Denote by \(H(\mathbb D)\) the set of all holomorphic functions in the unit disk \(\mathbb D\) and by \(H^p(\mathbb D)\) the usual Hardy space. Let \(\omega:[0,2]\mapsto\mathbb R\) be a continuous increasing function such that \(\omega(0)=0\) and so that \(\omega(t)/t\geq \omega(s)/s\) whenever \(0\leq t\leq s\). Moreover, let
\[ \Lambda_\omega(\mathbb D)= \biggl\{f:\mathbb D\to\mathbb C,\;\sup_{z\not=w} \frac{| f(z)-f(w)| }{\omega(| z-w| )} <\infty\biggr\}. \]
K. Dyakonov [Acta Math. 178, 143–167 (1997; Zbl 0898.30049)] studied the functions in \( \Lambda_\omega(\mathbb D)\cap H(\mathbb D)\) in terms of their moduli. The paper under review continues this line of research by considering the same problem for the Hardy-Lipschitz spaces \[ \Lambda_\omega^p(\mathbb D)= \biggl\{f:\mathbb D\to\mathbb C \text{ Borel measurable, } \sup_{z\not=w}\frac{\|f_w-f_z\|_p}{\omega(| w-z| )}<\infty\biggr\}, \]
where \(f_w\) is given by \(f_w(\xi)=f(w\xi)\) for \(| \xi| <1/| w| \), \(w\in\mathbb D\), and where \(\|g\|_p\) is the usual \(H^p(\mathbb D)\)-norm. The author also addresses the question when the product of a function \(f\in H^p(\mathbb D)\bigcap \Lambda_\omega^p(\mathbb D)\) with an inner function \(I\) is in \(\Lambda_\omega^p(\mathbb D)\) again.
Reviewer’s remark. Formula (3) and the definition of the oscillation \(\text{osc}(u;z,\varepsilon)\) is not printed correctly.