In this lecture, given in 1996, Yves Meyer presented improvements of the usual Sobolev inequality. Let be the homogeneous Besov space and In the first part, it is shown, when , by a real interpolation method, the existence of a constant C such that
where is the norm in it is pointed out the invariance of (1) under the affine and the Weyl-Heisenberg groups. In the second part, when , it is obtained, by an elegant proof which does not use interpolation spaces, the existence of a constant such that
for all functions such that The core of the proofs is the theory of wavelets, cf. [Y. Meyer, “Ondelettes et opérateurs. I” (Actualités Mathématiques, Hermann, Paris) (1990; Zbl 0694.41037)].