Recently, I have stumbled upon the following non-local operator in $\mathbb{R}^{2}$ $$ \frac{\log{\Delta}}{\Delta} \ , $$ where $\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}$ represents the Laplacian operator. According to https://arxiv.org/pdf/1710.03416.pdf , the logarithm of the Laplacian operator should be defined as $$ \left. \log\Delta=\frac{d}{ds}\Delta^s \right\rvert_{s=0} \ . $$
I read https://arxiv.org/pdf/1710.03416.pdf and https://arxiv.org/pdf/1104.4345.pdf and I am now aware of the integral representations in two dimensions of the inverse Laplacian $\Delta^{-1}$ (up to my understanding, this corresponds to the Laplacian's Green's function), of the fractional Laplacian $\Delta^s$ and of the Logarithmic Laplacian $\log \Delta$.
I am very new to this whole subject, so I would like to ask
- if an integral representation of the non-local operator $\frac{\log{\Delta}}{\Delta}$ is known in two dimensions;
- for any references that could allow me to understand how to derive such an integral representation.