2026/07/17 by Neal Bez, Keith M. Rogers, Shunya Toyoshima
#math.CA #math-ph #math.MP #math.SP
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 -Δ-μ, where μ 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.