Abstract
Let E be an arbitrary closed set on the unit circle ∂ⅅ and u be a harmonic function on the unit disk ⅅ 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 \(\tilde u\) 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.
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Chyzhykov, I., Kosaniak, Y. Estimates of conjugate harmonic functions with given set of singularities and application. Anal Math 47, 795–809 (2021). https://doi.org/10.1007/s10476-021-0093-7
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DOI: https://doi.org/10.1007/s10476-021-0093-7