Question

I note here $\mathcal{D}'$ the space of all distributions and $\mathcal{S}'$ the space of tempered distributions, I am considering the following question:

Let $u \in \mathcal{D}'$ or $\mathcal{S}'$, I want to know general conditions such that we know there exists an inverse of $u$ for the convolution operation, meaning a distribution $v$ such that $u*v$ and $v*u$ can be defined and: $$u*v = v*u = \delta$$

When does a solution exist? When is that solution unique and can we describe all the solutions when it is not?

Does it change the problem to only consider right or left-inverse of $u$?

Thanks in advance.

Answer 1

The problem of division, for $u$ having compact support, was solved by L. Ehrenpreis in the 1950's. The equation $u*v=\delta$ is solvable if and only if the Fourier transform of $u$ is slowly decreasing, which means that an estimate $$\sup_{|\eta|< \log(e+|\xi|)} |\hat u(\xi+\eta)|\geq C(1+|\xi|)^{-N}$$ holds. See, for example Theorem 16.5.22 in volume II of Hörmander's treatise on the Analysis of LPDO.

Answer 2

Your question is related to the famous and notoriously difficult division problem. If $u\in\mathscr{S}'$, and $\hat{u}$ is its Fourier transform, you ask when it is possible to define $\frac{1}{\hat{u}}$. Checl L. Schwartz's book Theorie des Distributions, Chapter V, Sections 4 and 5.