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The trace inequality and eigenvalue estimates for Schrödinger operators. (English) Zbl 0591.47037
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.

47F05 General theory of partial differential operators
47A10 Spectrum, resolvent
26D10 Inequalities involving derivatives and differential and integral operators
35J10 Schrödinger operator, Schrödinger equation
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