Let \(H= \sum_{|\alpha|=k} a_\alpha z^\alpha\) be a homogeneous polynomial of degree \(k\) and define the operator \(P_H\) as \(P_H (f) := \sum_{|\alpha|=k} a_\alpha D^\alpha f\). Then, for a given domain \(D\subset \mathbb C^n\), set \[ K^H_D (z):= \sup \big\{ |P_H (f) (z)|^2 \colon f^{(j)}(z)=0, j=0,\dots,k-1, f \in L^2_h (D), \|f\|_{L^2(D)} =1 \big\}, \] where \( L^2_h (D)\) denotes the space of holomorphic functions in \(L^2 (D)\). For \(H=1\), one recovers the definition of the Bergman kernel (restricted to the diagonal) \(K_D\).
Assume that \(0\in D\) and denote by \(G\) the (pluri)complex Green function of (the complement of) \(D\) with pole at \(0\). The Azukawa indicatrix (at \(0\)) is defined as \[ I_D(0) := \{ X \in \mathbb C^n \colon \limsup_{\lambda \to 0} G (\lambda X ) - \log |\lambda| <0 \}. \]
The following is the main result of the paper.
Theorem. Let \(0\in D\subset \mathbb C^n\) be a pseudoconvex domain and \(H\) be a homogeneous polynomial of degree \(k\). Then the function \[ [-\infty,0]\ni a \mapsto K^H_{D_a} (0) \] is non-decreasing, where \(D_a := e^{-a} \{G <a\}\) and \(D_{-\infty}:= I_D (0)\). In particular, \(K^H_{I_D(0)} (0) \leq K_D^H (0)\). This gives a generalization of the (higher-dimensional counterpart of the) Suita conjecture. The proof follows the approach developed in [the first author, Lect. Notes Math. 2116, 53–63 (2014; Zbl 1321.32003); Bull. Math. Sci. 4, No. 3, 433–480 (2014; Zbl 1310.32004); the authors, New York J. Math. 21, 151–161 (2015; Zbl 1317.32024)]. The authors deduce some consequences on the dimension of the Bergman space, which lead to a partial solution to a problem of J. J. O. O. Wiegerinck [Math. Z. 187, 559–562 (1984; Zbl 0534.32001)]. They also introduce other sufficient conditions that guarantee a positive solution to this problem. Finally, they study the regularity of the function given by the volume of the Azukawa indicatrix.