The trace inequality and eigenvalue estimates for Schrödinger operators

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.