  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.

Analysis of PDEs