Almost critical local well-posedness for the space-time Monopole equation in Lorenz gauge

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in $H^s$ with $s>\frac14$. The equation is $L^2$-critical, and hence a $\frac14$ derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier-Lebesgue space $\hat{H_p^s}$ for $1<p\le 2$ which coincides with $H^s$ when $p=2$ but scales like lower regularity Sobolev spaces for $1<p< 2$. In particular, we will see that as $p\rightarrow 1^+$, the critical exponent $s^c_p\rightarrow 1^-$, in which case $\hat{\dot H_{1+}^{1-}}$ is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to $p$ close to 1.