On the monotonicity of positive invariant harmonic functions in the unit ball

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula

Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded…

Radial growth of harmonic functions in the unit ball

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.

Harmonic functions on the real hyperbolic ball I : Boundary values and atomic decomposition of Hardy spaces

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.

Boundary multifractal behaviour for harmonic functions in the ball

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 $d-β$.

球上$H$-调和莫比乌斯不变内积的唯一性

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。

Two monotonic functions involving gamma function and volume of unit ball

In present paper, we prove the monotonicity of two functions involving the gamma function $Γ(x)$ and relating to the $n$-dimensional volume of the unit ball $\mathbb{B}^n$ in $\mathbb{R}^n$.

Complete monotonicity of a function involving the gamma function and applications

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $Γ(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$.

Harmonic functions on the real hyperbolic ball II Hardy and Lipschitz spaces

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.

球上不属于 $H^s$（$s>1/2$）的连续调和函数

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。

Complete monotonicity of a function involving a ratio of gamma functions and applications

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

Characterization of harmonic and subharmonic functions via mean-value properties

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.

球上谐波的快速展开

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。

Spherical Harmonics

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.

Weighted Estimates for the Berezin Transform and Bergman Projection on the Unit Ball in $\mathbb{C}^{n}$

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.

Uncertainty Principles on weighted spheres, balls and simplexes

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.

Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations

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.