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

摘要

Abstract Let E be an arbitrary closed set on the unit circle ?? and u be a harmonic function on the unit disk ? 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 u?documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$tilde u$$end{document} of the same type which is sharp in some sense, and in the case E = ?? 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.