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

Abstract

Let E be an arbitrary closed set on the unit circle ∂D, u be a harmonic function on the unit disk D satisfying |u(z)| (1 − |z|) γ ρ −q (z) where ρ(z) = dist(z, E), γ, q are some real constants, γ ≤ q. We establish an estimate of the conjugate ũ of the same type which is sharp in some sense and in the case E = ∂D 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.