Sharp $C^{1,ar1}$ 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.

💡 Research Summary

The paper by Zbigniew Błocki and Tamás Darvas establishes sharp (C^{1,\bar1}) estimates for weighted Bergman kernels and applies them to Kähler quantization and the quantization of non‑pluripolar Radon measures.

Local Bergman kernel estimates.
For a bounded pseudoconvex domain (\Omega\subset\mathbb C^{n}) and a plurisubharmonic weight (u), the authors define the diagonal Bergman kernel (K_{\Omega,u}(z)) and the directional extremal quantity (eK_{\Omega,u}(z;v)). Using Hörmander’s (L^{2}) estimate together with a cutoff function, they prove that if (u\in C^{1,\bar1}(B_{r})) then both (K_{\Omega,u}) and (eK_{\Omega,u}) are bounded above by constants involving (\sup_{B_{r}}\Delta u). Conversely, if (\partial\bar\partial u\ge a,\omega) (strongly psh) on a ball, they obtain matching lower bounds. Combining these yields two‑sided estimates for the Bergman metric \