Hi I'm looking for some help with the function $$f(s,t)=a(2a-1)|t-s|^{2a-2}.$$ In particular I'm trying to calculate the integral $$\int_\mathbb{R}\int_\mathbb{R}f(s,t)dsdt=a(2a-1)\int_\mathbb{R}\int_\mathbb{R}|t-s|^{2a-2}dsdt.$$
As Eric Wofsey pointed out this indefinite integral is always infinite, unless a(2a−1)=0.
This integral is always infinite (unless $a(2a-1)=0$). Indeed, let $A\subset\mathbb{R}^2$ be the region where $|t-s|\geq 1$ and let $B\subset\mathbb{R}^2$ be the region where $|t-s|<1$. Note that $A$ and $B$ both have infinite area. If $2a-2\geq 0$, then the integrand $|t-s|^{2a-2}$ is always at least $1$ on $A$, and so the integral is infinite on $A$. If $2a-2<0$, then $|t-s|^{2a-2}$ is always at least $1$ on $B$, and again the integral is infinite.