Gagliardo–Nirenberg Inequalities with a BMO Term

We give a simple direct proof of the interpolation inequality‖ ∇ f ‖L 2 p2 ⩽ C‖ f ‖BMO‖ f ‖W 2 , p, where 1 < p < ∞. For p = 2 this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo–Nirenberg inequalities where the norm of the function is taken in L ∞ and other norms are in L q for appropriate q > 1. 2000 Mathematics Subject Classification 46E35 (primary), 46B70 (secondary).