On relations between moduli of continuity in different metrics (Q753050)

Let f be a periodic function defined on \({\mathbb{R}}^ N\) with period 1 with respect to each variable. For \(f\in L^ p(I^ N)\) \((I=[0,1]\), \(1\leq p<\infty)\) and \(\delta\equiv [0,1]\), the modulus of continuity \(\omega_ p(f;\delta)\) is given by \(\omega_ p(f;\delta)=\sup \{\| f(x+h)- f(x)\|;\quad | h_ i| \leq \delta,\quad i=1,...,N\}.\) The author proves that \[ (1)\quad (\int^{1}_{\delta}(t^{\theta - 1}\omega_ q(f;t))^ p(dt/t))^{1/p}\leq c\delta^{\theta - 1}(\int^{\delta}_{0}(t^{-\theta}\omega_ p(f;t))^ q(dt/t))^{1/q} \] provided that \(f\in L^ p(I^ N)\) \((1<p<\infty\), \(N\geq 1)\), \(p<q<\infty\), \(\theta:=N(1/p-1/q)<1\) and \(\delta\in (0,1]\). Moreover, it is shown that the inequality (1) holds with \(0<p\leq 1\), \(f\in H^ p(D_ 0)\) \((D_ 0=\{z\in {\mathbb{C}}\); \(| z| <1\})\), \[ \omega_ p(f;\delta)=\sup_{0\leq h\leq \delta}\{\int^{2\pi}_{0}| f(e^{i\phi})-f(e^{i(\phi +h)})|^ p d\phi \}^{1/p}, \] p\(<q<p/(1-p)\) and \(\theta =1/p-1/q\).