Sharp $C^{1,\bar1}$ estimates in Kähler quantization and non-pluripolar Radon measures

Let $K_φ$ denote the weighted Bergman kernel associated to a plurisubharmonic function $φ$. We obtain upper bounds and positive lower bounds for the Bergman metric $i\partial \bar{\partial} \log K_φ$, expressed solely in terms of upper bounds and positive lower bounds of $i\partial \bar{\partial}φ$. Our approach applies in both local and compact Kähler settings. As an immediate application we obtain the optimal $C^{1,α}$-convergence for the quantization of Kähler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact Kähler manifold admits a quantization.