论文标题
Schwarz引理单位球中的双曲线谐波映射
Schwarz lemma for hyperbolic harmonic mappings in the unit ball
论文作者
论文摘要
假设[1，\ infty] $和$ u = p_ {h} [ϕ] $， 其中$ ϕ \ in L^{p}（\ Mathbb {s}^{n-1}，\ Mathbb {r}^n）$和$ u（0）= 0 $。然后，我们获得了尖锐的不等式$ | u（x）| \ le g_p（| x |）\ | ϕ \ | _ {l^{p}} $，用于某些平滑函数$ g_p $ nishing $ 0 $。此外，我们在不等式中获得了尖锐常数$ c_p $的明确形式。这两个结果从谐波映射理论（\ cite [theorem 2.1] {kalaj2018}）和双曲线谐波理论（\ cite [theorem 1] {bur}）中概括并扩展了一些已知结果。
Assume that $p\in[1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $|u(x)|\le G_p(|x|)\|ϕ\|_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\|Du(0)\|\le C_p\|ϕ\|_{L^{p}}$. These two results generalize and extend some known result from harmonic mapping theory (\cite[Theorem 2.1]{kalaj2018}) and hyperbolic harmonic theory (\cite[Theorem 1]{bur}).
