Hyperbolic Harmonic Functions Research Articles

Modulus of continuity of normal derivative of a harmonic functions at a boundary point

We give sufficient conditions which ensure that harmonic extension u = P[f] to the upper half space {(x, y) | x ? Rn, y > 0} of a function f ? Lp(Rn) satisfies estimate ?u/?y (x, y) ? C?(y)/y for every x in E ? Rn, where ? is a majorant. The conditions are expressed in terms of behaviour of the Riesz transforms Rjf of f near points in E. We briefly investigate related questions for the cases of harmonic and hyperbolic harmonic functions in the unit ball.

Schwarz–Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic harmonic functions on the unit ball of $${\mathbb {R}^n}$$ and we disprove a recent conjecture of Liu (Int Math Res Not, 2021).

A Khavinson type conjecture for hyperbolic harmonic functions on the unit ball

In this paper, we prove a Khavinson type conjecture for hyperbolic harmonic functions on the unit ball. This conjecture was partially solved in [5]. For harmonic functions, the conjecture was recently proved by Liu [14].

Integral representations for hyperbolic harmonic function in the clifford algebra Cl n+1,0

The main goal of this paper is to give the basic principles of hyperbolic function theory. First, we discuss the Green identity for . Second, we also deal with the integral representations for hyperbolic harmonic function in the Clifford algebra

Hyperbolic Function Theory in the Skew-Field of Quaternions

We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called alpha -hyperbolic harmonic, that are harmonic with respect to the Riemannian metric dsα2=dx02+dx12+dx22+dx32x3α\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} ds_{\\alpha }^{2}=\\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\\alpha }} \\end{aligned}$$\\end{document}in the upper half space {mathbb {R}}_{+}^{4}={left( x_{0},x_{1},x_{2} ,x_{3}right) in {mathbb {R}}^{4}:x_{3}>0}. If alpha =2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function x^{m},(min {mathbb {Z}}), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental alpha -hyperbolic harmonic functions, depending only on the hyperbolic distance and x_{3}, we verify a Cauchy type integral formula for conjugate gradient of alpha -hyperbolic harmonic functions. We also compare these results with the properties of paravector valued alpha -hypermonogenic in the Clifford algebra {{,mathrm{{mathcal {C}}ell },}}_{0,3}.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

General Integral Formulas for k-hyper-mono-genic Functions

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

Hyperbolic harmonic functions and the associated integral equations

Abstract In this paper we consider a class of integral equations associated with the invariant mean value property of hyperbolic harmonic functions. We have shown that nonconstant solutions of these integral equations are functions of unbounded variations and do not attain their supremum or infimum on [0; 1): We present an algorithm to obtain numerical solutions of these integral equations. We also consider the equivalent ordinary differential equations and used MATLAB to obtain numerical solutions of these differential equations.

Radial growth and Hardy-Littlewood-type theorems on hyperbolic harmonic functions

In this paper, we first show that a result of Girela et al. on analytic functions can be extended to hyperbolic-harmonic functions, and then we establish Hardy-Littlewood-type theorems on hyperbolic harmonic functions.

On hypermonogenic functions

We research a function theory in higher dimensions based on the hyperbolic metric . The complex numbers are extended by the Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . In 1992, H. Leutwiler noticed that the power function (x 0 + x 1 e 1 + ··· + x n e n ) m is the generalized conjugate gradient of the functions . In the complex field (n = 1) this function h is harmonic in the usual sense, but in the higher dimensional case it is harmonic with respect to the Laplace–Beltrami operator with respect to the Riemannian hyperbolic metric . He started to study these type of functions, called H-solutions, that include positive and negative powers and elementary functions. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined by H. Leutwiler and the first author in 2000. The integral formula has been proved by the first author. In this article, we present a simple way to find hyperbolic harmonic functions depending on the hyperbolic distance. We use these functions to determine a better presentation of the kernel, that is surprisingly the shifted Euclidean Cauchy kernel. We prove a power series expansion of hypermonogenic functions and present a version of the Maximum Modulus theorem.

A hyperbolic Dirac operator and its kernels

We consider the hyperbolic generalization of the classical complex analysis to higher dimensions based on the hyperbolic metric . The field of complex numbers is generalized by the associative Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . H. Leutwiler has noticed around 1992 that the power function (x 0 + x 1 e 1 + ··· x n e n ) m is the conjugate gradient of a hyperbolic harmonic function. He started to study these types of functions that include positive and negative powers and elementary functions, defined similarly as in classical complex analysis. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined in terms of the modified Dirac operator introduced by H. Leutwiler and the author in 2000. The integral formula for hypermonogenic functions has been proved by the author. In this article we compute the same kernels using the hyperbolic distance. The kernel is surprisingly the shifted Euclidean Cauchy kernel. Using this we are able to prove mean-value properties for hypermonogenic functions and hyperbolic harmonic functions.

WEIGHTED LIPSCHITZ CONTINUITY, SCHWARZ–PICK'S LEMMA AND LANDAU–BLOCH'S THEOREM FOR HYPERBOLIC-HARMONIC MAPPINGS IN ℂN

In this paper, we discuss some properties on hyperbolic-harmonic functions in the unit ball of ℂ n . First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz–Pick type theorem for hyperbolic-harmonic functions and apply it to prove the existence of Landau-Bloch constant for functions in α-Bloch spaces.

Weighted Dirichlet spaces of harmonic functions on the real hyperbolic ball

Let 𝔹 n denote the unit ball in ℝ n , n ≥ 2, with boundary S, and let τ, ∇ h and Δ h denote the volume measure, gradient and Laplacian with respect to the hyperbolic metric on 𝔹 n . In the article we consider the weighted Dirichlet space 𝒟γ and Hardy space ℋ p , p ≥ 1, of hyperbolic harmonic functions on 𝔹 n for which D γ(f) and ‖ f ‖ p are finite, where and One of the main results of the article is the following theorem. THEOREM Let f be a hyperbolic harmonic function on 𝔹 n . (a) If f ∈ 𝒟γ for some γ, (n − 3) < γ ≤ (n − 1) (0 < γ ≤ 1 when n = 2), then f ∈ ℋ p for all p, 0 < p ≤ 2(n − 1)/γ, with where C 1 is a positive constant independent of f. (b) If f ∈ ℋ p for some p, 1 < p ≤ 2, then f ∈ 𝒟γ for all γ ≥ 2(n − 1)/p with where C 2 is a positive constant independent of f. In this article, we also prove two results concerning the integrability of |∇ h f | of eigenfunctions f of Δ h with nonnegative eigenvalues and prove the Littlewood–Paley inequalities for hyperbolic harmonic functions on 𝔹 n .

On Hyperbolic Function Theory

The hyperbolic version of the standard Clifford analysis will be considered. In this modification the power function x m becomes a solution. In more details, the Dirac operator \(Df = \sum^n_{i=0} e_i \frac{\partial f} {\partial x_i}\) with e 0 = 1, defined with respect to the Clifford algebra Cl n , is replaced by the operator \(M_kf(x) = Df (x) + \frac{k}{x_n} Q^{\prime}f(x)\), where ′ denotes the main involution in Cl n and Qf is given by the unique decomposition f(x) = Pf(x) + Qf(x)e n with Pf(x),Qf(x) ∈Cl n-1. The operator M k (k ∈R) will mainly be considered for k = 0, k = n − 1 and k = 1 − n. In case k = 0 the equation Mkf = 0 yields the well-known monogenic functions, in case k = n − 1 one obtains the so-called hypermonogenic functions introduced in [5]. Besides M k we also study the operator \(\overline{M_k}M_k = M_k\overline{M_k}\), a natural generalization of the Laplace operator Δ. Solutions of the equation \(\overline{M_{n-1}}M_{n-1}f =0\) are called hyperbolic harmonic functions. The main goal of this article is to give integral representations for hypermonogenic and hyperbolic harmonic functions in the upper half space \({\mathbb{R}}^{n+1}_+\).

Hyperbolic Harmonic Functions: Weak Approach with Applications in Function Spaces

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Qp-spaces.

Hyperbolic Harmonic Function in Minkowski Space

The paper discusses the properties of four-dimensional Minkowski space by using Clifford algebra, then gives the concept of hyperbolic harmonic function by constructing a system (P4) in the four-dimensional Minkowski space, and obtains several properties and a sufficient and necessary condition for the solvability of the system (P4) .

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC 1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝn be a bounded convex domain with C2 boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space Hkp,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, Hkp,q,φ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.

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

In this article we study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic space Bn. 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 Bn. We then study Hardy spaces H p (Bn), 0 < p < 1, whose elements appear as the hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere H p (S n 1 ). In particular, we obtain an atomic decomposition of this spaces.