论文标题
khavinson问题在耐力空间中的双曲线谐波映射
Khavinson problem for hyperbolic harmonic mappings in Hardy space
论文作者
论文摘要
\ begin {Abstract}在本文中，我们在强大空间中双曲线谐波映射的环境中部分解决了广义的Khavinson猜想。假设$ u = \ mathcal {p}_Ω[ϕ] $和 $ ϕ \ in l^{p}（\partialΩ，\ mathbb {r}）$，其中$ p \ in [1，\ infty] $，$ \ mathcal {p}_Ω[ϕ] $表示$ $ $ $ $ $ $ $ $ $ $ $ poisson的poisson poisson的poisson contemal表示单位球$ \ mathbb {b}^{n} $或半空间$ \ mathbb {h}^{n} $。对于任何$ x \ inω$和$ l \ in \ mathbb {s}^{n-1} $，让$ \ mathbf {c} _ {c} _ {ω，q}（x）$和$ \ m athbf {c} _ {c} _ {Ω | \ nabla u（x）| \ leq \ mathbf {c} _ {ω，q}（x）\ | ϕ \ | _ {l^{p}（\partialΩ，\partialΩ u（x），l \ rangle | \ leq \ mathbf {c} _ {ω，q}（x; l）\ | ϕ \ | _ {l^{l^{p}（\partialΩ，\ mathbb {r}） 分别。 这里$ Q $是$ p $的共轭。 如果$ q = \ infty $或$ q \ in [\ frac {2k_ {0} -1} -1} {n-1} +1，\ frac {2k_ {0}}} {n-1} +1} +1] \ cap [1，\ cap [1，\ infty）$然后$ \ MATHBF {C} _ {\ MATHBB {B}^{n}，Q}（x）= \ MathBf {C} _ {\ MathBb {\ Mathbb {b}^{n}，q}，q}，q}（x; \ pm pm \ frac \ frac {x}} $ x \ in \ Mathbb {b}^{n} \ backSlash \ {0 \} $，和$ \ \ \ \ \ mthbf {c} _ {\ MathBB {h}^{n}^{n}，q}，q}，q}，q}（x）（x）= \ mathbf {c} _ {c} _ {c} _ { e_ {n}）$ for任何$ x \ in \ mathbb {h}^{n} $，其中$ e_ {n} =（0，\ ldots，0,1）\ in \ mathbb {s}}^{n-1} $。但是，如果$ q \ in（1，\ frac {n} {n-1}）$，则$ \ mathbf {c} _ {\ mathbb {b}^{n}，q}，q}（x）= \ \ mathbf {c} $ x \ in \ mathbb {b}^{n} \ backslash \ {0 \} $，and $ \ MATHBF {C} _ {\ MATHBB {h}^{n}，q}（x）= \ MathBf {c} _ {\ Mathbb {\ Mathbb {h}^{n}，q}，q}，x; t_; t_ {e_ {n}} $这里$ t_ {w} $表示$ \ mathbb {r}^{n} $中的任何单位向量 这样的$ \ langle t_ {w}，w \ rangle = 0 $ for $ w \ in \ mathbb {r}^{n} \ setMinus \ {0 \} $。 \ end {摘要}
\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_Ω[ϕ]$ and $ϕ\in L^{p}(\partialΩ, \mathbb{R})$, where $p\in[1,\infty]$, $\mathcal{P}_Ω[ϕ]$ denotes the Poisson integral of $ϕ$ with respect to the hyperbolic Laplacian operator $Δ_{h}$ in $Ω$, and $Ω$ denotes the unit ball $\mathbb{B}^{n}$ or the half-space $\mathbb{H}^{n}$. For any $x\in Ω$ and $l\in \mathbb{S}^{n-1}$, let $\mathbf{C}_{Ω,q}(x)$ and $\mathbf{C}_{Ω,q}(x;l)$ denote the optimal numbers for the gradient estimate $$ |\nabla u(x)|\leq \mathbf{C}_{Ω,q}(x)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})} $$ and gradient estimate in the direction $l$ $$|\langle\nabla u(x),l\rangle|\leq \mathbf{C}_{Ω,q}(x;l)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})}, $$ respectively. Here $q$ is the conjugate of $p$. If $q=\infty$ or $q\in[\frac{2K_{0}-1}{n-1}+1,\frac{2K_{0}}{n-1}+1]\cap [1,\infty)$ with $K_{0}\in\mathbb{N}=\{0,1,2,\ldots\}$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;\pm\frac{x}{|x|})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;\pm e_{n})$ for any $x\in \mathbb{H}^{n}$, where $e_{n}=(0,\ldots,0,1)\in\mathbb{S}^{n-1}$. However, if $q\in(1,\frac{n}{n-1})$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;t_{x})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;t_{e_{n}})$ for any $x\in \mathbb{H}^{n}$. Here $t_{w}$ denotes any unit vector in $\mathbb{R}^{n}$ such that $\langle t_{w},w\rangle=0$ for $w\in \mathbb{R}^{n}\setminus\{0\}$. \end{abstract}
