Geodesic
Fractional Differences and Lizorkin--Triebel Spaces
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 845-854.

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In this paper, we obtain inequalities for maximal functions of fractional differences and fractional derivatives and use them to prove that in the Lizorkin–Triebel spaces $F^s_{pq}=F^s_{pq}(\mathbb R^n)$, $0$, $0$, given sufficient smoothness $s > 0$, we can introduce equivalent quasinorms expressible in terms of fractional differences of order $\alpha>s$. 
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N. L. Kudryavtsev. Fractional Differences and Lizorkin--Triebel Spaces. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 845-854. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a4/

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