  We consider the homogeneous Landau equation with Coulomb potential and
general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$.
We show the local-in-time existence and uniqueness of smooth solutions for such
initial data. The constraint $p > 3/2$ has appeared in several related works
and appears to be the minimal integrability assumption achievable with current
techniques. We adapt recent ODE methods and conditional regularity results
appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$
smoothing estimates. These estimates enable us to construct local-in-time
smooth solutions for large $L^p$ initial data, and allow us to show directly
conditional regularity results for solutions verifying \emph{unweighted}
Prodi-Serrin type conditions. As a consequence, we obtain additional stability
and uniqueness results for the solutions we construct.
