Inverse of laplacian operator I recently read a paper, the author treats 
$$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$
up to a constant in $\mathbb{R}^d$.
I am not familiar with unbounded operator, so my question is: Under what condition can one take the inverse of an unbounded operator like above? Can anyone refer some references? Thanks!

To make sense of this sort of problem, it's best to work with Distributions also known as generalized functions. The sort of solution you gave above is sometimes called  Greens function or a fundamental solution.
See for example Friedlander--Joshi Theory of Distributions.
