Michael Dorff, Maria Nowak, Wojciech Szapiel
We consider a class $\THO$ of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.
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In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.
Using Real-Variable Techniques to Study Zeros of Complex-Valued Harmonic Functions
本文研究两类单参数调和函数族的零点，利用零点位于特定射线的性质，通过实变量技术计数各射线上的零点，揭示零点数目与参数的依赖关系。
Real non-attractive fixed point conjecture for complex harmonic functions
本文证明复多项式及有理调和函数的实非吸引不动点猜想，给出满足乘子实部条件的 \mathfrak{h}-不动点，并补充示例、可视化及超越情形讨论。
Several properties of a class of generalized harmonic mappings
研究定义一类二阶齐次偏微分方程解的实核 α-调和映射，并分析其表示定理、Lipschitz 连续性、单叶性等性质。
$\mathcal H$-Harmonic Bergman-Besov Spaces on the Real Hyperbolic Ball
通过多种微分算子刻画，将实双曲球上$\mathcal H$-调和函数的Bergman空间族从$\alpha>-1$延拓到所有$\alpha\in\mathbb R$，并推广了投影、对偶、原子分解和包含关系等性质。
Harmonic Bloch Space on the Real Hyperbolic Ball
本文研究实双曲球上调和函数的 Bloch 空间，证明 Bergman 投影的满射性，确定双曲调和 Bergman 空间的对偶与预对偶，并给出 Bloch 函数的原子分解。
We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.
Harmonic Functions on Four Dimensions
本文通过推广二维实调和函数与复解析函数的关联，建立了四维实输入的双曲双复调和函数理论，定义了其双曲实部、调和共轭函数，并给出了对应的双复Poisson核与积分表示。
Note on real and imaginary parts of harmonic quasiregular mappings
本文证明在单位圆盘上的K-拟正则调和映射的实部和虚部的积分均值具有相同的增长阶，并进一步表明它们在边界上具有相同的光滑度。
Motivated by some recent results of Kalaj and Vuorinen (Proc. Amer. Math. Soc., 2012), we prove that positive harmonic functions defined in the upper half--plane are contractions w.r.t. hyperbolic metrics of half--plane and positive part of the real line, respectively
We present a family of sense-preserving harmonic mappings in the unit disk related to the classical generalized (analytic) Koebe functions. We prove that these are precisely the mappings that maximize simultaneously the real part of every Taylor coefficient as well as the growth and distortion of functions in affine and linear invariant families of complex-valued harmonic functions.
We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we discus some Riesz type results for holomorphic functions.
Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature
本文证明了在度量具有非负高斯曲率时，单位圆盘上实 $ρ$-调和函数满足特定梯度不等式，推广了已有结果并解决了2014年的猜想。
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
Zygmund type spaces of harmonic functions on the real unit ball
研究实单位球上的调和齐格蒙德型空间，通过齐格蒙德型条件刻画\(\mathcal{HZ}^\alpha\)，建立\(\mathcal{HZ}\)的类似刻画及\(\mathcal{HZ}^\alpha\)的积分刻画，推广相关定理并得到类似布洛赫空间结果。
We prove the following theorem. Let $U$ be a pseudo-harmonic function on a surface $M^2$. For a real valued continuous function $V : M^2 \to {\mathbb R}$ to be a conjugate pseudo-harmonic function of $U$ on $M^2$ it is necessary and sufficient that $V$ is open on level sets of $U$.
This article gives a ``fundamental solution'' based energy-norm harmonic interpolation approach for two half-space settings of interest: the upper-half $\mathbb{R}^n$ plane, where fundamental solutions satisfy Laplace's equation, and the upper-half complex plane, where simple poles are of interest. This approach can handle higher-order pole fits, as well as logarithmic source fits, in the complex setting and it can handle higher-order multipole fits in the general real $\mathbb{R}^n$ setting. Higher-order multipoles in the real $\mathbb{R}^n$ half-space setting are of particular interest since…
By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on the constant function equal to one. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in $\mathbb{C}$. At the end we compare formulas, obtain…
This note reviews complex and real techniques in harmonic analysis. We describe a common source of both approaches rooted in the covariant transform generated by the affine group. Keywords: wavelet, coherent state, covariant transform, reconstruction formula, the affine group, ax+b-group, square integrable representations, admissible vectors, Hardy space, fiducial operator, approximation of the identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal function, vertical maximal functions, non-tangenti…
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