Kjersti Solberg Eikrem, Eugenia Malinnikova
We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets.
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We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put forward different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball. We then study Hardy spaces of hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere. In particular, we obtain an atomic decomposition of these spaces.
We provide variational estimates for Bloch functions on the unit ball of $\mathbb{R}^d$ extending previous work on the Anderson conjecture for conformal maps on the unit disc.
It is well known that if $h$ is a nonnegative harmonic function in the ball of $\RR^{d+1}$ or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any $β\in [0,d]$, the Hausdorff dimension of the set of points $ξ$ on the sphere such that $h(rξ)$ looks like $(1-r)^{-β}$ is equal to $d-β$.
Multiple Orthogonal Polynomials on the Ball and Radial Extensions
本文通过结合Jacobi型径向分量与球谐函数角部，利用Jacobi-Piñeiro多重正交多项式，在单位球上引入第一类和第二类多重正交多项式，并建立单变量多重正交性若干基本结果的多变量推广，最后将构造扩展到更一般的径向权函数域。
A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.
Uniqueness of $H$-harmonic Moebius invariant inner products on the ball
该研究证明了实n维空间单位球上双曲调和函数的莫比乌斯不变半内积具有唯一性，明确了被双曲拉普拉斯算子零化的函数空间上该内积的唯一特性。
Weighted mean value identities over balls are considered for harmonic functions and their derivatives. Logarithmic and other weights are involved in these identities for functions. Some applications of weighted identities are presented. Also, new analytic characterizations of balls are proved; each of them requires the volume mean of a single weight function over the domain under consideration to be equal to a prescribed number depending on the weight.
In this paper we present the necessary modifications of the proof by Mozolyako and Havin in to work for Lipschitz domains instead of $\mathcal{C}^2$ domains.
In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started by the second named author. Our focus here is on the theory of Hardy, Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman-Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball.
Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring
本文研究三维单位球上受切向约束的$\mathbb{S}^2$值极小调和映射，获得尊重切向性的单调性公式以证明边界正则性，并给出极小元具有对称性的新充分条件，最终刻画其恰有两个位于边界相对点的点奇点。
On non-centered maximal operators related to a non-doubling and non-radial exponential measure
本文研究指数测度下非中心 Hardy–Littlewood 极大算子，证明立方体与菱形情形 p>1 的 L^p 有界性并否定弱 (1,1) 型，欧氏球情形在 d≤4 时得到相应肯定结论。
This is Chapter 1 of the book {\it Approximation Theory and Harmonic Analysis on Spheres and Balls} by the authors. It provides a self-contained introduction to spherical harmonics. The book will be published as a title in {\it Springer Monographs in Mathematics} by Springer in 2013. The table of contents of the book is attached at the end of this file.
Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in $\mathbb{C}^n$. The estimates are in terms of the Bekolle-Bonami constant of the weight.
This paper studies the uncertainty principle for spherical $h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl-Laplace-Beltrami operator on the weighted sphere.
The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced.
Continuous harmonic functions on a ball that are not in $H^s$ for $s>1/2$
本文基于 Hadamard 的无穷能量调和函数思想，利用球面调和函数级数构造了在任意维数球上连续到边界但不属于充分大指标 Sobolev 空间的调和函数，并验证了非线性传输边值问题解的正则性最优性。
Fast expansion into harmonics on the ball
本文提出快速且精度可证的算法，在三维函数的笛卡尔体素表示与球谐函数展开之间高效转换，将时间复杂度从 $O(N^6)$ 降至 $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$，并验证了其准确性。
Consider the pseidounitary group $G=U(p,q)$ and its compact subgroup $K=U(p)$. We construct an explicit unitary intertwining operator from the tensor product of a holomorphic representation and a antiholomorphic representation of $G$ to the space $L^2(G/K)$. This implies the existense of a canonical action of the group $G\times G$ in $L^2(G/K)$. We also give a survey of analysis of Berezin kernels and their relations with special functions.
We give a characterization of harmonic and subharmonic functions in terms of their mean values in balls and on spheres. This includes the converse of an inequality of Beardon's for subharmonic functions. We also obtain integral inequalities of Harnack type between these two means in general domains.
We present a fast and numerically accurate method for expanding digitized $L \times L$ images representing functions on $[-1,1]^2$ supported on the disk $\{x \in \mathbb{R}^2 : |x|<1\}$ in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in $O(L^2 \log L)$ operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagona…
We study harmonic Besov spaces $b^p_α$ on the unit ball of $\mathbb{R}^n$, where $0<p<1$ and $α\in\mathbb{R}$. We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman-Besov spaces. We show that the dual of harmonic Besov space $b^p_α$ is weighted Bloch space $b_β^{\infty}$ under certain volume integral pairing for $0<p<1$ and $α,β\in\mathbb{R}$. Our other results are about growth at the boundary and atomic decomposition.
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