论文标题
在耐力空间中双曲线谐波映射的最佳估计值
Optimal estimates for hyperbolic harmonic mappings in Hardy space
论文作者
论文摘要
假设$ p \ in（1，\ infty] $和$ u = p_ {h} [ϕ] $，其中$ ϕ \ in l^{p}（\ Mathbb {\ Mathbb {s}^{n-1}，\ Mathbb {r}不等式$$ | u（x）| \ leq \ frac {\ mathbf {c} _ {q}^{\ frac {1} {q}} {q}}}（x）} {（1- | x |^2）^{\ frac {n-1} {n-1} {p}}}}}}}}}}}}}}}}}}} \ | n-c \ | n \ | n l^} | u（x）| \ leq \ frac {\ mathbf {c} _ {q}^{\ frac {\ frac {1} {q}}}}} {（1- | x |^2）^{\ frac {n-1} {n-1} {p}}}}}}}}}}}}}}}}} $ \ mathbf {c} _ {q}（x）$和常数$ \ mathbf {c} _ {q} $在Gauss Hyperdeometric和Gamma函数方面，其中$ q $是$ p $的共轭。该结果从谐波映射理论（[5，定理1.1和1.2]和[1，命题6.16]）中推广并扩展了一些已知结果。
Assume that $p\in(1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}}(x)}{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p}} \quad\text{and}\quad |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}} }{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p} } $$ for some function $\mathbf{C}_{q}(x)$ and constant $\mathbf{C}_{q}$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$. This result generalize and extend some known result from harmonic mapping theory ([5, Theorems 1.1 and 1.2] and [1, Proposition 6.16]).
