In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the Coulomb energy in $d=2$? $$D(f,g):=-\displaystyle\int\_{\mathbb{R}^2}\int\_{\mathbb{R}^2}\overline{f(x)}g(y)\log{\|x-y\|}~dxdy$$ Is $\int f=0$, $f\in L^1(\mathbb{R}^2)$ sufficient for $D(f,f)\geq 0$? References would be appreciated.



Positivity of the Coulomb energy in two dimensions
This is true when the support of $f$ is contained in the unit disc. If the support is contained in a disc $|z|<R$, then $(f,f)$ is bounded from below by a constant that depends
on $R$. This minor nuisance makes the logarithmic potential somewhat different from the Newtonian
potential, however most statements of potential theory are similar for these two cases,
or can be easily modified.
For the details, the standard reference is
MR0350027 Landkof, N. S. Foundations of modern potential theory. Springer-Verlag, 
New York-Heidelberg, 1972.


