The direct theorem of the theory of approximation of periodic functions with monotone Fourier coefficients in different metrics

We study the problem of order optimality of an upper bound for the best approximation in $L_{q}(\mathbb{T})$ in terms of the $l$th-order modulus of smoothness (the modulus of continuity for $l=1$) in
$$L_{p}(\mathbb{T})\colon E_{n-1}(f)_{q}\le C(l,p,q)\big(\textstyle\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q},\ \ n\in\mathbb{b}N,$$
on the class $M_{p}(\mathbb{T})$ of all functions $f\in L_{p}(\mathbb{T})$ whose Fourier coefficients satisfy the conditions
$$a_{0}(f)=0,\ a_{n}(f)\downarrow 0,\ \text {and}\ b_{n} (f)\downarrow 0\ (n\uparrow \infty),\ \text{where}\ l\in\mathbb{N},\ 1<p<q<\infty,\ l>\sigma=1/p-1/q,\ \text{and}\ \mathbb{T}=(-\pi,\pi].$$
For $l=1$ and $p\ge 1$, the bound was first established by P. L. Ul'yanov in the proof of the inequality of different metrics for moduli of continuity; for $l>1$ and $p\ge 1$, the proof of the bound remains valid in view of the $L_{p}$-analog of the Jackson–Stechkin inequality. Below we formulate the main results of the paper. A function $f\in M_{p}(\mathbb{T})$ belongs to $L_{q}(\mathbb{T})$, where $1<p<q<\infty$, if and only if $\sum_{n=1}^{\infty}n^{q\sigma-1}\omega_{l}^{q}(f;\pi/n)_{p}<\infty$, and the following order inequalities hold: (a) $E_{n-1}(f)_{q}+n^{\sigma}\omega_{l}(f;\pi/n)_{p}\asymp\big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q} (f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb{N}$; (b) $n^{-(l-\sigma)}\big(\sum_{\nu=1}^{n}\nu^{p(l-\sigma)-1}E_{\nu-1}^{p}(f)_{q}\big)^{1/p}\asymp \big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb{N}$. \noindent In the lower bound in inequality (a), the second term $n^{\sigma}\omega_{l}(f;\pi/n)_{p}$ generally cannot be omitted. However, if the sequence $\{\omega_{l}(f;\pi/n)_p\}_{n=1}^{\infty}$ or the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ satisfies Bari's $(B_{l}^{(p)})$-condition, which is equivalent to Stechkin's $(S_{l})$-condition, then
$$E_{n-1}(f)_{q}\asymp\bigg(\sum_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\bigg)^{1/q},\ \ n\in\mathbb{N}.$$
The upper bound in inequality (b), which holds for any function $f\in L_{p}(\mathbb{T})$ if the series converges, is a strengthened version of the direct theorem. The order inequality $(b)$ shows that the strengthened version is order-exact on the whole class $M_{p}(\mathbb{T})$.