Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $ L^2 $-based Sobolev spaces $ H^s $ and $ H^l $ for the electromagnetic field $ \phi $ and the potential $ A $, respectively, the minimal regularity assumptions are $ s > \frac{1}{2} $ and $ l > \frac{1}{4} $, which leaves a gap of $ \frac{1}{2} $ and $ \frac{1}{4} $ to the critical regularity with respect to scaling $ s_c = l_c = 0 $. This gap can be reduced for data in Fourier-Lebesgue spaces $ \widehat{H}^{s, r} $ and $ \widehat{H}^{l, r} $ to $ s> \frac{21}{16} $ and $ l > \frac{9}{8} $ for $ r $ close to $ 1 $, whereas the critical exponents with respect to scaling fulfill $ s_c \to 1 $, $ l_c \to 1 $ as $ r \to 1 $. Here $ \|f\|_{\widehat{H}^{s, r}} : = \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $ \phi $ as well as $ A $ in both gauges.