A generalization of the Littlewood-Paley inequality for the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$

We prove a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$, where $\alpha \in (0,2)$.