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Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

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Frontiers of Mathematics in China 2022年第3期17卷 455-471页

作者： Wanwan YANG Bo LI Center for Applied Mathematics Tianjin UniversityTianjin 300072China

Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X×R_(+),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition supx_(B),r_(B) ∫_(0)^(r_(B) ) f_(B(x_(B),r_(B))) |t■u(x,t)|^(2)dμ(x)dt/t≤Charmonic functions whose traces are in BMO space.

关键词： harmonic function metric measure space BMO Carleson measure

HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS

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Acta Mathematica Scientia 2018年第6期38卷 1751-1758页

作者：林勇宋宏业 Department of Mathematics Renmin University of China School of General Education Beijing International Studies University

We prove a Harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions ... 详细信息

关键词： harmonic function subharmonic function Harnack inequality mean valueinequality graph

RIEMANN BOUNDARY VALUE PROBLEMS FOR SOME K-REGULAR functionS IN CLIFFORD ANALYSIS

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Acta Mathematica Scientia 2012年第5期32卷 2029-2049页

作者：姜乐杜金元 School of Mathematics and Statistics Wuhan University

In this paper,we study the R m（m〉0） Riemann boundary value problems for regular functions,harmonic functions and bi-harmonic functions with values in a universal clifford algebra C（Vn,n）.By using Plemelj formula,we get the solutions of R m（m〉0） Riemann boundary value problems for regular functions.Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions,we obtain the solutions of R m（m〉0） Riemann boundary value problems for harmonic functions and bi-harmonic functions.

关键词： Riemann boundary value problem harmonic function bi-harmonic function Plemelj formula

Growth of certain harmonic functions in an n-dimensional cone

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Frontiers of Mathematics in China 2013年第4期8卷 891-905页

作者： Lei QIAO Guantie DENG Department of Mathematics and Information Science Henan University of Economics and Law Zhengzhou 450002 China School of Mathematical Science Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 China

We give the growth properties of harmonic functions at infinity in a cone, which generalize the results obtained by Siegel-Talvila.

关键词： Growth property harmonic function cone

SINGULAR BOUNDARY PROPERTIES OF harmonic functionS AND FRACTAL ANALYSIS

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Chinese Annals of Mathematics,Series B 1997年第3期18卷 337-344页

作者： WEN ZHIYING ZHANG YIPING Department of Mathematics Wuhan UniversityWuhan 430072China

This paper shows an important relation between the fractal analysis and the boundary properties of harmonic functions.It is proved that the multifractal analysis of a finite measureμonIR^(d)determines the(non-tangential)boundary increasing properties ofPμ,the Poisson integral ofμwhich is harmonic onIR^(d+1)_(+).Some examples are given.

关键词： Fractal harmonic function Multifractal analysis

STOKES FLOW DUE TO FUNDAMENTAL SINGULARITIES BEFORE A PLANE BOUNDARY

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Applied Mathematics and Mechanics(English Edition) 2004年第7期25卷 799-805页

作者： N.Aktar F.Rahman S.K.Sen Department of Mathematics Department of MathematicsDepartment of Mathematics Shahjalal University of Science and Technology Sylhet Bangladesh Bangladesh Institute of Technology Khulna Bangladesh University of Chittagong Chittagon-4331 Bangladesh

A representation for the velocity and pressure fields in three-dimensional Stokes flow was presented in terms of a biharmonic function A and a harmonic function B.This representation was used to establish a general theorem for the calculation of Stokes flow due to fundamental singularities in a region bounded by a stationary no-slip plane boundary.Collins's theorem for axisymmetric Stokes flow before a rigid plane follows as a special case of the theorem.A few illustrative examples are given to show its usefulness.

关键词： Stokes flow Stokeslet Fouries transform harmonic function biharmonic function

ON THE LOWER BOUND FOR A CLASS OF harmonic functionS IN THE HALF SPACE

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Acta Mathematica Scientia 2012年第4期32卷 1487-1494页

作者：张艳慧邓冠铁高洁欣 Department of Mathematics Beijing Technology and Business UniversityBeijing 100048China School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China Department of Mathematics Faculty of Science and TechnologyUniversity of MacaoChina

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize the Carleman＇s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna＇s representation for harmonic functions in the half sphere by using HSrmander＇s theorem.

关键词： harmonic function Carleman＇s formula Nevanlinna＇s representation for halfsphere lower bound

SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

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Acta Mathematica Scientia 2012年第6期32卷 2096-2104页

作者：曹丽 School of Mathematics Hefei University of Technology School of Mathematics and Statistics Wuhan University

We study the asymptotic behavior near the boundary of u（x, y） = Ky ＊ μ （x）, defined on the half-space R^＋ x RN by the convolution of an approximate identity Ky （.）（y 〉 0） and a measure μ on IRN. The Poisson and the heat kernel are unified as special cases in our setting. We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ. Then a boundary limit theorem of Fatou＇s type for singular measures is proved. Meanwhile, the asymptotic behavior of a quotient of Kμ and Ku is also studied, then the corresponding Fatou-Doob＇s boundary relative limit is obtained. In particular, some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours. At the end, an application in fractal geometry is given.

关键词： fractal density harmonic function convolution singular measure

DERIVATIVES OF harmonic MIXED NORM AND BLOCH-TYPE SPACES IN THE UNIT BALL OF R^n

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Acta Mathematica Scientia 2011年第1期31卷 81-92页

作者：唐笑敏胡璋剑吕小芬 Department of Mathematics University of Science and Technology of China Department of Mathematics Huzhou Teachers College

Let H（B） be the set of all harmonic functions f on the unit ball B of Rn. For 0 p, q ≤ ∞ and a normal weight φ, the mixed norm space Hp,q, φ（B） consists of all functions f in H（B） for which the mixed norm p,q, φ ∞. In this article, we obtain some characterizations in terms of radial, tangential, and partial derivative norms in Hp,q, φ（B）. The parallel results for the Bloch-type space are also obtained. As an application, the analogous problems for polyharmonic functions are discussed.

关键词： harmonic function mixed norm space Bloch-type space norm derivatives