  We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb
potential and initial data in polynomially weighted $L^{3/2}$. We show that
there exists a smooth solution that is bounded for all positive times. The
proof is based on short-time regularization estimates for the Fisher
information, which, combined with the recent result of Guillen and Silvestre,
yields the existence of a global-in-time smooth solution. Additionally, if the
initial data belongs to $L^p$ with $p>3/2$, there is a unique solution.
  At the crux of the result is a new $\varepsilon$-regularity criterion in the
spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in
weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical
quantity for the Landau-Coulomb equation, using this norm to measure the
regularity of solutions presents significant complications. For instance, the
$L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the
competing reaction and diffusion coefficients. These analytical challenges
caused prior methods relying on the parabolic structure of the Landau-Coulomb
to break down.
  Our new framework is general enough to handle slowly decaying and singular
initial data, and provides the first proof of global well-posedness for the
Landau-Coulomb equation with rough initial data.
