing this we get that $y\,u(x,y) = A(y)\,\delta(x)$ for some distribution $A(y).$ Then $u(x,y) = B(y)\,\delta(x) + C(x)\,\delta(y),$ where $A(y)=y\,B(y)$ and $C(x)$ is some distribution. Thus, $$ u(x,y) = \delta(x) \otimes B(y) + C(x) \otimes \delta(y), $$ where $B(y)$ and $C(x)$ are some distributions.

We want to find solutions that also satisfy $(x^2-y^2)\,u(x,y)=0,$ i.e. $$ 0 = (x^2-y^2)(\delta(x) \otimes B(y) + C(x) \otimes \delta(y)) = \delta(x) \otimes (-y^2)\,B(y) + x^2\,C(x) \otimes \delta(y) $$ since $x^2\,\delta(x)=0=y^2\,\delta(y).$

Thus we shall have $\delta(x) \otimes y^2\,B(y) = x^2\,C(x) \otimes \delta(y)$ meaning that $$ \begin{cases} \delta(x) = \lambda \, x^2\,C(x) \\ y^2\,B(y) = \lambda^{-1} \delta(y) \\ \end{cases} $$ for some constant $\lambda.$

The solutions to these two equations are, $$\begin{align} B(y) &= \frac12 \lambda^{-1}\, \delta''(y) + E\,\delta'(y) + F\,\delta(y) \\ C(x) &= \frac12 \lambda \,\delta''(x) + G\,\delta'(x) + H\,\delta(x) \\ \end{align}$$ where $E,F,G,H$ are constants.

Thus, $$ u(x,y) = \delta(x) \otimes (\frac12 \lambda^{-1}\, \delta''(y) + E\,\delta'(y) + F\,\delta(y)) + (\frac12 \lambda \,\delta''(x) + G\,\delta'(x) + H\,\delta(x)) \otimes \delta(y) . $$