  We are interested in the singular behaviour at the origin of solutions to the
equation $\mathscr{H} \rho = e$ on a half-axis, where $\mathscr H$ is the
one-sided Hilbert transform, $\rho$ an unknown solution and $e$ a known
function. This is a simpler model problem on the path to understanding wave
field singularities caused by curve-shaped scatterers in a planar domain.
  We prove that $\rho$ has a singularity of the form $\mathscr M[e](1/2) /
\sqrt{t}$ where $\mathscr M$ is the Mellin transform. To do this we use
specially built function spaces $\mathscr M'(a,b)$ by Zemanian, and these allow
us to precisely investigate the relationship between the Mellin and Hilbert
transforms. Fourier comes into play in the sense that the Mellin transform is
simpy the Fourier transform on the locally compact Abelian multiplicative group
of the half-line, and as a more familiar operator it guides our investigation.
