论文标题
波尔加因常数进行谐波测量的下限
Lower bounds on Bourgain's constant for harmonic measure
论文作者
论文摘要
对于每一个$ n \ geq 2 $，波尔加因的常数$ b_n $是最大的数字，因此（上）谐波度量的（上）Hausdorff尺寸最多是$ \ Mathbb {r Mathbb {r}^n $中每个域中的$ n-b_n $，定义了哪个谐波度量。 Jones and Wolff（1988）证明了$ b_2 = 1 $。当$ n \ geq 3 $时，布尔加因（1987）证明$ b_n> 0 $ and Wolff（1995）制作了示例，显示了$ b_n <1 $。精炼波尔加因的原始大纲，我们证明\ [b_n \ geq c \，n^{ - 2n（n-1）}/\ ln（n）\]对于所有$ n \ geq 3 $，其中$ c> 0 $是独立于$ n $的常数。我们进一步估计$ b_3 \ geq 1 \ times 10^{ - 15} $和$ b_4 \ geq 2 \ times 10^{ - 26} $。
For every $n\geq 2$, Bourgain's constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb{R}^n$ on which harmonic measure is defined. Jones and Wolff (1988) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain's original outline, we prove that \[ b_n\geq c\,n^{-2n(n-1)}/\ln(n)\] for all $n\geq 3$, where $c>0$ is a constant that is independent of $n$. We further estimate $b_3\geq 1\times 10^{-15}$ and $b_4\geq 2\times 10^{-26}$.
