Some properties of conjugate harmonic functions in a half-space
The following theorem is a result of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [J. Reine Angew. Math. 167, 405--423 (1932; Zbl 0003.20203)]. Let \(0<p\leq 1\) and let \(f(z)=u(z)+iv(z)\) be an analytic function in the unit disc, such that \(v(0)=0\) and \[ M_p(r,u):=\Big( \frac{1}{2\pi}\int_{0}^{2\pi}|u(re^{i\theta})|^pd\theta\Big)^{1/p}<C, \;0\leq r<1. \] Then \(M_p(r,v)\leq AC+AC\Big(\log\frac{1}{1-r}\Big)^{1/p}\). The authors prove an analog of the above theorem for conjugate harmonic functions in \(\mathbb{R}^{n+1}_{+}\). They also give sufficient conditions for a harmonic vector to belong to \(H^p(\mathbb{R}^{n+1})\), \(0<p\leq 1\).
scientific article; zbMATH DE number 5757910