A generalization of the Hardy-Littlewood theorem and estimates for the Riesz system in \(\mathbb{R}_+^{n+1}\) (Q1587170)

Given a harmonic function \(u\) in the unit disk such that \(\int_{-\pi}^\pi|u(re^{i\varphi})|d\varphi\leq \mu(\frac{1}{1-N})\), bounds are found for its harmonic conjugate, provided the function \(\mu\) satisfies certain regularity conditions. Similar results are obtained for harmonic vector fields in a half-space of \(\mathbb{R}^{n+1}\) as well. (There are few misprints in the paper. In particular, the function \(\psi:= \log\varphi\) is not defined in the statement of Theorem 1, and the third assertion of that theorem, however proved, is not formulated).