Inequality between moduli of smoothness of different orders in different metrics (Q1178094)

Let \(L_ p(1\leq p<\infty)\) be the space of all \(f:\mathbb{R}\mapsto\mathbb{R}\) measurable, \(2\pi\)-periodic functions for which \(\| f\|_ p<\infty\), \(W^ r_ p(r\in\mathbb{N})\) be the class of functions \(f\in L_ p\) for which \(f^{(r-1)}\) is absolutely continuous and \(f^{(r)}\in L_ p\) \((f^{(0)}\equiv f\), \(W^ 0_ p\equiv L_ p)\), \(\omega_ \ell(f;\delta)_ p\) the modulus of smoothness of \(1\in\mathbb{N}\) order associate to the function \(f\in L_ p\), \(\Omega_ \ell\) the class of the functions \(\omega\) defined on \((0,\pi]\) and which satisfy the following conditions: \(\omega(\delta)>0\), \(\omega(\delta)\downarrow 0\) for \(\delta\downarrow 0\), \(\delta^{-\ell}\omega(\delta)\downarrow\) for \(\delta\uparrow\). In this paper the author proves two theorems considering estimates for \(\omega_ k(f^{(r)};{\pi\over n})_ 2\) and \(\sup\{\omega_ k(f^{(r)};{\pi\over n})_ 2:f\in H^ \ell_ p[\omega]\}\) where \(H^ \ell_ p[\omega]:=\{f\in L_ p;\omega_ \ell(f,\delta)_ p\leq\omega(\delta),\delta\in(0,\pi]\}\), \(1\leq p<q<\infty\); \(\ell,k\in\mathbb{N}\).