We give a simple direct proof of the interpolation inequality $ \|\nabla f\|^2_{L^{2p}} \le C \|f\|_{\rm BMO} \|f\|_{W^{2,p}}$, where $ 1<p<\infty$. 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 $L^\infty$ and other norms are in $L^q$ for appropriate $q>1$.
