关键词: Boundary Schwarz lemma, pluriharmonic mapping, unit polydisk, unit ball

Abstract:

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P(Dⁿ, B^N) is C^1+α at z₀ ∈ E_r ⊂ ∂Dⁿ with f(0)=0 and f(z₀)=w₀ ∈ ∂B^N for any n, N ≥ 1, then there exist a nonnegative vector λ_f=(λ₁,0, …, λ_r, 0, …,0)^T ∈ R²ⁿ satisfying λ_i ≥ 1/2^2n-1 for 1 ≤ i ≤ r such that
(Df(z'₀))^T w'₀=diag(λ_f)z'₀,
where z'₀ and w'₀ are real versions of z₀ and w₀, respectively.

Key words: Boundary Schwarz lemma, pluriharmonic mapping, unit polydisk, unit ball