The trace inequality and eigenvalue estimates for Schrödinger operators
Annales de l'Institut Fourier, Tome 36 (1986) no. 4, p. 207-228
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)dyf0 et xR n . Étant donné 1<p< et v0, nous démontrons qu’il existe C>0 de sorte que R n (Tf)(x) p v(x)dxC R n f(x) p dx pour tout f0, si et seulement si, C >0 existe avec Q T(x Q v)(x) p dxC 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.
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 f0 and xR n . Given 1<p< and v0, we show there exists C>0 so that R n (Tf)(x) p v(x)dxC R n f(x) p dx for all f0, if and only if C >0 exists with Q T(x Q v)(x) p dxC 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.
@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},
     address = {28 - Luisant},
     volume = {36},
     number = {4},
     year = {1986},
     pages = {207-228},
     doi = {10.5802/aif.1074},
     zbl = {0591.47037},
     mrnumber = {88b:35150},
     language = {en},
     url = {http://www.numdam.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, Tome 36 (1986) no. 4, pp. 207-228. doi : 10.5802/aif.1074. https://www.numdam.org/item/AIF_1986__36_4_207_0/

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