On imbedding in classes $\varphi(L)$

In this paper necessary and sufficient conditions are found for imbeddings of the form $H_p^{\omega_1,\dots,\omega_k}\subset L^p\Phi(L)$. It is proved that in the one-dimensional case the corresponding condition on the modulus of continuity of a monotone function $f\in L^p(0,1)$ is not only sufficient but also necessary for $f\in L^p\Phi(L)$. In connection with this the existence is established of a monotone function in $L^p$ with preassigned order of modulus of continuity.
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