GAGLIARDO–NIRENBERG INEQUALITIES WITH A BMO TERM

Abstract

We give a simple direct proof of the interpolation inequality ‖ ∇ f ‖ L 2 p 2 ⩽ 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 Lq for appropriate q > 1. 2000 Mathematics Subject Classification 46E35 (primary), 46B70 (secondary).