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Let X = X(t); t ≥ 0 be the hyperbolic Brownian motion on the real hyperbolic space ℍⁿ = x ∈ ℝⁿ:xₙ > 0. We study the Green function and the Poisson kernel of tube domains of the form D × (0,∞)⊂ ℍⁿ, where D is any Lipschitz domain in ${ℝ}^{n-1}$. We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in ${ℝ}^{2n}$. We give formulas and uniform estimates for the set $D_a=x\in ℍⁿ:x₁\in (0,a)$. The constants in the estimates depend only on the dimension of the space.