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In this paper we prove an optimal local well-posedness result for the 1+2dimensional system of nonlinear wave equations (NLW) with quadratic null-formderivative nonlinearities $Q_{\mu\nu}$. The Cauchy problem for these equationsis known to be ill-possed for data in the Sobolev space $H^s$ with $s<5/4$ forall the basic null-forms, except $Q_0$. However, the scaling analysis predictslocal 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 considerinitial data in the Fourier-Lebesgue spaces $\^{H}_s^r$, which coincide withthe Sobolev spaces of the same regularity for $r=2$, but scale like lowerregularity Sobolev spaces for $1<r<2$. Here we obtain local well-posedness forthe 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 establisheslocal well-posedness for the model null-form problem for the almost criticalFourier-Lebesgue space $\^{H}_{2+}^{1+}$. Using appropriate multiplicativeproperties of the solution spaces and relying on bilinear estimates for the$Q_{\mu\nu}$ forms, we prove almost critical local well-posedness for the Wardwave map problem as well.

Almost critical well-posedness for nonlinear wave equation with $Q_{\mu\nu}$ null forms in 2D