# ANNALES DE L'INSTITUT FOURIER

The trace inequality and eigenvalue estimates for Schrödinger operators
Annales de l'Institut Fourier, Volume 36 (1986) no. 4, p. 207-228
Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^{n}$, which is nonincreasing in $|x|$. Set $\left(Tf\right)\left(x\right)={\int }_{{R}^{n}}\Phi \left(x-y\right)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^{n}$. Given $1 and $v\ge 0$, we show there exists $C>0$ so that ${\int }_{{\mathbf{R}}^{n}}\left(Tf\right)\left(x{\right)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^{n}}f\left(x{\right)}^{p}dx$ for all $f\ge 0$, if and only if ${C}^{\prime }>0$ exists with ${\int }_{Q}T\left({x}_{Q}v\right)\left(x{\right)}^{{p}^{\prime }}dx\le {C}^{\prime }{\int }_{Q}v\left(x\right)dx<\infty$ for all dyadic cubes Q, where ${p}^{\prime }=p/\left(p-1\right)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.
Soit $\Phi$ une fonction radiale, non négative, localement intégrable sur ${\mathbf{R}}^{n}$, qui ne s’accroît pas en $|x|$. Posons $\left(Tf\right)\left(x\right)={\int }_{{\mathbf{R}}^{n}}\Phi \left(x-y\right)f\left(y\right)dy$$f\ge 0$ et $x\in {\mathbf{R}}^{n}$. Étant donné $1 et $v\ge 0$, nous démontrons qu’il existe $C>0$ de sorte que ${\int }_{{\mathbf{R}}^{n}}\left(Tf\right)\left(x{\right)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^{n}}f\left(x{\right)}^{p}dx$ pour tout $f\ge 0$, si et seulement si, ${C}^{\prime }>0$ existe avec ${\int }_{Q}T\left({x}_{Q}v\right)\left(x{\right)}^{{p}^{\prime }}dx\le {C}^{\prime }{\int }_{Q}v\left(x\right)dx<\infty$ pour tout cube dyadique $Q$, où ${p}^{\prime }=p/\left(p-1\right)$.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.
@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},
publisher = {Imprimerie Durand},
volume = {36},
number = {4},
year = {1986},
pages = {207-228},
doi = {10.5802/aif.1074},
mrnumber = {88b:35150},
zbl = {0591.47037},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_1986__36_4_207_0}
}

Kerman, R.; Sawyer, Eric T. The trace inequality and eigenvalue estimates for Schrödinger operators. Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 207-228. doi : 10.5802/aif.1074. https://aif.centre-mersenne.org/item/AIF_1986__36_4_207_0/

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