Green functions on Riemann surfaces

Question

Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ be the Green function of $\Delta_g$ that is to say $$\Delta_g G_g( .  , y) =\delta_y -\frac{1}{vol(M)}$$ Formally, we have $$G_g(x,y)= \sum_{i>0} \frac{\phi_i(x) \phi_i(y) }{\lambda_i}.$$ My questions are: How strong is the convergence on the right hand side? Do someone have good references about this subject?

Answer 1

You may want to clarify your question. What do you mean by how strong is the convergence? Are you referring to some norm, and if so, what kind of norm are you interested in? The operator $G_y$ acts between various Sobolev spaces. In fact it is a pseudo-differentuial operator of order $-2$.

Consider the the difference

$$ R^L:=G_y-\sum_{0<\lambda_i\leq L}\frac{1}{\lambda_i}\phi_i(x)\phi_i(y),\;\; L\to \infty ?$$

It is also a pseudodifferential operator of order $-2$ so it defines bounded operators

$$R^L: L^{s,2}(M)\to L^{s+2,2}(M),\;\;s\in\mathbb{R}, $$

where $L^{2,s}$ denotes the Sobolev space of distributions $s$-times differentiable with derivatives in $L^2$. Denote by $\Vert-\Vert_s$ the norm on $L^{s,2}$. We can define an operator norm

$$\Vert R^L \Vert_s = \sup_{\Vert\psi\Vert_s=1}\|R^L \psi\|_{s+2}. $$

Are you interested in how fast $\|R^L\|_s\to 0$ for some $s$?

You can also think of $G_y$ as a distribution on $M\times M$ are you interested in the convergence in Sobolev spaces of distributions on $M\times M$?

Two places you can look for the Green function.

M. Taylor: Pseudodifferential operators, Chap XII, or

Berligne-Getzler-Vergne: Heat Kernels and Dirac operators, Chap 3.