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

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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 Grünrock's result in 3D, we take the data in the Fourier-Lebesgue spaces Ĥₛʳ, 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+3/2r, 1<r≤ 2. On one end this recovers the sharp result on the Sobolev scale, H⁷/⁴⁺, while on the other end establishes the Ĥ₅/₂¹⁺ result, which scales like the Sobolev H³/²⁺, thus, corresponding to a 1/4 derivative improvement on the Sobolev scale.