Consider Schwartz's distribution on $\mathbb{R}^2$. Let $$L=a\partial_x^2+b\partial_x\partial_y+c\partial_y^2$$ and $A:=\{(x,y)\in\mathbb{R}^2|y\geq |x|\}$. The problem asks if $L\chi_A=\delta$ as distribution, determine $a,b,c$. (Here $\chi_{A}$ means characteristic function of $A$, and $\delta$ means the Dirac measure weighted at original point of the 2-d plane.)
I guess the core procedure shall lie in computing Fourier transform , but I don't quite know how to deal with $\chi_{A}$ since $A$ is no bounded and hence its Fourier transform is necessarily not in $L^2$, which seems to be obstacle.