Reference request: inverse of differential operators

Question

I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).

As an example consider the Laplacian operator. The inverse of the Laplacian is given by $$(-\Delta)^{-1} u(x) = C \int_{\mathbb{R}^n} u(x-y) \frac{1}{|y|^{n-2}} dy$$ where $n$ is the dimension of $\mathbb{R}^n$.

I would like to learn more about such operators, as I have often seen their formulas stated but not explained. For example, we can motivate fractional powers of the Laplacian by use of the Fourier transform. I am looking for a similar explanation but for the inverse.

Some of the questions I have been thinking about are:

1. How are these explicit inverses derived? In the case of the Laplacian, is there a reason its inverse is identical to its fundamental solution?
2. Under what conditions and for which function spaces are the inverses valid?
3. In the case of a differential operator, I assume that we gain some degree of regularity by applying its inverse, but how is this proven? What is the image of this operator?

I have struggled to find this information in any textbooks. Can anyone suggest some textbooks or references in this direction? Thanks.

Answer 1

There are many possible answers to this question, depending on the type of the differential operator, the domain of the functions, and whether you want to impose any additional conditions such as specifying boundary data.

For example, if you assume that the differential operator has constant coefficients, then the Malgrange-Ehrenpreis theorem shows that such a Green's function exists on $\mathbb{R}^n$.

In general, given a linear differential operator $P$ on a domain $D \subset \mathbb{R}^n$, the existence of an inverse operator is essentially equivalent to showing that there exists a constant $C$ such that for any compactly supported functions $f$ and $u$ on $D$, there is an a priori bound of the form $$ \|u\|_1 \le C\|_f\|_2, $$ where $\|\cdot\|_1$ and $\|\cdot\|_2$ are appropriately chosen norms, depending on what $P$ is.

If $P$ is elliptic on a domain $D$ and suitable assumptions are made on both $P$ and $D$, then an inverse operator exists (but is not unique) and is a pseudodifferential operator (a generalization of differential operators defined using the Fourier transform).

If $P$ is hyperbolic on a domain $D$ and suitable assumptions are made on both $P$ and $D$, then an inverse operator exists and is a Fourier integral operator (a generalization of pseudodifferential operators, also defined using the Fourier transform).

Analogous but different stories hold for parabolic, hypoelliptic, and many other types of differential operators. The problem is that there is no systematic approach that unifies all of these results. Each type is handled using an ad hoc approach tailored to the assumptions.

Back in the 60's and 70's, there was a big effort by many, including Hormander, Nirenberg, Treves, to identify which differential operators have inverse operators. However, the effort never completely succeeded, and everyone started focusing more on nonlinear PDEs.