The imbedding of certain classes of functions

Necessary and sufficient conditions on the modulus of continuity $\omega(\delta)$ are found such that the inclusion $\psi(x)\in H_p^\omega$, $p\in[1,\infty)$ should imply $\psi(x)\sim\psi^*(x)\in H_p^\omega(L^\infty=C)$; sufficient conditions on $\omega(\delta)$ are also found such that $\psi(x)\in H_p^\omega$, $p\in[1,\infty)$, should imply $\psi(x)\in H_q^{\omega^*}$, $p<q<\infty$.