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

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