A theorem of Hardy-Littlewood for harmonic functions satisfying Hölder's condition (Q1907749)

Let \(M\) denote a constant, not necessarily the same on any two occurrences, let \(B\) be the unit ball of \(\mathbb{R}^n\) \((n\geq 2)\), and let \(\nabla\) denote the gradient operator. The authors start by observing that known techniques can be adapted to prove the following result. Theorem A. Let \(u\) be harmonic on \(B\) and \(0< \alpha\leq 1\). Then \(|\nabla (x) |\leq M(1- |x|)^{\alpha-1}\) for all \(x\in B\) if and only if \(|u(x)- u(y) |\leq M|x-y |^\alpha\) for all \(x,y\in B\). Theorem A is generalized as follows. Let \(k: (0, \infty)\to (0, \infty)\) be increasing and satisfy \(k(2t)\leq Mk(t)\) for all \(t>0\), and define \(h(t)= tk (1/t)\) for \(t>0\) and \(h(0) =0\). Theorem 1. Suppose that \(t^\beta k(1/t)\) is increasing on \((0, \infty)\) for some \(\beta\in (0, 1)\), and let \(u\) be harmonic on \(B\). Then \(|\nabla u(x) |\leq Mk ((1- |x|)^{-1}\) for all \(x\in B\) if and only if \(|u(x)- u(y) |\leq Mh (|x-y |)\) for all \(x,y\in B\). Two extensions of Theorem A are also stated and similarly generalized: one of these extensions is a version of Theorem A for the case where \(\alpha= 0\); the other deals with the iterated gradient \(\nabla_2\).