It is known that, in $\mathbb{R}^2$, we can define the non-local operator $\frac{1}{\Delta}$ with the Green function of the Laplace operator $\Delta$. This provides the non-local operator with an integral representation: when it acts on a test function $f(z)$, we can write $$ \frac{1}{\Delta}f(z)=- \frac{1}{2 \pi}\int_{\mathbb{R}^{2}}d^2 w \ \log\left(|z-w| \right)f(w). $$ In my computations, I have recently stumbled upon the following integral representation of a non-local operator $$ \mathcal{D}_{\text{non-local}} f(z)=\int_{\mathbb{R}^{2}}d^2 w \ \left(\log\left(|z-w| \right)\right)^2 f(w). $$ I haven't been able to find any source in the literature acknowledging this operator, and I am very new to the whole subject. Hence, I would like to ask:
- if the operator is known in literature;
- if there is any possibility to derive the operator starting from the definition of the inverse Laplacian (given above).