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We construct a maximal function associated with a family of skewed cylinders.These cylinders, which are defined as tubular neighborhoods of trajectories ofa mollified flow, appear in the study of fluid equations such as theNavier-Stokes equations and the Euler equations. We define a maximal functionsubordinate to these cylinders, and show it is of weak type $(1, 1)$ and strongtype $(p, p)$ by a covering lemma. As an application, we give an alternativeproof for the higher derivatives estimate of smooth solutions to thethree-dimensional Navier-Stokes equations.

Construction of Maximal Functions associated with Skewed Cylinders Generated by Incompressible Flows and Applications