Orthonormal Sobolev estimates with fractal measures

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 $-\Delta-\mu$, where $\mu$ 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.