The trace inequality and eigenvalue estimates for Schrödinger operators

Kerman, R.; Sawyer, Eric T.

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

Soit $\Phi$ une fonction radiale, non négative, localement intégrable sur ${\mathbf{R}}^n$, qui ne s’accroît pas en $\left|x\right|$. Posons $\left(Tf\right)\left(x\right)={\int }_{{\mathbf{R}}^n}\Phi (x-y)f\left(y\right)dy$ où $f\ge 0$ et $x\in {\mathbf{R}}^n$. Étant donné $1<p<\infty$ et $v\ge 0$, nous démontrons qu’il existe $C>0$ de sorte que ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ pour tout $f\ge 0$, si et seulement si, $C^{\prime }>0$ existe avec ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\prime }}dx\le C^{\prime }{\int }_{Q}v\left(x\right)dx<\infty$ pour tout cube dyadique $Q$, où $p^{\prime }=p/(p-1)$.

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 $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1<p<\infty$ and $v\ge 0$, we show there exists $C>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\prime }>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\prime }}dx\le C^{\prime }{\int }_{Q}v\left(x\right)dx<\infty$ for all dyadic cubes Q, where $p^{\prime }=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

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