论文标题
在沃尔夫定理上重新审视
On a Theorem of Wolff Revisited
论文作者
论文摘要
我们研究$ p $ - 谐波功能，$ 1 <p \ neq 2 <\ infty $，in $ \ mathbb {r}^{2} _ + = \ {z = x + i y：y> 0， - \ infty <x <x <x <x <x <x <\ infty \} $ and $ b（0，1） <1 \} $。 We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{iθ} )$, which is $ 2π/N $ periodic in the $ θ$ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $ \ leq c n $ on $ \ partial b（0，1）$满足$ v（0）= 0 $和$ c^{ - 1} \ leq \ int _ { - π}^πv（e^{i^{iθ}）dθ\ leq c $。如果$ 2 <p <\ infty $，我们给出了$ v $的或多或少的明确示例，我们的工作是$ \ mathbb {r}^{2} {2} _+ $ to $ b（0，1）$的沃尔夫结果的扩展。使用我们的第一个结果，我们将沃尔夫（Wolff）的工作扩展到FATOU类型定理因$ \ mathbb {r}^{2} _+ $的失败，以$ b（0，1）$（0，1）$ for $ p $ - harmonic函数，$ 1 <p \ p \ neq 2 <\ neq 2 <\ infty $。最后，我们还概述了扩展Llorente，Manfredi和Wu的工作所需的修改，以在$ \ partial \ Mathbb {r}^{2} {2} _+ $到$ \ partial b（0，1）$上进行$ p $ harmonic措施的失败。
We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $ \mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{iθ} )$, which is $ 2π/N $ periodic in the $ θ$ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $ c^{-1} \leq \int_{-π}^π V ( e^{iθ} ) d θ\leq c$. In case $2<p<\infty $ we give a more or less explicit example of $V$ and our work is an extension of a result of Wolff on $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$. Using our first result, we extend the work of Wolff on failure of Fatou type theorems for $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$ for $p$-harmonic functions, $1< p\neq 2<\infty$. Finally, we also outline the modifications needed for extending the work of Llorente, Manfredi, and Wu regarding failure of subadditivity of $p$-harmonic measure on $ \partial \mathbb{R}^{2}_+ $ to $\partial B (0, 1)$.
