I am currently trying to show that the function $\log(|x-y|)$ is an asymptotic smooth kernel function, in the sense that: for $x,y \in \mathbb{R}^2$ there exist constants $C_{1},C_{2} > 0$ and an order of singularity $s \geq 0$ such that \begin{eqnarray*} \partial_{x_1}^{\alpha_{1}}\partial_{x_2}^{\alpha_{2}}\partial_{y_1}^{\beta_{1}} \partial_{y_2}^{\beta_{2}} \log(|x-y|) \leq C_1 (C_2|x-y|)^{-(s+\alpha_1+\alpha_2+\beta_1+\beta_2)} (\alpha_1+\beta_1)!(\alpha_2+\beta_2)! \end{eqnarray*}
I have already tried solving the problem by looking at an explicit formulation of the derivative using $\log(|x-y|)=\frac{1}{2} \log(|x-y|^2)$ and $f(x1,x2,y1,y2) := (x1-y1)^2+(x2-y2)^2$ and then using chain rule for $\frac{1}{2} \log \circ f$. One can show by induction that $D^n(\log(z))=(-1)^{n-1}(n-1)! z^{-n}$ for $z \in \mathbb{R}$. Then one can use a generalization of the formula of faà di bruno (see this paper: http://arxiv.org/abs/1410.3493, Theorem 5) which basically gives you for $\gamma = (\alpha_1,\alpha_2,\beta_1,\beta_2) \in \mathbb{N}^{4}_{0}$ \begin{eqnarray*} D^{\gamma}(\frac{1}{2} \log \circ f)=\frac{1}{2} \sum_{\ell=1}^{|\gamma|}\left( \sum_{[\gamma_1,...,\gamma_{\ell}] \in \Pi(\gamma,\ell)} \underbrace{(D^{\ell}\log)(f(x1,x2,y1,y2))}_{= (-1)^{\ell -1}(\ell-1)!|x-y|^{-2 \ell}}\prod_{k=1}^{\ell}D^{\gamma_{k}}(f(x1,x2,y1,y2)) \right) \end{eqnarray*} In the above formula I mean by $[\gamma_1,...,\gamma_{\ell}] \in \Pi(\gamma,\ell)$ all partitions of $\gamma$ of order $\ell$, in other words the sum runs over all $\gamma_1,...,\gamma_{\ell} \in \mathbb{N}^{4}_{0}: \gamma_1+...+\gamma_{\ell} = \gamma$. For the derivatives of $f$ of course only derivatives of maximum order $2$ can appear, since $f$ is quadratic function. Then I am unsure how to proceed.