I am dealing with some derivatives on Dirac delta function, $(x\partial_x+y\partial_y)\delta(x+y)$. Consider the following integral ($f(x,y)$ is bounded) and perform integration by parts, \begin{align} &\int f(x,y)(x\partial_x+y\partial_y)\delta(x+y) dxdy \\ =& -\int \left(2f(x,y)+x\partial_xf(x,y)+y\partial_yf(x,y)\right)\delta(x+y) dxdy \end{align}
Now I can also perform a coordinate transformation into $\{u, v\}$, $$x=\frac{u+v}{\sqrt{2}},\quad y=\frac{u-v}{\sqrt{2}}$$ \begin{align} &\int f(x,y)(x\partial_x+y\partial_y)\delta(x+y) dxdy \\ =& \int \tilde{f}(u,v)(u \partial_u+v\partial_v)\delta(\sqrt{2}u) dudv \\ =& \int \tilde{f}(u,v) \sqrt{2}u \frac{\partial}{\partial \sqrt{2}u} \delta(\sqrt{2}u) dudv \\ =& -\int \tilde{f}(u,v)\delta(\sqrt{2}u) dudv\\ =&-\int f(x,y)\delta(x+y) dxdy. \end{align}
Comparing both it seems that \begin{align} \int\left(2f(x,y)+x\partial_xf(x,y)+y\partial_yf(x,y)\right)\delta(x+y)&=\int f(x,y)\delta(x+y)\\ \int \left(f(x,y)+x\partial_xf(x,y)+y\partial_yf(x,y)\right)\delta(x+y)&=0 \end{align}
Is the above derivation correct? I have tested the above identity using many $f(x,y)$ and it seems it is correct. If it is true, I wonder if there is another proof (direct proof?) of this identity. Thank you very much!
Update: the above identity cannot be true for bounded functions. As Bertram pointed out, it does not work for $1$. However, does it work for functions with compact support?