Consider the torus $\mathbb T^2:=\mathbb C/(\mathbb Z+i \mathbb Z)$ and the operator $$ T = (2D_{\bar z}-\lambda)^{-1} $$ on the torus with periodic boundary conditions. This one is well-defined for $\lambda \notin 2\pi ( \mathbb Z+i \mathbb Z)$ where $2D_{\bar z}= (-i \partial_{x}+\partial_y).$

My question is if there is anything known about the Green's function $K$ satisfying $$ Tf(x) = \int_{\mathbb T^2} K(x,y) f(y) \ dy $$ for $f$ a periodic function.

* Addendum:* The case $\lambda=0$ is actually rather simple and can be easily deduced from the paper

*Class of hypocomplex structures on the two dimensional torus*. But is probably already well-known in this much simpler case. Then, in terms of the Jacobi $\Theta$ function $$ T f(x) = \int_{\mathbb T^2} M(x,y) f(y) \ dy $$ where $$ M(x,y) = \frac{\Theta'(y-x+\frac{1+i}{2})}{\Theta(y-x+\frac{1+i}{2})}. $$

diagonal_approximationto $M(x,y)$, for small $|x-y|$, is given on page 1073 of Dimers and famillies of Cauchy-Riemann operators. $\endgroup$