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.