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We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq \frac{1}{|x|}, |u''(x)|\leq c\frac{1}{|x|^2}$. But I can not find anything like that in the Internet. Could you provide me with a reference, please? Any information anout $c$?

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq \frac{1}{|x|}, |u''(x)|\leq c\frac{1}{|x|^2}$. But I can not find anything like that in the Internet. Could you provide me with a reference, please? Any information about $c$?

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq \frac{1}{|x|}, |u''(x)|\leq \frac{1}{|x|^2}$. But I can not find anything like that in the Internet. Could you provide me with a reference, please?

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq \frac{1}{|x|}, |u''(x)|\leq c\frac{1}{|x|^2}$. But I can not find anything like that in the Internet. Could you provide me with a reference, please? Any information anout $c$?

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq 1/|x|, |u''(x)|\leq 1/|x|^2$. But I can not find anything like that in the Internet. Could you provide me with a reference, please?

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator

Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)|\leq C \ln|x|,\Delta u = 1$ at $(0,0)$ and $\Delta u=0$ elsewhere. I am sure that somewhere it is proven that the discrete derivatives of $u$ behave exactly as we expect them : $|u'(x)|\leq \frac{1}{|x|}, |u''(x)|\leq \frac{1}{|x|^2}$. But I can not find anything like that in the Internet. Could you provide me with a reference, please?