Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces $\widehat{H}^{s,r}$ , where $\|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}}$ , $\frac{1}{r}+\frac{1}{r'} = 1$ . The assumed regularity for the data is almost optimal with respect to scaling as $r \to 1$ . This closes the gap between what is known in the case $r=2$ , namely $s > \frac{3}{4}$ , and the critical value $s_c = \frac{1}{2}$ with respect to scaling.