I am following Barry Simon on the semigroup $e^{tH}$, for $t$ a real number. The paper is called “Schrödinger semigroups”. In it he says casually that ’the function $| x — y |^{-1}$ arises in the definition of [some function space] because it is (up to a constant) the integral kernel of $(\Delta)^{-1}$.’
It sounds like this is related to the Riesz–Fréchet representation. Is that correct? Can I calculate these kernels in some way such that I can see what the integral kernel might be for some common operations, for example the inverse of the laplacian? Please recommend literature if possible.