Geodesic
On imbedding in classes~$\varphi(L)$
Izvestiya. Mathematics , Tome 9 (1975) no. 2, pp. 395-413

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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. Bibliography: 10 items. 
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     author = {V. I. Kolyada},
     title = {On imbedding in classes~$\varphi(L)$},
     journal = {Izvestiya. Mathematics },
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a9/}
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V. I. Kolyada. On imbedding in classes~$\varphi(L)$. Izvestiya. Mathematics , Tome 9 (1975) no. 2, pp. 395-413. http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a9/