I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, locally, one can write $\Gamma(x-y) = c\log(|x-y|) + h(x,y)$, where $h(x,y)$ is harmonic in $x$.
I think one could prove this in a quite simple way by
- fixing a cut-off function $\rho \in C^\infty(\mathbb R^2)$, supported close to $(0,0)$; $\rho$ can also be interpreted as a function on the torus, or rather, we can consider the periodization $\rho^{(p)}(x) = \sum_{k\in \mathbb Z^2} \rho(x-k) \in C^\infty(\mathbb T^2)$ (which I won't do below).
- Let $\psi \in C^\infty(\mathbb T^2)$ be solution to $$\Delta \psi = -c[2\nabla \rho(x-y)\cdot \nabla \log(|x-y|) + (\Delta \rho(x-y))\,\log(|x-y|)].$$
- By construction, the function $\Gamma(x,y) = \rho(x-y) \log(|x-y|) + \psi(x,y)$ solves $\Delta \Gamma = \delta(x-y)$; and it has the claimed form, locally.
However, I'd like to have a reference for this. Does anyone know of a good reference?