Imbedding Theorems in Metric Spaces $L_{ψ}$

Abstract

Let $L_0 (T^m)$ be the set of periodic measurable real-valued functions of $m$ variables, let $ψ: R_+^1  → R_+^1$ be the continuity modulus, and let $${L}_{\psi}\left({T}^m\right)=\left\{f\in {L}_0\left({T}^m\right):{\left\Vert f\right\Vert}_{\psi }:={\displaystyle {\int}_{T^m}\psi \left(\left|f(x)\right|\right)dx<\infty}\right\}.$$ The relationship between the modulus of continuity of functions from $L_{ψ} (T^m)$ and the corresponding $K$-functionals is analyzed and sufficient conditions for the imbedding of the classes of functions $H_{ψ}^{ω} (T^m)$ into $L_q (T^m),\; q ∈ (0; 1]$, are obtained.