Gagliardo-Nirenberg inequalities with a BMO term

We give a simple direct proof of the interpolation inequality parallel to del f parallel to(2)(L)(2P) <= C parallel to f parallel to(BMO)parallel to f parallel to(W2,p), where 1 < p < infinity. For p = 2 this inequality was obtained by Meyer and Riviere 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-infinity and other norms are in L-q for appropriate q > 1.