Khavinson Problem for Hyperbolic Harmonic Mappings in Hardy Space

In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that u=𝒫_Ω[ϕ ] and ϕ∈ L^p(∂Ω, ℝ) , where p∈ [1,∞ ] , 𝒫_Ω[ϕ ] denotes the Poisson integral of ϕ with respect to the hyperbolic Laplacian operator Δ h in Ω, and Ω denotes the unit ball 𝔹^n or the half-space ℍ^n . For any x ∈Ω and l∈𝕊^n-1 , let C Ω, q ( x ) and C Ω, q ( x ; l ) denote the optimal numbers for the gradient estimate |∇ u(x) |≤𝐂_Ω ,q(x)ϕ_L^p(∂Ω, ℝ) and the gradient estimate in the direction l |⟨∇ u(x),l⟩ |≤𝐂_Ω ,q(x;l)ϕ_L^p(∂Ω, ℝ), respectively. Here q is the conjugate of p . If q∈ [1,∞ ] , then 𝐂_𝔹^n,q(0)≡𝐂_𝔹^n,q(0;l) for any l∈𝕊^n-1 . If q=∞ , q = 1 or q∈ [2K_0-1/n-1+1,2K_0/n-1+1] with K_0∈ℕ , then 𝐂_𝔹^n,q(x)=𝐂_𝔹^n,q(x;±x/|x|) for any x∈𝔹^n\{0} , and 𝐂_ℍ^n,q(x)=𝐂_ℍ^n,q(x;± e_n) for any x∈ℍ^n . However, if q∈ (1,n/n-1) , then 𝐂_𝔹^n,q(x)=𝐂_𝔹^n,q(x;t_x) for any x∈𝔹^n\{0} , and 𝐂_ℍ^n,q(x)=𝐂_ℍ^n,q(x;t_e_n) for any x∈ℍ^n . Here t w denotes any unit vector in ℝ^n such that 〈 t w , w 〉 = 0 for w∈ℝ^n∖{0} .