  The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are
locally well-posed in suitable Fourier-Lebesgue spaces $\hat{H}^{s,r}$. Our aim
is to minimize $s=s(r)$ in the range $1<r \le 2$ . If $r \to 1$ we show that we
almost reach the critical regularity dictated by scaling. In the classical case
$r=2$ the results are due to Huh and Oh. Crucial is the fact that the decisive
quadratic nonlinearities fulfill a null condition.
