Order equalities in different metrics for moduli of smoothness of various orders

IIn this paper, we obtain order equalities for the $k$th order $L_{q}(T)$-moduli of smoothness $\omega_{k}(f;\delta)_{q}$ in terms of expressions that contain the $l$th order $L_{p}(T)$-moduli of smoothness $\omega_{ l }(f;\delta)_{p}$ on the class of periodic functions $f\in L_{p}(T)$ with monotonically decreasing Fourier coefficients, where $1<p<q<\infty,$ $k,l \in \mathbb{N},$ and $T=(-\pi,\pi].$