The trace inequality and eigenvalue estimates for Schrödinger operators (Q1074844)

Suppose \(\Phi\) is a nonnegative, locally integrable, radial function on \(R^ n\), which is nonincreasing in \(| x|\). Set \[ (Tf)(x)=\int_{R^ n}\Phi (x-y)f(y)dy \] when \(f\geq 0\) and \(x\in R^ n\). Given \(1<p<\infty\) and \(v\geq 0\), we show there exists \(C>0\) so that \[ \int_{R^ n}(Tf)(x)^ pv(x)dx\leq C\int_{R^ n}f(x)^ pdx \] for all \(f\geq 0\). If and only if \(C'>0\) exists with \[ \int_{Q}T(x_ Qv)(x)^{p'}dx\leq C'\int_{Q}v(x)dx<\infty \] 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.