The trace inequality and eigenvalue estimates for Schrödinger operators

Kerman, R.; Sawyer, Eric T.

doi:10.5802/aif.1074

Kerman, R. ; Sawyer, Eric T.

Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 207-228.

Résumé
Abstract

Soit $Φ$ une fonction radiale, non négative, localement intégrable sur $R^{n}$ , qui ne s’accroît pas en $| x |$ . Posons $(T f) (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ où $f \geq 0$ et $x \in R^{n}$ . Étant donné $1 < p < \infty$ et $v \geq 0$ , nous démontrons qu’il existe $C > 0$ de sorte que $\int_{R^{n}} (T f) (x)^{p} v (x) d x \leq C \int_{R^{n}} f (x)^{p} d x$ pour tout $f \geq 0$ , si et seulement si, $C^{'} > 0$ existe avec $\int_{Q} T (x_{Q} v) (x)^{p^{'}} d x \leq C^{'} \int_{Q} v (x) d x < \infty$ pour tout cube dyadique $Q$ , où $p^{'} = p / (p - 1)$ .

On se sert de ce résultat pour raffiner des approximations récentes de la part de C.L. Fefferman et D.H. Phong de la distribution de valeurs propres d’opérateurs de Schrödinger.

Suppose $Φ$ is a nonnegative, locally integrable, radial function on $R^{n}$ , which is nonincreasing in $| x |$ . Set $(T f) (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ when $f \geq 0$ and $x \in R^{n}$ . Given $1 < p < \infty$ and $v \geq 0$ , we show there exists $C > 0$ so that $\int_{R^{n}} (T f) (x)^{p} v (x) d x \leq C \int_{R^{n}} f (x)^{p} d x$ for all $f \geq 0$ , if and only if $C^{'} > 0$ exists with $\int_{Q} T (x_{Q} v) (x)^{p^{'}} d x \leq C^{'} \int_{Q} v (x) d x < \infty$ for all dyadic cubes Q, where $p^{'} = p / (p - 1)$ . This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

MR 88b:35150 | Zbl 0591.47037

DOI : https://doi.org/10.5802/aif.1074

@article{AIF_1986__36_4_207_0,
     author = {Kerman, R. and Sawyer, Eric T.},
     title = {The trace inequality and eigenvalue estimates for {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {207--228},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {36},
     number = {4},
     year = {1986},
     doi = {10.5802/aif.1074},
     mrnumber = {88b:35150},
     zbl = {0591.47037},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1074/}
}

Kerman, R.; Sawyer, Eric T. The trace inequality and eigenvalue estimates for Schrödinger operators. Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 207-228. doi : 10.5802/aif.1074. https://aif.centre-mersenne.org/articles/10.5802/aif.1074/

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