Imbedding Theorems in Metric Spaces L _ψ

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 ^ω _ψ (T ^m) into L _q (T ^m), q ∈ (0; 1], are obtained.