Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D

In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Gr"{u}nrock’s result in 3D, we take the data in the Fourier-Lebesgue spaces ${\hat{H}}_s^r$, which coincide with the Sobolev spaces of the same regularity for $r=2$, but scale like lower regularity Sobolev spaces for $1<r<2$. We show local well-posedness (LWP) for the range of exponents $s>1+\frac{3}{2r}$, $1<r\le 2$. On one end this recovers the sharp result on the Sobolev scale, $H^{\frac{7}{4}+}$, while on the other end establishes the ${\hat{H}}_{\frac{5}{2}}^{1+}$ result, which scales like the Sobolev $H^{\frac{3}{2}+}$, thus, corresponding to a $\frac{1}{4}$ derivative improvement on the Sobolev scale.