A remark on vorticity maximal function ✩

In this note the following inequality is proved. For any nonnegative measure μ ∈ H−1(R2), x ∈ R2 and 0 < r <1, there holds μ B(x, r) C ln 1 r −1 2 μ H−1 (1) where C is a positive constant. Using (1) an estimate for the vorticity maximal function similar to the estimate in Majda [A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J. 42 (1993) 921–939] is established without assuming the initial vorticity having compact support. From this a more simple proof of the Delort’s celebrated theorem [J.M. Delort, Existence de mappes de fourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991) 553–586] is presented. © 2007 Elsevier Inc. All rights reserved