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In this note, we provide an alternative way of describing the distinguished boundaries of symmetrized bidisc, tetrablock and pentablock. We show that each of these distinguished boundaries can be characterized as the union of orbits of two elements under their automorphism groups. [ABSTRACT FROM AUTHOR]

We give the condition that the Bergman kernel function of the second type of Cartan-Hartogs domain has zeros. When the Bergman kernel function of this type of domain has zeros, the topological properties of the zero cluster are proved, and the conditions are given to judge whether a boundary point on the definition domain of the Bergman kernel function is an aggregation point of the zero cluster. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of logarithmically convex domains in C n , called elementary Reinhardt domains, which can be regarded as a natural generalization of Hartogs triangles. The purpose of this paper is twofold. On one hand, we will compute the explicit forms of the Bergman kernel of weighted Hilbert space with radial symbols. On the other hand, by using the expressions of the weighted Bergman kernel, we will show the regularity of the Berezin transform on the elementary Reinhardt domains. [ABSTRACT FROM AUTHOR]

We discuss interrelations between H ∞ -convex domains and H ∞ -domains of holomorphy for various classes of domains in C n . [ABSTRACT FROM AUTHOR]

The Hartogs domain over homogeneous Siegel domain DN,s (s > 0) is defined by the inequality ∥ζ∥2 < KD(z, z)−s, where D is a homogeneous Siegel domain of type II, (z, ζ) ∈ D × ℂN and KD(z, z) is the Bergman kernel of D. Recently, Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains DN,s and D′N′,s′ are biholomorphisms for N ≥ 2. In this article, we find a counter-example to show that the rigidity result is not true for D1,s and obtain a classification of proper holomorphic mappings between D1,s and D′1,s′. [ABSTRACT FROM AUTHOR]

For given p : = (p 1 , … , p n) ∈ (Z +) n , the n-dimensional generalized Hartogs triangle is a bounded nonsmooth pseudoconvex domain defined by H p n : = { (z 1 , … , z n) ∈ C n : | z 1 | p 1 < ⋯ < | z n | p n < 1 }. In this paper, we give some sharp criteria for the L p - L q boundedness of the Toeplitz operators on H p n induced by a class of positive radial symbols with 1 < p ≤ q < + ∞ . Our main result generalizes many eariler results to a more general setting. [ABSTRACT FROM AUTHOR]

We discuss interrelations between H ∞ -convex domains and H ∞ -domains of holomorphy for various classes of domains in C n . [ABSTRACT FROM AUTHOR]

In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for the span metric near C 2 -smooth boundary points on such domains. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notion of n-generalized Hartogs triangles. Then, we characterize proper holomorphic mappings between some of these domains, and describe their automorphism groups. [ABSTRACT FROM AUTHOR]

The elementary Reinhardt domain, a generalization of the classical Hartogs triangle, is defined by H (k) : = { z ∈ D n : z k is defined and | z k | < 1 } , where k = (k 1 , ... , k n) ∈ Z n . In this paper, a sharp criteria for the L p - L q boundedness of the Toeplitz operator with a class of positive radial symbol is obtained on H (k) , which generalizes the previous results of Khanh et al. (Proc Am Math Soc 147:327–338, 2019) and Balay et al. ( L p Regularity of Toeplitz Operators on Generalized Hartogs Triangles, 2020) to a more general setting [ABSTRACT FROM AUTHOR]

The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros. If the Bergman kernel function of this type of domain has zeros, the zero set is composed of several path-connected branches, and there exists a continuous curve to connect any two points in the non-zero set. [ABSTRACT FROM AUTHOR]

Let D n , m = { (z , w) ∈ C n × C m : | w | 2 < e − μ | z | 2 } be the Fock–Bargmann–Hartogs domain, where μ > 0. For any fixed positive integer m, we prove that there exists a unique positive integer n 0 such that D n , m is not a Lu Qi-Keng domain if and only if n ≥ n 0 , which verifies a conjecture posed by Yamamori. Meanwhile, we show that the Bergman projection is bounded on L p (D n , m) if and only if p = 2. We also obtain the optimal rate of growth for holomorphic functions in L p (D n , m). [ABSTRACT FROM AUTHOR]

In this present paper, we study positivity and completeness of some invariants metrics on unbounded domains in C n and generalizations of some theorem in complex spaces. [ABSTRACT FROM AUTHOR]

In this present paper, we study positivity and completeness of some invariants metrics on unbounded domains in C n and generalizations of some theorem in complex spaces. [ABSTRACT FROM AUTHOR]

For appropriate domains Ω 1 , Ω 2 , we consider mappings Φ A : Ω 1 → Ω 2 of monomial type. We obtain an orthogonal decomposition of the Bergman space A 2 (Ω 1) into finitely many closed subspaces indexed by characters of a finite abelian group associated to the mapping Φ A . We then show that each subspace is isomorphic to a weighted Bergman space on Ω 2 . This leads to a formula for the Bergman kernel on Ω 1 as a sum of weighted Bergman kernels on Ω 2 . [ABSTRACT FROM AUTHOR]

In this paper the complex geodesics of a convex domain in ℂn are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂn. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed. [ABSTRACT FROM AUTHOR]

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2} , where $\unicode[STIX]{x1D707}>0$ , is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$ , where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$ , except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty)\setminus \{2\}$ , in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain. [ABSTRACT FROM AUTHOR]

Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities ρ j (| a 1 | , ... , | a m |) < 0 relatively moduli | a i | of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (1991) consisting in the fact that the Horn's formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a signle or two inequalities ρ (| a t | , | a s |) ≶ 0 , where ρ is a reduced discriminant. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the weak‐type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc, we provide a proof that the weak‐type behavior is of 'LlogL' type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle, we show that the operator is of weak‐type (4,4); settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak‐type (43,43) and provide evidence as to what the correct behavior at this endpoint might be. [ABSTRACT FROM AUTHOR]

We introduce a wide class of bounded Hartogs domains in C n , which contains some generalizations of the classical Hartogs triangle. A sharp criterion for the L p - L q boundedness of the Toeplitz operator with the symbol K - t is obtained on these domains, where K is the Bergman kernel on the diagonal and t ≥ 0 . It generalizes the results by Beberok and Chen in the case 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

Let Gi be closed Lie groups of U(n), Ωi be bounded Gi-invariant domains in ℂn which contains 0, and O(ℂn)Gi = ℂ, for i = 1, 2. It is known that if f: Ω1 →} Ω2 is a proper holomorphic mapping, and f−1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f. [ABSTRACT FROM AUTHOR]

An integral solution operator for ∂ ¯ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy L p estimates for all 1 ≤ p < ∞ , though with non-standard—relative to strongly pseudoconvex domains—bounding term. These estimates imply L p estimates for ∂ ¯ on the Hartogs triangle, with greater range of p than the canonical solution satisfies. [ABSTRACT FROM AUTHOR]

The Hardy space of vector valued analytic functions in tube domains in and with values in Banach space are defined. Analytic functions in tube domains in with values in Hilbert space which have functions with values in Hilbert space as distributional boundary value are proved to be in. A Poisson integral representation for such vector valued analytic functions is obtained. [ABSTRACT FROM AUTHOR]

We consider a certain class of unbounded nonhyperbolic Reinhardt domains which is called the twisted Fock-Bargmann-Hartogs domains. By showing Cartan’s linearity theorem for our unbounded nonhyperbolic domains, we give a complete description of the automorphism groups of twisted Fock-Bargmann-Hartogs domains. [ABSTRACT FROM AUTHOR]

We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball. [ABSTRACT FROM AUTHOR]

We show that proper holomorphic self-maps of smoothly bounded pseudoconvex quasi-balanced domains of finite type are automorphisms. This generalizes the classical Alexander's theorem on proper holomorphic self-maps of the unit ball. [ABSTRACT FROM AUTHOR]

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of $$L_h^2$$ -domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in $$\mathbb {C}^2$$ . This allows easily to decide which pseudoconvex balanced domain in $$\mathbb {C}^2$$ has a positive Bergman kernel and which admits the Bergman metric. [ABSTRACT FROM AUTHOR]

Let G be a closed Lie subgroup of U( n), Ω be a bounded G -invariant domain in C which contains 0, and $$O{\left( {{\mathbb{C}^n}} \right)^{{G_i}}} = \mathbb{C}$$ , for i = 1; 2. If f: Ω → Ω is a biholomorphism, and f(0) = 0, then f is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan's theorem. [ABSTRACT FROM AUTHOR]