  Consider the Landau equation with Coulomb potential in a periodic box. We
develop a new $L^{2}\rightarrow L^{\infty }$ framework to construct global
unique solutions near Maxwellian with small $L^{\infty }\ $norm. The first step
is to establish global $L^{2}$ estimates with strong velocity weight and time
decay, under the assumption of $L^{\infty }$ bound, which is further controlled
by such $L^{2}$ estimates via De Giorgi's method \cite{golse2016harnack} and
\cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$
spaces to control velocity derivatives to ensure uniqueness, which is based on
Holder estimates via De Giorgi's method \cite{golse2016harnack},
\cite{golse2015holder}, and \cite{mouhot2015holder}.
