Let
$$f(z,w)=\frac1{|z|^{2a}+|w|^{2b}}$$
be a function on $\mathbb C^2$, where $a$ and $b$ are positive reals.
How may I find all $(a,b)$ with $a>0$ and $b>0$ for which $f$ is locally Lebesgue integrable at $(0,0)$?
I have deduced that both $a$ and $b$ have to be greater than or equal to $1$, but is there a more accurate result?