This post has already been on StackExchange but hasn't received any answers. Since its origin is a research paper, I thought that maybe this forum might be a better place for it. If you disagree please be so kind to let me know and I'll delete it straight away.
So I am a complete beginner to Green's functions in stochastic and only need to use these for one proof: I need to show that the function $$ F(x,y) = G_x(y)-\frac 1 {2\pi} \ln d(x,y) \textrm{ where } G_x(y) = - \sum_{p\in 2\pi\mathbb Z^2\setminus \{0\}} \frac {e^{ip(x-y)}} {|p|^2} $$ is bounded uniformly when $y$ is in a small neighborhood of $x$. I came across that problem in this paper (p.7, Lem. 4).
Progress:
- Case $|p| > \frac 1 {|z|}$: So far, I do know that the sum $G_x(y)$ is bounded uniformly over $z:=x-y$ when $x$ and $y$ are in a small neighborhood of each other. That is the first step of the proof.
- Case $|p| \leq \frac 1 {|z|}$: The author then estimates that (and I can follow this step) $$ - G_x(y) \leq \sum_p \frac 1 {|p|^2} \left(1-\frac {(pz)^2}{4}\right) $$ and wants to show with this (that's where I struggle) that $$ \left|G_x(y) - \sum_{p\in 2\pi\mathbb Z^2\setminus\{0\}, |p| \leq |z|^{-1}}\frac 1 {|p|^2}\right| $$ is bounded uniformly for $y$ in a small neighborhood of $x$.
Question: I can't prove step 2 from above. Also, I don't know how to go from there: Why is important to the overall boundedness of the function $F(x,y)$ that we are actually interested in?
