Related papers: One dimensional estimates for the Bergman kernel a…
It is proved that for any domain in $\mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the "multidimensional…
We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space $\mathbb{C}^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then…
A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function…
Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is…
On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the…
We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman…
Let C be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of…
We obtain a quantitative estimate of Bergman distance when $\Omega \subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $\alpha_l(\Omega)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of…
We solve Chui's conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we prove that for every $N$, the simplest fractions with…
A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of…
We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to…
In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets…
We develop the theory for the Bergman spaces of generalized $L_p$-solutions of the bicomplex-Vekua equation $\overline{\boldsymbol{\partial}}W=aW+b\overline{W}$ on bounded domains, where the coefficients $a$ and $b$ are bounded…
Let $\Gamma\subset \mathrm{SU}((2,1),\mathbb{C})$ be a torsion-free cocompact subgroup. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric $\mu_{\mathrm{hyp}}$, and let…
We establish geometric upper and lower estimates for the Carath\'eodory and Kobayashi-Eisenman volume elements on the class of non-degenerate convex domains, as well as on the more general class of non-degenerate $\mathbb{C}$-convex…
The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic…
The theory of analytic function spaces in very general tubular domains over symmetric cones is a relatively new interesting research area. Tube domains are very general and very complicated domains. Recently several new results in this…
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $\mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational…
In this paper, we prove the boundedness of the Bergman projection on weighted mixed norm spaces of the upper-half space for some weights that are constructed using the logarithm function and growth functions. Our necessary and sufficient…
The bicomplex Bergman spaces are studied for any bounded bicomplex domain. Its Bergman kernel is computed in terms of the kernels of the complex projections of the domain. We also introduce two additional reproducing kernel Hilbert spaces…