Harmonic functions on planar and almost planar graphs and manifolds,via circle packings

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If is a bounded circle packing in whose contacts graph is a bounded valence triangulation of a disk, then, with probability , the simple random walk on converges to a limit point. Moreover, in this situation any continuous function on the limit set of extends to a continuous harmonic function on the closure of the contacts graph of ; that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs and manifolds, and show that under the assumptions of transience and bounded local geometry these possess non constant, harmonic, bounded, Dirichlet functions. Let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph. Oblatum 4-I-1995 & 23-IV-1996     

Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds

The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.     

加权调和Bergman核及其诱导的度量矩阵

在Rn中的有界域上建立加权调和Bergman核,并得出单位球的加权调和Bergman核的表达式;利用加权调和Bergman核在Rn的有界域上构造度量矩阵;得到关于调和映射的Jacobi矩阵与度量矩阵之间的一个不等式.     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR _IV(2)) to any Riemannian manifold with finite energy has to be constant.     

Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth

In this note we construct two quasi-isometric graphs. One admits an infinite dimensional space of nonconstant bounded harmonic functions, while the other admits only constant bounded harmonic functions. Translation of the construction to manifolds answers a problem due to T. Lyons.     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

某类双调和映射的Landau型定理

令f(z)=h(z)+g(z)为开单位圆盘U内的调和映射.本文首先建立了有界调和映射的精确系数估计,其次得到了满足有界性条件|h(z)|+|g(z)|≤M的正规化调和映射的精确系数估计.作为其应用,得到了双调和映射的一些Landau型定理.这些结果改进和推广了早期作者的相关结果.     

Landau and Bloch theorems for harmonic mappings

Chen, Gauthier and Hengartner obtained some versions of Landau's theorem for bounded harmonic mappings and Bloch's theorem for harmonic mappings which are quasiregular and for those which are open. Later, Dorff and Nowak improved their estimates concerning Landau's theorem. In this study, we improve these last results by obtaining sharp coefficient estimates for properly normalized harmonic mappings. Furthermore, our estimates allow us to improve Bloch constant for open harmonic mappings.     

Harmonic maps with finite total energy

We will give a criteria for a nonnegative subharmonic function with finite energy on a complete manifold to be bounded. Using this we will prove that if on a complete noncompact Riemannian manifold , every harmonic function with finite energy is bounded, then every harmonic map with finite total energy from into a Cartan-Hadamard manifold must also have bounded image. No assumption on the curvature of is required. As a consequence, we will generalize some of the uniqueness results on homotopic harmonic maps by Schoen and Yau.

     

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Boundary Behavior of Harmonic Functions on Manifolds

In this paper, we mainly set up a kind of representation theorem of harmonic functions on manifolds with Ricci curvature bounded below and study non-tangential limits of harmonic functions.     

Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D. The research of this author is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

The Refined Analysis on the Convergence Behavior of Harmonic Map Sequence from Cylinders

In this paper, we are concerned with the convergence behavior of a sequence of harmonic maps from long cylinders with uniformly bounded energy. If the bubbling phenomenon does not occur, we give the length formula of the limit map (i.e., geodesic in the target manifold). Furthermore, we provide a geometric explanation of the energy identity for a sequence of harmonic maps from degenerating Riemann surfaces with uniformly bounded energy, proved by M. Zhu.     

Maps preserving the harmonic mean or the parallel sum of positive operators

Let H be a complex Hilbert space. The symbol A!B stands for the harmonic mean of the positive bounded linear operators A,B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.     

Dirichlet空间上的Bergman型Toeplitz算子

本文给出了Dirichlet空间上以有界调和函数为符号的Bergman型Toeplitz算子是紧算子的充要条件.同时刻画了此类Bergman型Toeplitz算子在Dirichlet空间上的交换性.     

Algebras on the unit disk and Toeplitz operators on the Bergman space

This paper studies algebras of functions on the unit disk generated byH ^∞(D) and bounded harmonic functions. Using these algebras, we characterize compact semicommutators and commutators of Toeplitz operators with harmonic symbols on the Bergman space. Supported in part by the National Science Foundation and the University Research Council of Vanderbilt University.