This article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and Lp-decay of solutions of several interesting partial differential equations. In particular, there are applications to quasigeostrophic flows, in two space variables with critical viscosity, that model the Eckman pumping [see Baroud, Ch. N., Plapp, B. B., She, Z. S. & Swinney, H. L. (2002) Phys. Rev. Lett. 88, 114501 and Constantin, P. (2002) Phys. Rev. Lett. 89, 184501].