Gagliardo–nirenberg Inequalities with a Bmo Term

We give a simple direct proof of the interpolation inequality ‖∇f‖2 L2p 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.