On a theorem of cartwright in higher dimensions (Q2793759)

A classical theorem of \textit{M. L. Cartwright} [Q. J. Math., Oxf. Ser. 4, 246--257 (1933; Zbl 0008.11803)] states that if \(u\) is harmonic in the unit disc \(\mathbb D\) with \(u(0) =0\) and satisfies the one-sided growth condition \(u(z) \leq \omega(1-|z|),\, z\in\mathbb D\), where \(\omega(t) = 1/t^p\) for some \(p>1\), then the reverse inequality holds: \(u(z) \geq C \omega(1-|z|)\) where \(C\) depends only of \(p\). This result was subsequently extended to more general weights by \textit{M. L. Cartwright} herself [Q. J. Math., Oxf. Ser. 6, 94--105 (1935; Zbl 0011.35803)] and \textit{C. N. Linden} [Proc. Camb. Philos. Soc. 52, 49--60 (1956; Zbl 0070.07202); ibid. 58, 26--37 (1962; Zbl 0101.29603)]. In the present paper the authors extend these results to harmonic functions in the unit ball in \(\mathbb R^{n+1}\) for a large class of weights.NEWLINENEWLINELet \(\omega : \mathbb R_+ \to \mathbb R_+\) be a strictly decreasing \(C^2\) function with \(\omega(1) =1\), \(\lim_{y\to 0}\omega(y) = \infty\), which satisfies the following growth and regularity conditions: NEWLINE\[NEWLINE \lim_{y\to 0}\frac{\omega(y)}{\omega^\prime(y)} = 0 \quad\text{and}\quad \left(\frac{\omega(y)}{\omega^\prime(y)}\right)^\prime \geq - \frac{1-\delta}{n},\quad 0<y<1,\tag{1} NEWLINE\]NEWLINE for some positive \(\delta\). The main result of the paper is the following:NEWLINENEWLINENEWLINENEWLINE Theorem 1. Let \(U\) be a harmonic function in the unit ball \(B\subset \mathbb R^{n+1}\), \(U(0)=0\). Assume that \(U\) admits the growth condition \(U(z) \leq \omega(1-|z|), \, z\in B\), where \(\omega\) satisfies (1) above. Then the following two-sided estimate holds: \(|U(z)| \leq C \omega(1-|z|), \, z\in B\), where the constant \(C\) depends only on the parameter \(\delta\) and the dimension \(n\). The weight \(\omega(y) = y^{-p}\) satisfies (1) if and only if \(p>n\). For the case \(p=n\) the authors prove the following generalization of another theorem of Cartwright. NEWLINENEWLINENEWLINENEWLINE Theorem 2. If \(U\) is harmonic in \(B\), \(U(0) =0\), and \(U(z) \leq (1-|z|)^{-n},\, z\in B\), then NEWLINE\[NEWLINE |U(z)| \leq C(1-|z|)^{-n} \left(\log \frac1{1-|z|}\right)^{n+1},\, |z|>\tfrac12, NEWLINE\]NEWLINE where \(C\) depends only on \(n\).