Geodesic
Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals
Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 77-89
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This article gives a necessary and sufficient condition for a $p$-integrable function to have partial derivatives of specified orders which are $p$th power integrable over $R^n$. This condition is expressed using integrals of differences which in general converge conditionally in the $L_p$-norm. We also prove a Fubini theorem for these function spaces. Bibliography: 7 titles. 
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     title = {Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals},
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P. I. Lizorkin. Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals. Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 77-89. http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/

[1] P. I. Lizorkin, “Obobschennoe liuvillevskoe differentsirovanie i funktsionalnye prostranstva $L_p^r(E_n)$”, Matem. sb., 60(102) (1963), 325–353 | MR | Zbl

[2] P. I. Lizorkin, “Neizotropnye besselevy potentsialy. Teoremy vlozheniya dlya prostranstv tipa Soboleva $L_p^{(r_1,\dots,r_n)}$”, DAN SSSR, 170:3 (1966), 508–511 | MR | Zbl

[3] E. M. Stein, “The characterization of functions arising ias potentials. I”, Bull. Amer. Math. Soc., 67 (1961), 102–104 | DOI | MR

[4] O. V. Besov, P. I. Lizorkin, “Singulyarnye integralnye operatory i posledovatelnosti svertok v prostranstvakh $L_p$”, Matem. sb., 73(115) (1967), 65–88 | MR | Zbl

[5] P. I. Lizorkin, “$(L_p,L_q)$-multiplikatory integralov Fure”, DAN SSSR, 152:4 (1963), 808–811 | MR | Zbl

[6] R. S. Strichartz, “Multipliers on fractional Sobolev spaces”, J. Math. and Mech., 16:9 (1967), 1031–1060 | MR | Zbl

[7] P. I. Lizorkin, “Obobschennoe liuvillevskoe differentsirovanie i metod multiplikatorov v teorii vlozhenii klassov funktsii”, Trudy matem. in-ta im. V. A. Steklova AN SSSR, 105 (1969), 89–167 | MR | Zbl