Some properties of the moduli of continuity of periodic functions in metric spaces

Abstract

Let $L_0(T)$) be the set of real-valued periodic measurable functions, let $\Psi : R^{+} \rightarrow R^{+}$ be the modulus of continuity, and let $$L_{\Psi} \equiv L_{\Psi} (T) = \left\{ f \in L_0(T) : \| f\| _{\Psi} := \frac1{2\pi} \int_T \Psi (| f(x)| )dx < \infty \right\}.$$ We study the properties of multiple modules of continuity for the functions from $L_{\Psi}$.