A rate $L_p$-version for the Riesz criterion of the absolute convergence of trigonometric Fourier series

Rate versions of the Riesz criterion ($A=L_2(\mathbb T)*L_2(\mathbb T)$) are established in terms of best approximations ($A[\lambda]=E_2t[\lambda^{1/2}]*E_2[\lambda^{1/2}]$) and moduli of smoothness ($A[\omega]=H_2^l[\omega^{1/2}]*H_2^l[\omega^{1/2}]$) of the functions that compose the convolution, and conditions are found for $\lambda$ (necessary and sufficient in the case $1\le p<2$ and sufficient in the case $2<p<\infty$) under which the equality$A[\lambda]=E_p[\lambda^{1/2}]*E_p[\lambda^{1/2}]$ is valid, where $\lambda\in M_0$, $\omega\in\Omega_l$, $l\in\mathbb N$.