On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation Δ _hu(x) = ψ(x) , where ψ∈ L^∞(Bⁿ, Rⁿ) and (Formula Presented.)Δhu(x)=(1-|x|2)2Δu(x)+2(n-2)(1-|x|2)∑i=1nxi∂u∂xi(x)is the hyperbolic Laplace operator in the n-dimensional space Rⁿ for n≥ 2. We show that if n≥ 3 and u∈ C²(Bⁿ, Rⁿ) ∩ C(Bⁿ¯ , Rⁿ) is a solution to the hyperbolic Poisson equation, then it has the representation u= P_h[ ϕ] - G_h[ ψ] provided that u∣Sn-1=ϕ and ∫Bn(1-|x|2)n-1|ψ(x)|dτ(x)<∞. Here P_h and G_h denote Poisson and Green integrals with respect to Δ _h, respectively. Furthermore, we prove that functions of the form u= P_h[ ϕ] - G_h[ ψ] are Lipschitz continuous.