Sharp $L^p$-weighted Sobolev inequalities

Pérez, Carlos

Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 809-824.

Nous obtenons des estimations de la forme

$${\int }_{{\mathbf{R}}^n}|f\left(x\right){|}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}|q\left(D\right)\left(f\right)\left(x\right){|}^{p}N\left(v\right)\left(x\right)dx$$

dans des espaces de Sobolev avec poids. Nous montrons que le résultat est optimal. Ici $q\left(D\right)$ est un opérateur différentiel, $N$ étant le composé de plusieurs opérateurs de type maximal liés avec $q\left(D\right)$ et $p$.

We prove sharp weighted inequalities of the form

$${\int }_{{\mathbf{R}}^n}|f\left(x\right){|}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}|q\left(D\right)\left(f\right)\left(x\right){|}^{p}N\left(v\right)\left(x\right)dx$$

where $q\left(D\right)$ is a differential operator and $N$ is a combination of maximal type operator related to $q\left(D\right)$ and to $p$.

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