Pluriharmonic Functions Research Articles

Bounded Pluriharmonic Functions and Holomorphic Functions on Teichmüller Space

Abstract In this paper, we discuss the boundary behavior of bounded pluriharmonic functions and bounded holomorphic functions on the Teichmüller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the Teichmüller geodesic rays, and a version of the F. and M. Riesz theorem that the radial limit of a non-constant bounded holomorphic function is not constant on any non-null measurable set on the Bers boundary in terms of the pluriharmonic measure. As a corollary, we obtain the non-ergodicity of the action of the Torelli group for a closed surface of genus $g\ge 2$ on the space of projective measured foliations.

Clark Measures on Bounded Symmetric Domains

Given a bounded symmetric domain D, we study (positive) pluriharmonic functions on D and investigate a possible analogue of the family of Clark measures associated with a holomorphic function from D into the unit disc in C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document}.

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions

Abstract We study a “p-powered” version K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} of the well-known Bohr radius problem for the family F ⁢ ( R ) {F(R)} of holomorphic functions f : R → X {f:R\to X} satisfying ∥ f ∥ < ∞ {\lVert f\rVert<\infty} , where ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} is a norm in the function space F ⁢ ( R ) {F(R)} , R ⊂ ℂ n {R\subset{\mathbb{C}}^{n}} is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 {p>0} , we describe in full detail the asymptotic behavior of K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} , where F ⁢ ( R ) {F(R)} is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 𝔻 n {{\mathbb{D}}^{n}} ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ ( l t n ) {B(l_{t}^{n})} of the Minkowski space l t n {l_{t}^{n}} . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results.

Pluripotential theory on Teichmüller space II – Poisson integral formula

This is the second paper in a series of investigations of the pluripotential theory on the Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on the Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type theorem on the boundary behavior of the Poisson integral. We will see a relationship between the pluriharmonic measures and the Patterson-Sullivan measures discussed by Athreya, Bufetov, Eskin and Mirzakhani.

Bott-Chern hypercohomology and bimeromorphic invariants

AbstractThe aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.

Core sets in Kähler manifolds

The primary objective of this paper is to study core sets in the setting of m-subharmonic functions on the class of (non-compact) Kähler manifolds. Core sets are investigated in different aspects by considering various classes of plurisubharmonic functions. One of the crucial concepts in studying the structure of this kind of sets is the pseudoconcavity. In a more general way, we will have the structure of core defined with respect to the m-subharmonic functions, which we call m-core in our setting, in terms of m-pseudoconcave sets. In the context of m-subharmonic functions, we define m-harmonic functions and show that, in Cn(n≥2) and more generally in any Kähler manifold of dimension at least 2, m-harmonic functions are pluriharmonic functions for m≥2.

A NOTE ON A NEW SHARP RESULT IN SPACES OF PLURIHARMONIC FUNCTIONS AND RELATED PROBLEMS

We provide a new sharp result for classical Bergman spaces of pluriharmonic functions related with the trace operator extending previously known assertions. Related new estimates for other pluriharmonic new function spaces in product domains will be also discussed and provided.

О ЧИСЛЕННЫХ МЕТОДАХ РЕШЕНИЯ БЕСКОНЕЧНЫХ СИСТЕМ ЛИНЕЙНЫХ АЛГЕБРАИЧЕСКИХ УРАВНЕНИЙ

We provide a new sharp result for classical Bergman spaces of pluriharmonic functions related with the trace operator extending previously known assertions. Related new estimates for other pluriharmonic new function spaces in product domains will be also discussed and provided.

Extensions of Analytic and Pluriharmonic Functions in Given Directions by the Chirka Method: a Survey

In this paper, we provide a survey of results on analytic and plurisubharmonic extensions of functions that have a thin set of singularities along a fixed direction. We show the advantages of using the pluripotential theory and the Jacobi–Hartogs series for description of the singular set of such functions.

A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn

A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn

CR Q-curvature and CR Pluriharmonic Functions

In this paper, we show that the CR $Q$-curvature is orthogonal to the space of CR pluriharmonic functions on any closed strictly pseudoconvex CR manifold of dimension at least five. To this end, we obtain a cohomological expression of the integral of the product of the CR $Q$-curvature and a CR pluriharmonic function.

A geometric criterion for prescribing residues and some applications

Let X be a compact complex manifold and D a ℂ-linear finite formal sum of divisors of X. A theorem of Weil and Kodaira says that if X is Kähler, then there is a closed logarithmic 1-form with residue divisor D if and only if D is homologous to zero in H 2n-2 (X,ℂ). We generalized their theorem to general compact complex manifolds. The necessary and sufficient condition is described by a new invariant called 𝒬-flat class. In the second part of the paper, we classify all the pluriharmonic functions on a compact algebraic manifold with mild singularities.

The Patterson–Sullivan Reconstruction of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces

The Patterson–Sullivan Reconstruction of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces

Complex Einstein-Finsler doubly twisted product metrics

Let F be a doubly twisted product metric of two strongly pseudoconvex complex Finsler metrics F1 and F2. In this paper, we obtain a necessary and sufficient condition that F to be a complex Einstein-Finsler metric, and get the conclusion that F is a complex Einstein-Finsler metric if and only if F1 and F2 are weakly Einstein-Finsler metrics when logarithm functions of twisted functions both are pluriharmonic functions. We also proved that the holomorphic curvature of F vanishes identically under the condition that F is a complex generalized Einstein-Finsler metric.

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

Brown–Halmos characterization of multi-Toeplitz operators associated with noncommutative polyhyperballs

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $$ \kappa_{\bf m}(z,w):=\prod_{i=1}^k \frac{1}{(1-\bar z_i w_i)^{m_i}},\qquad z,w\in {\bf D}^k, $$ where $m_i\geq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

Linear Integrable Systems on Quad-Graphs

Abstract In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb{C}$, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluri-harmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

The pluricomplex Poisson kernel for strongly pseudoconvex domains

In this paper we introduce, via a Phragmén-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the pluricomplex Poisson kernel because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new “intrinsic” version of the classical Julia's Lemma and Julia-Wolff-Carathéodory's Theorem.