  In this paper we prove an optimal local well-posedness result for the 1+2
dimensional system of nonlinear wave equations (NLW) with quadratic null-form
derivative nonlinearities $Q_{\mu\nu}$. The Cauchy problem for these equations
is known to be ill-possed for data in the Sobolev space $H^s$ with $s<5/4$ for
all the basic null-forms, except $Q_0$. However, the scaling analysis predicts
local well-posedness all the way to the critical regularity of $s_c=1$.
Following Gr\"{u}nrock's result for the quadratic derivative NLW, we consider
initial data in the Fourier-Lebesgue spaces $\^{H}_s^r$, which coincide with
the Sobolev spaces of the same regularity for $r=2$, but scale like lower
regularity Sobolev spaces for $1<r<2$. Here we obtain local well-posedness for
the range $s>1+{1}{r}$, $1<r\leq 2$, which at one extreme coincides with
$H^{{3}{2}+}$ Sobolev space result, while at the other extreme establishes
local well-posedness for the model null-form problem for the almost critical
Fourier-Lebesgue space $\^{H}_{2+}^{1+}$. Using appropriate multiplicative
properties of the solution spaces and relying on bilinear estimates for the
$Q_{\mu\nu}$ forms, we prove almost critical local well-posedness for the Ward
wave map problem as well.
