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

TL;DR

This paper derives sharp estimates for conjugate harmonic functions with prescribed singularities on the unit circle and explores their growth behavior in relation to the smoothness of associated measures.

Contribution

It provides new sharp estimates for conjugate harmonic functions with singularities on arbitrary closed sets on the unit circle, extending known results and linking growth conditions to measure smoothness.

Findings

Established sharp estimates for conjugate harmonic functions with singularities.

Connected growth classes of harmonic functions to the smoothness of associated measures.

Generalized known estimates to arbitrary closed sets on the unit circle.

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 $u$.