Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent $$ R(s)=(\Delta_{hyp}-s(1-s))^{-1}, $$ where $s$ is in a suitable open subset of $\mathbf{C}$. It is known that $R(s)$ is an integral operator which has a kernel which equals to: $$ G_s(z,w):=-\frac{\Gamma(s)^2}{4\pi\cdot\Gamma(2s)}\cdot(1-\tanh^2 \frac{r}{2}))^s\cdot {}_{2}F_1\left(s,s,2s;1-\tanh^2 \frac{r}{2}\right), $$ where $r:=d_{hyp}(z,w)$, $z=x+iy,w\in\mathfrak{h}$, and $d_{hyp}$ is the hyperbolic distance. Here one may think of $w$ as the kernel variable over which one integrates. In particular, if we set $s=1$ in $G_s(z,w)$, then we obtain the classical hyperbolic Green function on $\mathfrak{h}$ which reduces in this case to $-\frac{1}{2\pi}\cdot\log |\frac{z-w}{z-\overline{w}}|$. Since the operator $\Delta_{hyp}$ is invariant under the group of hyperbolic rotations (isomorphic to $SO(2)$) it follows that the kernel of the resolvent (for a fixed $s$) must satisfy an ODE of order 2 in the variable $r$, where $r$ corresponds to the hyperbolic radius. The fact that we are also looking at a function which has a logarithmic singularity at $z=w$ (so it should behave like $-\frac{1}{2\pi}\log r$ when $r$ is small) singles out a special line in the two dimensional solution space of this ODE.

Replace now $\mathfrak{h}$ by $\mathfrak{h}^2=\mathfrak{h}\times \mathfrak{h}$ (more generally we could consider say $g$ copies of $\mathfrak{h}$) with its product metric and let $\Delta_{hyp,2}$ be the corresponding Laplacian acting on $\mathfrak{h}^2$.

**Question**: Do we have an explicit expression for the kernel of the resolvent of $\Delta_{hyp,2}$ on $\mathfrak{h}^2$ ?

If we try to follow the same strategy as in the case where $g=1$, I guess that the invariance of the Laplacian under $SO(2)\times SO(2)$ would reduce the problem to solving a certain PDE in TWO variables, namely $r_1$ and $r_2$, the two radii corresponding to each copy of $\mathfrak{h}$. I don't know much about PDE's and in general they seem to be underdetermined which frightens me a little bit... Any help or advice would be more than appreciated.