Find all distributions $T$ belonging to $\mathcal{D}(\mathbb{R})$ such that $(x^2)T=0$

Question

My exercise is to find all distributions $T$ on $\mathbb{R}$, such that $(x^2)T=0$. On the lecture we solved the equation $xT=0$, where the solution where all distributions $T$, such that $T=C\delta.$

At that moment i think the solution of $(x^2)T=0$ is the same, but I'm not certain. If You know better and can acknowledge that or give any hint, I'll be grateful.

Have a nice day.

Answer 1

Set $S = xT$. Then the equation is $xS = 0$ which has solutions $S = C\delta$. Then you shall solve $xT = C\delta.$ Do you know any solution to $xT = \delta$? What are the solutions to the corresponding homogeneous equation? Add them.