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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     author = {P. I. Lizorkin},
     title = {Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals},
     journal = {Sbornik. Mathematics},
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     volume = {10},
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     year = {1970},
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     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_1_a5/}
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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/