  We consider the Landau equation with Coulomb potential in the spatially
homogeneous case. We show short time propagation of smallness in $L^p$ norms
for $p>3/2$ and instantaneous regularization in Sobolev spaces. This yields new
short time quantitative a priori estimates that are unconditional near
equilibrium. We combine these estimates with existing literature on global
well-posedness for regular data to extend the well-posedness theory to small
$L^p$ data with $p$ arbitrarily close to $3/2$. The threshold $p = 3/2$ agrees
with previous work on conditional regularity for the Landau equation in the far
from equilibrium regime.
  In light of the monotonicity of the Fisher information shown in the recent
preprint [arXiv:2311.09420], our primary nonlinear regularization estimate
holds even in the far-from-equilibrium regime. As a consequence, we obtain
exponential convergence to equilibrium for suitably localized solutions in
every Sobolev norm.
