How to determine the followings:
$$[\Box,\frac{1}{\Box}]\mathcal{O}=?$$
$$[\nabla,\frac{1}{\nabla}]\mathcal{O}=?$$
$$[\nabla^2,\frac{1}{\nabla^2}]\mathcal{O}=?$$
$$[\partial^{2}_{r},\frac{1}{\partial^{2}_{r}}]\mathcal{O}=?$$
Note: In the case of $\Box$ we know they do NOT commute. But is this also true for partial derivative case? We know in some very specific form of $\mathcal{O}$ they do commute, but generally it seems they do not?
How one define $\Box^{-1}$,$\nabla^{-1}$ and etc in terms of integral? What would be the boundaries of the integral?
and $\mathcal{O}$ is an operator in general (one can define between scalar, vector, tensor) (the easiest is scalar of course).
Regarding the integral:
As $\Box = \partial_\mu \partial^\mu$ and $\nabla = \partial_k \partial^k$ I would expect the usual Fourier transform $i \partial^\mu = p^\mu$ to map the differential operators to algebraic expressions which can be inverted and back-transformed.