In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E\subset \mathbb C$ is compact and $\mu$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies $$ \gamma(E) \geq c\,\frac{\mu(E)^2}{\int_I \|P_\theta\mu\|_2^2\,d\theta}, $$ where $c$ is some positive constant, $I\subset [0,\pi)$ is an arbitrary interval, and $P_\theta\mu$ is the image measure of $\mu$ by $P_\theta$, the orthogonal projection onto the line $\{re^{i\theta}:r\in\mathbb R\}$. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity. We also prove a generalization of the above inequality to higher dimensions which involves related capacities associated with signed Riesz kernels.

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分析能力和预测

在本文中，我们研究了集合的解析能力与其正交投影大小之间的联系。更准确地说，我们证明如果$E\subset \mathbb C$ 是紧致的并且$\mu$ 是$E$ 上支持的Borel 测度，那么$E$ 的解析容量满足$$ \gamma(E) \geq c\,\frac{\mu(E)^2}{\int_I \|P_\theta\mu\|_2^2\,d\theta}, $$ 其中 $c$ 是某个正常数，$I\子集 [0,\pi)$ 是任意区间，$P_\theta\mu$ 是 $\mu$ 通过 $P_\theta$ 的图像测度，正交投影到线 $\{re^{i \theta}:r\in\mathbb R\}$。这一结果与维图什金关于法瓦德长度和分析能力之间关系的旧猜想有关。我们还证明了将上述不等式推广到更高维度，其中涉及与带符号 Riesz 核相关的相关容量。