Let E be an arbitrary closed set on the unit circle ∂ⅅ and u be a harmonic function on the unit disk ⅅ satisfying |u(z)| ≾ (1 − |z|)^γρ−q(z)where ρ(z) = dist (z, E), γ, q are some real constants, γ ≤ q. We establish an estimate of the conjugate \(\tilde u\) of the same type which is sharp in some sense, and in the case E = ∂ⅅ coincides with known estimates. As an application we describe growth classes defined by the non-radial condition |u(z)| ≾ _ρ−q(z) in terms of smoothness of the Stieltjes measure associated to the harmonic function u.

中文翻译：

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给定奇点集的共轭调和函数估计及其应用

设E是单位圆 ∂ⅅ 上的任意闭集，u是单位圆盘ⅅ 上满足 | 的调和函数。u ( z )| ≾ (1 − | z |) ^γρ−q ( z ) 其中ρ ( z ) = dist ( z, E ), γ, q是一些实常数，γ ≤ q。我们建立了在某种意义上是尖锐的相同类型的共轭\(\tilde u\)的估计，并且在E = ∂ⅅ的情况下与已知估计重合。作为一个应用程序，我们描述了由非径向条件定义的生长类 | u ( z)| ≾ _ρ−q ( z ) 在与调和函数u相关的斯蒂尔杰斯测量的平滑度方面。