Geodesic
Functions of bounded $q$-integral $p$-variation and imbedding theorems
Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 279-288 
A. P. Terekhin. Functions of bounded $q$-integral $p$-variation and imbedding theorems. Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 279-288. http://geodesic.mathdoc.fr/item/SM_1972_17_2_a8/
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     title = {Functions of bounded $q$-integral $p$-variation and imbedding theorems},
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a function of one real variable there is defined a notion of $q$-integral $p$-variation generalizing Wiener $p$-variation. In terms of this notion there is given a necessary and sufficient condition that a function in $L_q$ have a higher derivative in $L_p$ ($p\leqslant q$), and also that the derivative have a definite smoothness in $L_p$. In addition, embedding theorems with inversion are proved in the periodic case for generalized Lipschitz classes in $L_p$. Bibliography: 9 titles.

[1] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, Moskva, 1957 | MR

[2] A. P. Terekhin, “Priblizhenie funktsii ogranichennoi $p$-variatsii”, Izv. VUZov, Matematika, 1965, no. 2, 171–187 | Zbl

[3] N. Wiener, “The quadratic variation of a function and its Fourier coefficients”, Massachusett's J. Math., 3 (1924), 72–94 | Zbl

[4] G. Hardy, J. Littlewood, “Some properties of fractional integrals. I”, Math. Z., 27 (1928), 565–606 | DOI | MR | Zbl

[5] A. F. Timan, Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, Moskva, 1960

[6] A. P. Terekhin, “Integralnye svoistva gladkosti periodicheskikh funktsii ogranichennoi $p$-variatsii”, Matem. zametki, 2:3 (1967), 289–300 | Zbl

[7] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Novosibirsk, 1962

[8] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, Moskva, 1969 | MR

[9] L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration”, Acta Math., 67 (1936), 251–282 | DOI | MR | Zbl