Khavinson Problem for Hyperbolic Harmonic Mappings in Hardy Space (original) (raw)

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 ∆ 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 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 = ∞ or q ∈ [ 2K0−1 n−1 + 1, 2K0 n−1 + 1] ∩ [1, ∞) with K 0 ∈ N = {0, 1, 2,. . .}, 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 , where e n = (0,. .. , 0, 1) ∈ S n−1. 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 en) 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}.