A \(\mathcal{C}^2\)-plurisubharmonic weight \(\varphi: \mathbb{C}^n \to \mathbb{R}\) is said to be admissible if
\[ \sup \big\{ \triangle \varphi (w): w\in B(z,2r) \big\} \leq D \sup \big\{ \triangle \varphi (w): w\in B(z,r) \big\} \] for each \(z\in \mathbb{C}^n\), \( r>0\), for some constant \(D\) independent of \(z\) and \(r\) (\(L^\infty\) doubling condition), and if there exists \(c>0\) such that
\[ \inf_{z\in \mathbb{C}^n} \sup \big\{ \triangle \varphi (w): w\in B(z,c) \big\}>0. \]
A function \(\rho : \mathbb{R}^d \to (0,+\infty)\) is a radius function if it is Borel and there exists a constant \(C>0\) such that for every \(x\in \mathbb{R}^d\) one has \(C^{-1} \rho(x) \leq \rho(y) \leq C \rho(x)\) for every \(y\in B(x, \rho(x)).\) To any radius function \(\rho,\) one can associate a Riemannian metric \(d_\rho(x,y)= \inf_\gamma L_\rho (\gamma),\) where \(L_\rho(\gamma):= \int_0^1 \frac{|\gamma'(t)|}{\rho(\gamma(t))}\,dt\) and the infimum is taken over all piecewise \(\mathcal{C}^1\)-curves connecting \(x\) and \(y.\) The maximal eigenvalue radius function associated to the admissible weight is defined by
\[ \rho (z) := \sup \bigg\{ r>0 : \sup_{w\in B(z,r)} \triangle \varphi (w) \leq r^{-2} \bigg\}. \]
The author considers the \(\overline\partial\)-complex for \(L^2(\mathbb{C}^n, e^{-2\varphi})\) and the corresponding complex Laplacian \(\square_\varphi = \overline\partial \overline\partial^*_\varphi + \overline\partial^*_\varphi \overline\partial\). Suppose that \(\varphi\) is an admissible weight and that there exists a bounded radius function \(\kappa\) such that the maximal eigenvalue radius function \(\rho\) statisfies \(\kappa(z) \geq \rho(z)\), for every \(z\in \mathbb{C}^n\). In addition assume that \(\square_\varphi \) is \(\kappa^{-1}\)-coercive, which means that \(\square_\varphi \geq \kappa^{-2}\) as self-adjoint operators on appropriate Hilbert spaces. It is shown that the Bergman kernel \(K_\varphi (z,w)\) of the space of all entire functions \(f:\mathbb{C}^n \to \mathbb{C}\) such that \(\int_{\mathbb{C}^n} |f(z)|^2 e^{-2\varphi (z)}\,d\lambda(z) < \infty\) can be estimated pointwise: \[ |K_\varphi(z,w)| \lesssim e^{\varphi(z)+\varphi(w)} \frac{\kappa(z) e^{-\epsilon d_\kappa(z,w)}}{\rho(z) \rho(z)^n \rho(w)^n}, \] for every \(z,w\in \mathbb{C}^n\), where \(\epsilon>0\) is a constant depending only on \(\varphi, \kappa\) and the dimension \(n\). This result is a generalization of the estimate if \(n=1\), due to M. Christ [J. Geom. Anal. 1, No. 3, 193–230 (1991; Zbl 0737.35011)].
In the proof, the author uses methods of S. Agmon [Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of \(N\)-body Schrödinger operators. Princeton, New Jersey: Princeton University Press; Tokyo: University of Tokyo Press (1982; Zbl 0503.35001)] and the weighted \(\overline\partial\) calculus. He also points out that the estimate can considerably be simplified if the eigenvalues of the Levi matrix \((\frac{\partial^2 \varphi}{\partial z_j \partial \overline z_k})_{j,k=1}^n\) are comparable. H. Delin [Ann. Inst. Fourier 48, No. 4, 967–997 (1998; Zbl 0918.32007)] obtained a similar estimate in several variables using different methods.