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

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.