no. 4
The trace inequality and eigenvalue estimates for Schrödinger operators
Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 207-228.

Suppose Φ is a nonnegative, locally integrable, radial function on R n , which is nonincreasing in |x|. Set (Tf)(x)=∫ R n Φ(x-y)f(y)dy when f≥0 and x∈R n . Given 1<p<∞ and v≥0, we show there exists C>0 so that ∫ R n (Tf)(x) p v(x)dx≤C∫ R n f(x) p dx for all f≥0, if and only if C ′ >0 exists with ∫ Q T(x Q v)(x) p ′ dx≤C ′ ∫ Q v(x)dx<∞ 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.

Soit Φ une fonction radiale, non négative, localement intégrable sur R n , qui ne s’accroît pas en |x|. Posons (Tf)(x)=∫ R n Φ(x-y)f(y)dy où f≥0 et x∈R n . Étant donné 1<p<∞ et v≥0, nous démontrons qu’il existe C>0 de sorte que ∫ R n (Tf)(x) p v(x)dx≤C∫ R n f(x) p dx pour tout f≥0, si et seulement si, C ′ >0 existe avec ∫ Q T(x Q v)(x) p ′ dx≤C ′ ∫ Q v(x)dx<∞ 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.

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     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 = {Institut Fourier},
     address = {Grenoble},
     volume = {36},
     number = {4},
     year = {1986},
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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/articles/10.5802/aif.1074/

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