Problem: Find all distributions $f$ in $\mathcal D'(\mathbb R^2)$ so that $xyf=0$ and $(x^2-y^2)f=0.$ My Thoughts: Note that $c\delta_{(0,0)}$ where $c\in\mathbb C$ satisfies the distributional equations above. It appears that $\operatorname{supp}(u)=\{x=0\}\cap\{y=0\}=(0,0)$ which would imply that $u$ is a finite linear combination of $\delta_0$ and its derivatives. Is my reasoning on the right track? Apologies if I am not showing enough details, I am trying to make sure that my intuition is on point. Thank you for your time.
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.
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