  Relying on the recent work of Liu-Sz\'ekelyhidi we give a weak asymptotic
estimate for the Bergman kernels of polarized K\"ahler manifolds with Ricci
lower bound and Sobolev constant upper bound. We will also give a simple proof
for the partial $C^0$ estimate along the (generalized) K\"ahler-Ricci flow on
Fano manifolds.
