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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 $s 1+{1}{r}$, $1<r\leq 2$, 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.

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