Pluriharmonic Functions Research Articles

Functional determinant on pseudo-Einstein 3-manifolds

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{\theta}$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $\theta\mapsto e^{w}\theta$ with $w\in \P$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in \cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in \cite{CY}.

Об операторах Лапласа и Хуа Ло-Кена

We define holomorphic and pluri-harmonic functions in classical E. Cartan domains of the first type, and research Laplace and Hua Luogeng operators. We find a connection between these operators.

Continuation of Analytic and Pluriharmonic Functions in the Given Direction by the ChirkaMethod: a Survey

In this paper, we provide a survey of results on analytic and plurisubharmonic continuations of functions that have this 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.

Weighted Integral Representations of Pluriharmonic Functions in the Siegel Domain of $$C^n$$

The Siegel domain in the space $$C^n$$ is defined as follows: $$\begin{aligned} \Omega _n=\left\{ \eta =(\eta _1,\eta _2,\dots ,\eta _n)\in C^n: Im \eta _1>\sum _{k=2}^n |\eta _k|^2 \right\} . \end{aligned}$$In the paper the weighted spaces $$L_{\alpha }^p(\Omega _n)$$ with the weight function of the type $$(Im\eta _1-\sum _{k=2}^n |\eta _k|^2)^{\alpha }$$ are introduced. For pluriharmonic functions u from the spaces $$L_{\alpha }^p(\Omega _n)$$ weighted integral representations are established. Inequalities between weighted $$L^p$$-norms of conjugate pluriharmonic functions are shown.

Reflection Principle for the Complex Monge\u2013Amp\xe8re Equation and Plurisubharmonic Functions

We study reflection principle for several central objects in pluripotential theory. First we show that the odd reflected function gives an extension for pluriharmonic functions over a flat boundary. Then we show that the even reflected function gives an extension for nonnegative plurisubharmonic functions. In particular cases odd and/or even reflected functions give extensions for classical solutions of the homogeneous complex Monge–Ampère equation. Finally, we state reflection principle for the generalized complex Monge–Ampère equation and maximal plurisubharmonic functions.

Multi-Toeplitz operators and free pluriharmonic functions

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains Dqm(H)⊂B(H)n, m,n≥1, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain Dqm(H). We prove that the set of all weighted multi-Toeplitz operators coincides withA(Dqm)⁎+A(Dqm)‾WOT, where the domain algebra A(Dqm) is the norm-closed unital non-selfadjoint algebra generated by the universal model (W1,…,Wn) of the noncommutative domain Dqm(H). These results are used to study the class of free pluriharmonic functions on Dqm(H)∘. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions. Our study of weighted multi-Toeplitz operators on Fock spaces is a blend of multi-variable operator theory, noncommutative function theory, operator spaces, and harmonic analysis.

Schwarz-Pick Type Estimates for Gradients of Pluriharmonic Mappings of the Unit Ball

The main purpose of this paper is to find generalized Schwarz-Pick type inequalities for pluriharmonic functions f of the unit ball $${\mathbb {B}}^n$$ . The estimates for the gradient of |f| and f are given as follows: $$\begin{aligned} \big |\nabla |f|(z)\big |\le \frac{4\sqrt{n}}{\pi (1-|z|)}, \ \ \ z\in {\mathbb {B}}^n \end{aligned}$$ and $$\begin{aligned} |\nabla f(z)|^2\le \frac{2n(1-|f(z)|^2)}{1-|z|}, \ \ \ z\in {\mathbb {B}}^n. \end{aligned}$$ Moreover, by additional assuming the quasiregularity of f, some additional estimates are given in this paper.

О регулярных тканях, определенных плюригармоническими функциями

Как известно, функция двух переменных z=f(x, y) задает на плоскости (x, y) в окрестности регулярной точки некоторую три-ткань, образованную слоениями x=const, y=const и f(x, y)=const.
 Три-ткань называется регулярной, если она эквивалентна (локально диффеоморфна) три-ткани, образованной тремя семействами параллельных прямых.
 В этом случае уравнение ткани имеет вид $z=f\left(\alpha(x)+\beta(y)\right)$.
 В одной из работ авторов этой статьи были найдены все регулярные три-ткани, определяемые некоторыми известными уравнениями в частных производных, в частности, определяемые гармоническими функциями.
 В настоящей работе результаты обобщаются для плюригармонических функций вида
 u=f(x_1, ...., x_r, y_1, ... , y_r).
 Во-первых, функция такого вида определяет на многообразии размерности 2r (2r + 1)-ткань, образованную слоениями коразмерности 1 вида x_i=const, y_i=const, i=1, 2, ..., r и u=const.
 (2r + 1)-ткань называется регулярной, если в некоторых локальных координатах ее уравнение может быть записано в виде
 $$ 
 u=f\left(\varphi_1(x_1)+\ldots + \varphi_1(x_r)+\psi_1(y_1)+\ldots +\psi_r( y_r)\right).
 $$
 В этой статье мы находим все плюригармонические функции, задающие регулярные (2r + 1)-ткани (теорема 1).
 С другой стороны, каждая плюригармоническая функция u=f(x_1,..., x_r, y_1, ... , y_r)
 определяет на 2r-мерном многообразии три-ткань W(r,r,2r-1), образованную двумя r-мерными слоениями x_i=const и y_i=const и слоением u=const коразмерности 1.
 Эта ткань называется регулярной, если в некоторых локальных координатах ее уравнение может быть записано в виде
 $$
 u=f\left(\varphi(x_1, x_2,\ldots, x_r)+\psi(y_1, y_2,\ldots, y_r)\right).
 $$
 В этой работе найдены все плюригармонические функции, определяющие регулярные три-ткани W(r,r,2r-1) (теорема 2)

Prescribing the <inline-formula><tex-math id="M1">\begin{document}$ Q' $\end{document}</tex-math></inline-formula>-curvature in three dimension

In this note, we consider the problem of prescribing $ \overline{Q}' $-curvature on a three-dimensional pseudohermitian manifold. Given a positive CR pluriharmonic function $ f $, we construct a contact form on the three-dimensional pseudo-Einstein manifold with $ \overline{Q}' $-curvature being equal to $ f $, under some natural positivity conditions. On the other hand, we prove a Kazdan-Warner type identity for the problem of prescribing $ \overline{Q}' $-curvature on the standard CR three sphere.

Introduction to tropical series and wave dynamic on them

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs Bn(H) in B(H)n1+⋯+nk, where n=(n1,…,nk)∈Nk and B(H) is the algebra of all bounded linear operators on a Hilbert space H. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on Bn(H), study their properties, and obtain hyperbolic versions of Schwarz–Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case n1=⋯=nk=1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk) of all free holomorphic automorphisms of Dk(H), and the δH-topology induced on Dk(H) is the usual operator norm topology. The restriction of δH to the scalar polydisk Dk is equivalent to the Kobayashi distance on Dk. Most of the results of this paper are presented in the more general setting of Harnack (resp. Poisson) parts of the closed polyball Bn(H)−.

Some properties of Grauert type surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

CR geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions. The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.

$Q$-prime curvature and scattering theory on strictly pseudoconvex domains

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. In this paper, we generalize these objects to the boundaries of asymptotically complex hyperbolic Einstein (ACHE) manifolds, which are partially integrable, strictly pseudoconvex CR manifolds, by using the scattering matrix for ACHE manifolds. In this setting, the $P$-prime operator is a self-adjoint pseudodifferential operator acting on smooth functions and the $Q$-prime curvature is globally determined by the ACHE manifold and the choice of a contact form on the boundary. We prove that the integral of the $Q$-prime curvature is a conformal primitive of the $Q$-curvature; in particular, it defines an invariant of ACHE manifolds whose boundaries admit a contact form with zero $Q$-curvature. We also apply the generalized $Q$-prime curvature to compute the renormalized volume of strictly pseudoconvex domains whose boundaries may not admit pseudo-Einstein structure.

Free pluriharmonic functions on noncommutative polyballs

In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in a certain class, which we determine. We provide a Naimark type dilation theorem for direct products of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball, with operator-valued coefficients. We define the noncommutative the Berezin (resp. Poisson) transform of a completely bounded linear map on the C*-algebra generated by the universal operator model and give necessary an sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz-Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as Korannyi-Pukanszky version in scalar polydisks.

Schwarz lemma for pluriharmonic functions

In this note the Schwarz theory for pluriharmonic functions is studied, including the Schwarz lemma, the Julia lemma, and the behavior of invariant metric for pluriharmonic functions in the unit ball.

The CR Paneitz operator and the stability of CR pluriharmonic functions

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in C2 are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.

A remark on the kernel of the CR Paneitz operator

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kernel of the CR Paneitz operator contains a supplementary space to the CR pluriharmonic functions.

On CR Paneitz operators and CR pluriharmonic functions

Let $$(X,T^{1,0}X)$$ be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let $$\mathrm{P}$$ be the associated CR Paneitz operator. In this paper, we show that (I) $${\mathrm{P}}$$ is self-adjoint and $$\mathrm{P}$$ has $$L^2$$ closed range. Let $$N$$ and $$\Pi $$ be the associated partial inverse and the orthogonal projection onto $$\mathrm{Ker}\,\mathrm{P}$$ respectively, then $$N$$ and $$\Pi $$ enjoy some regularity properties. (II) Let $$\hat{\mathcal {P}}$$ and $$\hat{\mathcal {P}}_{0}$$ be the space of $$L^2$$ CR pluriharmonic functions and the space of real part of $$L^2$$ global CR functions respectively. Let $$S$$ be the associated Szego projection and let $$\tau $$ , $$\tau _0$$ be the orthogonal projections onto $$\hat{\mathcal {P}}$$ and $$\hat{\mathcal {P}}_{0}$$ respectively. Then, $$\Pi =S+\overline{S}+F_0$$ , $$\tau =S+\overline{S}+F_1$$ , $$\tau _0=S+\overline{S}+F_2$$ , where $$F_0, F_1, F_2$$ are smoothing operators on $$X$$ . In particular, $$\Pi $$ , $$\tau $$ and $$\tau _0$$ are Fourier integral operators with complex phases and $$\hat{\mathcal {P}}^\perp \bigcap {\hbox { Ker }}\mathrm{P}$$ , $$\hat{\mathcal {P}}_{0}^\perp \bigcap \hat{\mathcal {P}}$$ , $$\hat{\mathcal {P}}_{0}^\perp \bigcap {\hbox { Ker }}\mathrm{P}$$ are all finite dimensional subspaces of $$C^\infty (X)$$ (it is well-known that $$\hat{\mathcal {P}}_{0}\subset \hat{\mathcal {P}}\subset {\hbox { Ker }}\mathrm{P}$$ ). (III) $$\hbox {Spec }\mathrm{P}$$ is a discrete subset of $${\mathbb {R}}$$ and for every $$\lambda \in \hbox {Spec }\mathrm{P}$$ , $$\lambda \ne 0$$ , $$\lambda $$ is an eigenvalue of $$\mathrm{P}$$ and the associated eigenspace $$H_\lambda (\mathrm{P})$$ is a finite dimensional subspace of $$C^\infty (X)$$ .