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

Abstract

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 $|u(z)|\lesssim (1-|z|)^\gamma \rho^{-q}(z)$ where $\rho(z)= \mathop{\rm dist}(z, E)$, $\gamma$, $q$ are some real constants, $\gamma\le q$. 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 $|u(z)|\lesssim \rho^{-q}(z)$ in terms of smoothness of the Stieltjes measure associated to the harmonic function $u$.