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 $\rho (z)=\mathrm{dist}(z,E)$, $\gamma$, $q$ 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 $u$.

15 pages

References in corpus (1)

• A Blaschke-type condition and its application to complex Jacobi matrices