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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-Lebesguespaces $\^{H}_s^r$, which coincide with the Sobolev spaces of the sameregularity 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$s>1+\frac{3}{2r}$, $1<r\leq 2$. On one end this recovers the sharp result onthe Sobolev scale, $H^{\frac{7}{4}+}$, while on the other end establishes the$\^{H}_{\frac{5}{2}}^{1+}$ result, which scales like the Sobolev$H^{\frac{3}{2}+}$, thus, corresponding to a $\frac{1}{4}$ derivativeimprovement on the Sobolev scale.

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