Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 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.

share|improve this answer

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.