On the order of decrease of uniform moduli of smoothness for the classes of periodic functions $H_{p}^{l}[\omega],\ l\in \mathbb N,\ 1\le p < \infty$

S. B. Stechkin posed the following problem: for given $1\le p<q\le\infty$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, and $\omega \in\Omega_{l}(0,\pi]$, find the exact order of decrease of the $L_{q}(\mathbb T)$-modulus of smoothness of the $k$th order $\omega_{k}(f^{(r)};\delta)_{q}$ on the classes of $2\pi$-periodic functions $H_{p}^{l}[\omega]=\{f\in L_{p}(\mathbb T):$ $\omega_{l}(f;\delta)_{p}\le\omega(\delta),\,\delta\in(0,\pi]\}$, where $\mathbb T=(-\pi,\pi]$, $L_{\infty}(\mathbb T)\equiv C(\mathbb T)$, and $\Omega_{l}(0,\pi]$ is the class of functions $\omega=\omega(\delta)$ defined on $(0,\pi]$ and satisfying the conditions $0<\omega(\delta)\downarrow 0\ (\delta\downarrow 0)$ and $\delta^{-l}\omega(\delta)\downarrow (\delta\uparrow)$. Earlier the author solved this problem in the case $1\le p<q<\infty$. In the present paper, we give a solution in the case $1\le p<q=\infty$; more exactly, we prove the following theorems.

Theorem 1. Suppose that $1\le p<\infty$, $f\in L_{p}(\mathbb T)$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, $l>\sigma=r+1/p$, $\rho=l-(k+\sigma)$, and $\sum_{n=1}^{\infty}n^{\sigma-1}\omega_{l}(f;\pi/n)_p<\infty$. Then $f$ is equivalent to some function $\psi\in C^{r}(\mathbb T)$and the following estimate holds:$\omega_{k}(\psi^{(r)};\pi/n)_{\infty} \le C_{1}(l,k,r,p)\Big\{\sum_{\nu=n+1}^{\infty}\nu^{\sigma-1}\omega_{l}(f;\pi/\nu)_{p}+ \chi(\rho)n^{-k}\sum_{\nu=1}^{n}\nu^{k+\sigma-1}\omega_{l}(f;\pi/\nu)_{p}\Big\}$, $n\in\mathbb N$, where $\chi(t)=0$ for $t\le 0$, $\chi(t)=1$ for $t>0$, and $C^{r}(\mathbb T)$ is the class of functions $\psi \in C(\mathbb T)$ that have the usual $r$th-order derivative $\psi^{(r)}\in C(\mathbb T)$ $($we assume that $\psi^{(0)}=\psi$ and $C^{(0)}(\mathbb T)=C(\mathbb T))$.

Note that this estimate covers all possible cases of relations between $l$ and $k+r$.

Theorem 2. Suppose that $1\le p<\infty$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, $l>\sigma=r+1/p$, $\rho=l-(k+\sigma)$, $\omega \in\Omega_{l}(0,\pi]$, and $\sum_{n=1}^{\infty}n^{\sigma-1}\omega(\pi/n)<\infty$. Then $\sup\{\omega_{k}(\psi^{(r)};\pi/n)_{\infty}:$ $f\in H_{p}^{l}[\omega]\}\asymp\sum_{\nu=n+1}^{\infty}\nu^{\sigma-1}\omega(\pi/\nu)+\chi(\rho)n^{-k} \times \sum_{\nu=1}^{n}\nu^{k+\sigma-1}\omega(\pi/\nu)$, $n\in\mathbb N$, where $\psi$ denotes the corresponding function from $C^{r}(\mathbb T)$ equivalent to $f\in H_{p}^{l}[\omega]$.

In Theorems $1$ and $2$, the case $l=k+\sigma=k+r+1/p$ $(\Rightarrow \chi(\rho)=0)$ is of the most interest. This case is possible only for $p=1$, since $r\in\mathbb Z_{+}$ and $l,k\in\mathbb N$. In this case, the proof of the estimate in Theorem $1$ employs the inequality $n^{-l}\|T_{n,1}^{(l)}(f;\cdot)\|_{\infty} \le C_{2}(l)n\omega_{l+1}(f;\pi/n)_{1}$, where $T_{n,1}(f;\cdot)$ is a best approximation polynomial for the function $f\in L_{1}(\mathbb T)$. The latter inequality is derived from the strengthened version of the inequality of different metrics for derivatives of arbitrary trigonometric polynomials $\|t_{n}^{(l)}(\cdot)\|_{\infty}\le 2^{-1}\pi\|t_{n}^{(l+1)}(\cdot)\|_{1}$, $n\in\mathbb N$.