  In this manuscript we investigate the regularization of solutions for the
spatially homogeneous Landau equation. For moderately soft potentials, it is
shown that weak solutions become smooth instantaneously and stay so over all
times, and the estimates depend only on the initial mass, energy, and entropy.
For very soft potentials we obtain a conditional regularity result, hinging on
what may be described as a nonlinear Morrey space bound, assumed to hold
uniformly over time. This bound always holds in the case of moderately soft
potentials, and nearly holds for general potentials, including Coulomb. This
latter phenomenon captures the intuition that for moderately soft potentials,
the dissipative term in the equation is of the same order as the quadratic term
driving the growth (and potentially, singularities). In particular, for the
Coulomb case, the conditional regularity result shows a rate of regularization
much stronger than what is usually expected for regular parabolic equations.
The main feature of our proofs is the analysis of the linearized Landau
operator around an arbitrary and possibly irregular distribution. This linear
operator is shown to be a degenerate elliptic Schr\"odinger operator whose
coefficients are controlled by $A_p$-weights.
