Fractional Differences and Lizorkin–Triebel Spaces

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<p<\infty $, $0<q\le\infty$, given sufficient smoothness $s > 0$, we can introduce equivalent quasinorms expressible in terms of fractional differences of order $\alpha>s$.