  The convolution inequality $h*h(\xi) \leq B |\xi|^\theta h(\xi)$ defined on
$\Rn$ arises from a probabilistic representation of solutions of the
$n$-dimensional Navier-Stokes equations, $n \geq 2$. Using a chaining argument,
we establish the nonexistence of strictly positive fully supported solutions of
this inequality if $\theta \geq n/2$, in all dimensions $n \geq 1$. We use this
result to describe a chain of continuous embeddings from spaces associated with
probabilistic solutions to the spaces $BMO^{-1}$ and $BMO_T^{-1}$ associated
with the Koch-Tataru solutions of the Navier-Stokes equations.
