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Radial variation of Bloch functions on the unit ball of $\mathbb{R}^d$
Arkiv för Matematik ( IF 0.7 ) Pub Date : 2020-04-23 , DOI: 10.4310/arkiv.2020.v58.n1.a10
Paul F. X. Müller 1 , Katharina Riegler 1
Affiliation  

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that\[\int \limits_{[0, x]} \lvert \nabla b(\zeta) \rvert e^{b(\zeta)} \: d \lvert \zeta \rvert \lt \infty \; \textrm{.}\]In the second part of the paper, we show that the area or volume integral\[\int \limits_{B^d} \lvert \nabla u(w) \rvert p(w,\theta) \: dA(w)\]for positive harmonic functions $u$ is bounded by the value $cu(0)$ for at least one $\theta$. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

中文翻译:

Bloch函数在$ \ mathbb {R} ^ d $单位球上的径向变化

在[9]中,安德森的猜想是通过将Bloch函数的值与函数的变化进行比较来证明的。我们将Bloch函数的结果从二维扩展到任意维,并证明\ [\ int \ limits _ {[[0,x]} \ lvert \ nabla b(\ zeta)\ rvert e ^ {b(\ zeta)} \ \: d \ lvert \ zeta \ rvert \ lt \ infty \; \ textrm {。} \]在本文的第二部分中,我们显示了面积或体积积分\ [\ int \ limits_ {B ^ d} \ lvert \ nabla u(w)\ rvert p(w,\ theta )\:dA(w)\]对于正谐波函数$ u $的值由至少一个$ \ theta $的值$ cu(0)$界定。积分也被转移到简单连接的域,并从随机性的角度进行解释。提出了几个新出现的开放性问题。
更新日期:2020-04-23
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