Khavinson Problem for Hyperbolic Harmonic Mappings in Hardy Space

Abstract

In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that u= PΩ[ϕ] and ϕ∈ Lp(∂Ω , ℝ) , where p∈ [1 , ∞] , PΩ[ϕ] denotes the Poisson integral of ϕ with respect to the hyperbolic Laplacian operator Δh in Ω, and Ω denotes the unit ball Bn or the half-space ℍn. For any x ∈Ω and l∈ Sn−1, let CΩ,q(x) and CΩ,q(x;l) denote the optimal numbers for the gradient estimate|∇u(x)|≤CΩ,q(x)∥ϕ∥Lp(∂Ω,ℝ) and the gradient estimate in the direction l|〈∇u(x),l〉|≤CΩ,q(x;l)∥ϕ∥Lp(∂Ω,ℝ), respectively. Here q is the conjugate of p. If q∈ [1 , ∞] , then CBn,q(0)≡CBn,q(0;l) for any l∈ Sn−1. If q= ∞, q = 1 or q∈[2K0−1n−1+1,2K0n−1+1] with K∈ ℕ, then CBn,q(x)=CBn,q(x;±x|x|) for any x∈ Bn∖ { 0 } , and Cℍn,q(x)=Cℍn,q(x;±en) for any x∈ ℍn. However, if q∈(1,nn−1), then CBn,q(x)=CBn,q(x;tx) for any x∈ Bn∖ { 0 } , and Cℍn,q(x)=Cℍn,q(x;ten) for any x∈ ℍn. Here tw denotes any unit vector in ℝn such that 〈tw,w〉 = 0 for w∈ ℝn∖ { 0 }.