Convolution equations in spaces of distributions with one-sided bounded support

Abstract:

Let $\mathcal {D}\prime (0,\infty )$ be the space of distributions on $R$ with support in $[0,\infty )$ and $\mathcal {S}\prime (0,\infty )$ its subspace consisting of tempered distributions. We characterize the distributions $S \in \mathcal {D}\prime (0,\infty )$ for which $S \ast \mathcal {D}\prime (0,\infty ) = \mathcal {D}\prime (0,\infty )$, where $\ast$ is the convolution. We also characterize the distributions $S \in \mathcal {S}\prime (0,\infty )$ for which $S \ast \mathcal {S}\prime (0,\infty ) = \mathcal {S}\prime (0,\infty )$.