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

Abstract

Let $K_{\ddot{I}}†$ denote the weighted Bergman kernel associated to a plurisubharmonic function $\ddot{I}†$. We obtain upper bounds and positive lower bounds for the Bergman metric $i\partial \bar{\partial }\log K_{\ddot{I}}†$, expressed solely in terms of upper bounds and positive lower bounds of $i\partial \bar{\partial }\ddot{I}†$. Our approach applies in both local and compact Kähler settings. As an immediate application we obtain the optimal $C^{1,\hat{I}\pm }$-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.