Convexity of the Bergman Kernels on Convex Domains

Let $Ω$ be a convex domain in $\mathbb{C}^n$ and $φ$ a convex function on $Ω$. We prove that $\log{K_{Ω,φ}(z)}$ is a convex function (might be identically $-\infty$) on $Ω$, where $K_{Ω,φ}$ is the weighted Bergman kernel. When $φ\equiv0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_Ω(z)^{-\frac{1}{2n}}$ is a convex function if $Ω$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.

💡 Research Summary

The paper investigates the convexity properties of weighted Bergman kernels on convex domains in complex Euclidean space. Let Ω⊂ℂⁿ be a convex domain and φ:Ω→ℝ∪{−∞} a convex function (in the usual real‑valued sense). The weighted Bergman space A²(Ω,φ) consists of holomorphic functions f on Ω that are square‑integrable with respect to the weight e^{−φ}. Its reproducing kernel K_{Ω,φ}(ζ,z) is holomorphic in ζ and anti‑holomorphic in z; the diagonal function K_{Ω,φ}(z)=K_{Ω,φ}(z,z) is real‑analytic wherever it is finite. A classical fact is that log K_{Ω,φ} is always plurisubharmonic (or identically −∞). The main result (Theorem 1.1) strengthens this: if Ω is convex and φ is convex, then log K_{Ω,φ} is a convex function on Ω (convexity being defined by the usual linear interpolation inequality). The proof does not rely on PDE techniques; instead it uses Berndtsson’s subharmonicity theorem for Bergman kernels together with a simple convexity criterion (Lemma 2.1) that translates convexity into subharmonicity after a family of affine‑complex transformations t↦t_λ(s)=s+λ²\bar s (|λ|<1). By applying Berndtsson’s theorem to a lifted domain in ℂ^{n+1} and then invoking Lemma 2.1, the authors obtain convexity of the parameter‑dependent kernel and consequently convexity of log K_{Ω,φ}.

The paper further studies strict convexity. Theorem 1.2(1) shows that if Ω is bounded and φ is convex, then log K_{Ω,φ} is strictly convex. The argument uses a lower bound for the unweighted kernel K_Ω due to Nikolov‑Plug, which shows that K_Ω(z) grows like the inverse square of the distance to the boundary, multiplied by a factor involving successive orthogonal sections. This growth forces log K_{Ω,φ} to be non‑linear on any line segment, yielding strict convexity. Part (2) of Theorem 1.2 treats the unweighted case φ≡0: log K_Ω is strictly convex if and only if Ω does not contain a real line. If Ω contains a real line, translation along that line is an automorphism of Ω, making K_Ω invariant and thus log K_Ω constant on the line, destroying strict convexity.

From these results the authors derive Brunn–Minkowski‑type inequalities. Corollary 1.4 states that for convex Ω₀,Ω₁⊂ℂⁿ and points z₀∈Ω₀, z₁∈Ω₁, \