How can I compute the fundamental solution for $$\frac{1}{4}\partial_x^2\partial_y^2=0$$ Can anyone assist me in computing this.
So far I have: $(\frac{1}{4}\partial_x^2\partial_y^2)u=\delta$ $\implies$ $(\frac{1}{4}|x|^2|y|^2)\hat{u}=1$ $\implies$ $\hat{u}=\frac{4}{|x|^2|y|^2}$, how can I get the inverse FT?
Solving $\partial_x^2\partial_y^2u=\delta$ by removing one differentiation at a time:
$$ \partial_x^2\partial_y^2u(x,y)=\delta(x)\delta(y) \\ \partial_x\partial_y^2u(x,y)=\frac12\operatorname{sign}(x)\delta(y)+A''(y) \\ \partial_y^2u(x,y)=\frac12|x|\delta(y)+xA''(y)+B''(y) \\ \partial_yu(x,y)= \frac12|x|\frac12\operatorname{sign}(y)+xA'(y)+B'(y)+C(x)\\ u(x,y)=\frac12|x|\frac12|y|+xA(y)+B(y)+C(x)y+D(x) $$ where $A,B,C,D$ are arbitrary distributions.
Note: To reduce the amount of introduced names, first introducing one name for a distribution, then another for its primitive distribution, and yet another for the second primitive distribution, I "backtraced" the names and use derivatives early on.