Orthonormal Sobolev estimates with fractal measures

Abstract

We prove a fractal version of Lieb's Hardy-Littlewood-Sobolev inequality for orthonormal functions. On the one hand, this can be viewed as a trace theorem for orthonormal functions. On the other, it allows us to recover the Rozenblum-Tashchiyan bound for the number of negative eigenvalues of $-\hat{I}”-\hat{I}¼$, where $\hat{I}¼$ is a shell potential. We also recover Rozenblum's bound for the sum of negative eigenvalues via a Lieb-Thirring kinetic inequality. Our proof is direct, avoiding both Schatten classes and variational arguments. We first reprove Adams' fractal Hardy-Littlewood-Sobolev inequality (for single functions) via Fourier analysis. This yields the required endpoint estimate as well as a bound for the interaction energy of Frostman measures.

14 pages