Mean Lipschitz spaces and bounded mean oscillation
Assume that \(f(z)\) is analytic in the unit disk and has a non-tangential limit \(f(e^{i\theta})\) at a.e. point of the unit circle. The integral modulus of continuity of order \(p\), \(1\leq p < +\infty\), of the boundary function \(f(e^{i\theta})\) is \(\omega (\delta , f) = \sup_{0<|t|\leq \delta} \bigl({1 \over {2\pi}} \int_{-\pi}^{\pi} |f(e^{i(\theta + t)}) - f(e^{i\theta})|^{p}d\theta \bigr)^{1/p}\). The essential supremum is used in the usual fashion for \(\omega_{\infty}(\delta , f)\). The mean Lipschitz space \(\Lambda_{\alpha}^{p}\) consists of those functions \(f(z)\) for which \(\omega_{p}(\delta , f) = O(\delta^{\alpha})\) as \(\delta \to 0\). For a continuous and increasing function \(\omega:[0,\pi]\to [0,\infty)\) with \(\omega(0)=0\), the generalized mean Lipschitz space \(\Lambda(p,\omega)\) consists of those functions \(f(z)\) for which \(\omega_{p}(\delta , f) = O(\omega(\delta))\) as \(\delta \to 0\). For a continuous and increasing function \(\omega:[o,\pi]\to [0,\infty]\) with \(\omega(0)=0\) the generalized mean Lipschitz space \(\Lambda(p,\omega)\) consists of those functions \(f(z)\) for which \(\omega_p(\delta,f)=O(\omega(\delta))\) as \(\delta\to 0\). In Theorem 1 conditions on \(\omega(\delta)\) are given under which the class \(\Lambda(p,\omega)\), \(1<p<\infty\), contains a function \(f(z)\) which is not a Bloch function. Further, in Theorem 2 conditions on \(\omega(\delta)\) are given under which the class \(\Lambda(p,\omega)\), \(1<p<\infty\), contains a function \(f(z)\) which is not a normal function. Since the class \(BMOA\) lies in the class of Bloch functions which in turn lies in the class of normal functions and \(\Lambda(p,\omega) \not\subset BMOA\) for \(\omega\) in Theorem 1, this demonstrates the sharpness of the known inclusion \(\Lambda_{1/p}^{p} \subset BMOA\) due to \textit{P. Bourdon}, \textit{J. Shapiro} and \textit{W. Sledd} [Lond. Math. Soc. Lect. Note Ser. 137, 81-110 (1989; Zbl 0672.42003)].
scientific article; zbMATH DE number 1029012