On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

Abstract

In this paper, we investigate solutions of the hyperbolic Poisson equa-is the hyperbolic Laplace operator in the n-dimensional space R n for n ≥ 2. We show that if n ≥ 3 andHere P h and G h denote Poisson and Green integrals with respect to ∆ h , respectively. Furthermore, we prove that functions of the form uwhere η ∈ B n and σ is the (n − 1)-dimensional Lebesgue measure normalized so that σ(S n−1 ) = 1. Moreover, Kalaj [16] also proved the Lipschitz continuity of 2000 Mathematics Subject Classification. Primary: 31B05; Secondary: 31C05.

Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations

“…In fact, P [ψ] ∈ L ω (D, C) is Lipschitz continuous if and only if the Hilbert transform of dψ(e iθ )/dθ belongs to L ∞ (S 1 ) (see [1] and [29]), where ω(t) = t. In [1], Arsenović et al established the following result for harmonic mappings of B n into R n : For a boundary function which is Lipschitz continuous, if its harmonic extension is quasiregular, then this extension is also Lipschitz continuous. Recently, the relationship of the Lipschitz continuity between the boundary functions and their harmonic extensions has attracted much attention (see [4,5,16,19,22]). Li and Ponnusamy [22] discussed the Lipschitz characteristic of solutions to the inhomogeneous biharmonic equation (1.1) for n = 2.…”

Generalized Bloch spaces, integral means of hyperbolic harmonic mappings in the unit ball

“…We refer to [4,15,19,28,35,36,37] for basic properties of this class of mappings. For convenience, in the following of this paper, we always use the notation ∆ h u = 0 to mean that u = (u 1 , · · · , u n ) is hyperbolic harmonic in B n .…”

“…T at x, where T is the transpose and the gradients ∇f j (x) are understood as column vectors (cf. [4]). For two column vectors x, y ∈ R n , we use x, y to denote the inner product of x and y.…”

Khavinson problem for hyperbolic harmonic mappings in Hardy space

“…(I) First, we prove Theorems 1.2(2) and 1.5 (2). Assume that q ∈ ), where K 0 ∈ N. By (2.8), (3.2) and (3.4), we know that for any ρ ∈ [0, 1),…”

“…2 and [15,Lemma 1] (or[2, Theorem G]) thatC H n ,2 (x; l) ≤ C H n ,2 (x; ±e n ) = 1 2 2n−η, e n | 2 |1 − η, e n | 2(n−1) dS n−1 (η) = π n−1 2n−3 Γ( n−1 2 ) − t) n−1 dt. Let t = cos 2s, where t ∈ [0, π 2 ].…”