Estimates of conjugate harmonic functions with given set of singularities and application (Q2052770)

Let \(\mathbb{D}\) be the unit disk in the complex plane \(\mathbb{C}\) and \(\partial\mathbb{D}\) denote its boundary (the unit circle). For a harmonic function \(u\) on the unit disk \(\mathbb{D}\), denote by \(\tilde{u}\) the harmonic conjugate function of \(u\). Let \(E\) be an arbitrary closed set on the unit circle \(\partial\mathbb{D}\). If \(u\) satisfies the inequality \(|u(z)|\leq C_0(1-|z|)^\gamma\rho^{-q}(z)\) where \(\rho(z)={\text{ dist}}\,(z,E)\), \(C_0\), \(\gamma\), \(q\) are some real constants, \(\gamma\leq q\), then there exists \(C_1=C_1(C_0,q,\gamma)>0\) such that \[\begin{aligned} |\tilde{u}(z)|\leq C_1 (1-|z|)^\gamma\rho^{-q}(z),\quad & \text{when }q>0>\gamma \\ C_1 (-\log(1-|z|)\rho^{\gamma-q}(z)),\quad & \text{when }q>\gamma\geq0 \\ C_1 (-\log\rho(z)), \quad & \text{when } q=\gamma>0 \end{aligned}\] for \(2^{-1}\leq|z|<1\).\par Let \(0<\gamma<1\), \(\alpha\geq0\), and \(u(z)\) have the form \[u(re^{i\theta})=\int_{\partial\mathbb{D}}P_\alpha(z\bar\zeta)\,d\mu(\zeta),\] where \(\mu\) is a complex finite Borel measure on \(\partial\mathbb{D}\), \[P_\alpha(z)=\Re\left(\Gamma(1+\alpha)\left(\frac2{(1-z)^{1+\alpha}}-1\right)\right).\] As an application, the authors describe growth classes defined by the non-radial condition \(|u(z)|\leq C_0\rho^{-q}(z)\) in terms of the smoothness of the Stieltjes measure \(\mu\) associated to the harmonic function \(u\).