On the \(\phi\)-modulus of continuity (Q1092265)

\textit{P. Ul'yanov} [Usp. Mat. Nauk 27, No.2(164), 3-52 (1972; Zbl 0274.46027)] extended the definition of the \(L^ p\)-modulus of continuity to the space \(\phi\) (L) of measurable functions f on [0,1] satisfying \(\int^{1}_{0}\phi (f(t))dt<\infty\) where \(\phi\) is a nonnegative even function defined on (-\(\infty,\infty)\), nondecreasing on [0,\(\infty)\) and such that \(\lim_{t\to \infty}\phi (t)=\infty\). Ul'yanov called \(\phi\)- modulus the quantity \(\omega_{\phi}(f,\delta)=\sup_{0<h\leq \delta}\int^{1-h}_{0}\phi (f(x+h)-f(x))dx\) and proved that in order for it to be finite for every \(f\in \phi (L)\) it is necessary and sufficient for \(\phi\) to satisfy the \(\Delta_ 2\) condition, namely \(\phi (2t)=O(\phi (t))\), \(t\to \infty\). The author extends Ul'yanov's definition to functions of several variables and investigates the properties of this \(\phi\)-modulus. The reviewer could not follow some of the proofs and has some doubts about the validity of some of the results.