Geodesic
Necessary and sufficient conditions for the imbedding of some classes of functions
Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 388-400

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Necessary and sufficient conditions are found for membership of a function $f(x)\in L^p(0,1)$ in the class $L^q\Phi (L)$, $1\le p\le q\infty$. A connection is established between the modulus of continuity $\omega(\delta,f^*)$ and, on the one hand, the Fourier coefficients $a_n(f^*)$ and $b_n(f^*)$, and on the other hand the modulus of continuity $\omega(\delta,f)$. 
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     author = {\`E. A. Storozhenko},
     title = {Necessary and sufficient conditions for the imbedding of some classes of functions},
     journal = {Izvestiya. Mathematics },
     pages = {388--400},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a8/}
}
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È. A. Storozhenko. Necessary and sufficient conditions for the imbedding of some classes of functions. Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 388-400. http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a8/