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Sharp Sobolev inequalities. (Inégalités de Sobolev précisées.) (French) Zbl 1066.46501

In this lecture, given in 1996, Yves Meyer presented improvements of the usual Sobolev inequality. Let ${\stackrel{˙}{C}}^{-\alpha }\left({ℝ}^{n}\right),\alpha >0,$ be the homogeneous Besov space $B={\stackrel{˙}{B}}_{\infty }^{-\alpha ,\infty }\left({ℝ}^{n}\right),$ and ${\Lambda }=\sqrt{-{\Delta }}·$ In the first part, it is shown, when $n=3$, by a real interpolation method, the existence of a constant C such that

${\parallel f\parallel }_{6}\le {C\parallel \nabla f\parallel }_{2}^{\frac{1}{3}}{\parallel f\parallel }_{*}^{\frac{2}{3}}$

where ${\parallel f\parallel }_{*}$ is the norm in ${\stackrel{˙}{C}}^{-\frac{1}{2}}\left({ℝ}^{3}\right);$ it is pointed out the invariance of (1) under the affine and the Weyl-Heisenberg groups. In the second part, when $1, $s=\alpha \left(\frac{q}{p}-1\right),$ it is obtained, by an elegant proof which does not use interpolation spaces, the existence of a constant $C=C\left(n,\alpha ,p,q\right)$ such that

${||f||}_{q}\le C||{{\Lambda }}^{s}{f||}_{p}^{\frac{p}{q}}{\parallel f\parallel }_{B}^{1-\frac{p}{q}}$

for all functions $f\in {\stackrel{˙}{C}}^{-\alpha }\left({ℝ}^{n}\right)$ such that ${{\Lambda }}^{s}f\in {L}^{p}\left({ℝ}^{n}\right)·$ 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)].

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46M35 Abstract interpolation of topological linear spaces 26D15 Inequalities for sums, series and integrals of real functions 26D10 Inequalities involving derivatives, differential and integral operators 42C40 Wavelets and other special systems
##### Keywords:
Sobolev inequalities; interpolation spaces; wavelets