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 $\alpha>s$. 
@article{MZM_2002_71_6_a4,
     author = {N. L. Kudryavtsev},
     title = {Fractional {Differences} and {Lizorkin--Triebel} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {845--854},
     publisher = {mathdoc},
     volume = {71},
     number = {6},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a4/}
}
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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/