About functions on the dyadic group and Walsh series (Q343254)

Let \(\omega\) be any modulus of continuity in \(C(\mathbb{R})\) satisfying \[ \overline{\lim}_{t \rightarrow +0} t^{-1} \omega(t) = \infty. \] The author of the paper under review showed in a previous publication the existence of a function \(f_{\omega} (x)\) that satisfies \[ C_2 \geq \overline\lim_{| h | \rightarrow +0} \frac{| f_{\omega} (x+h) - f_{\omega}(x) |}{\omega (| h |)} \geq C_1 > 0 \] for all \(x \in \mathbb{R}\). In the paper under review the author gives several results of this type for the case of the dyadic group \(G\). For example, one such result proved by the author is: Let \(1 \leq p \leq \infty\) and let \(\omega = \{ \omega_n \}\) be a strictly decreasing sequence of numbers that converge to zero. Then there exists a function \(f_{\omega} \colon G \rightarrow \mathbb{R}\) such that for every \(x \in G\) and some constant \(C' > 0\), \[ \lim_{n \rightarrow \infty} (\omega_n)^{-1} \sup_{h \in U_n} | f_{\omega} (x+h) - f_{\omega}(x) | \geq C' > 0, \] where \(U_n = (0, \dots, 0, x_{n+1}, \dots).\)