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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 \(\dot{C}^{-\alpha}(\mathbb{R}^n), \alpha >0, \) be the homogeneous Besov space \(B= \dot{B}^{-\alpha, \infty}_{\infty} (\mathbb {R}^n),\) 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 \[ \| f\|_6 \leq C \| \nabla f \|_2^{\frac 13} \| f\| _*^{\frac 23} \] where \( \| f\| _*\) is the norm in \(\dot{C}^{-\frac 12} (\mathbb{R}^3);\) it is pointed out the invariance of (1) under the affine and the Weyl-Heisenberg groups. In the second part, when \(1 < p < q < \infty\), \(s= \alpha(\frac qp -1),\) it is obtained, by an elegant proof which does not use interpolation spaces, the existence of a constant \(C=C(n,\alpha,p,q)\) such that \[ | | f| | _q \leq C | | \Lambda ^s f| | _p^{\frac pq} \| f\| _B^{1-\frac pq} \] for all functions \(f \in \dot{C} ^{-\alpha}(\mathbb{R}^n)\) such that \( \Lambda ^s f \in L^p (\mathbb{R}^n).\) 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)].

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46M35 Abstract interpolation of topological vector spaces
26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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