Estimates of conjugate harmonic functions with given set of singularities with application

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

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