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arXiv · 2607.15826

Orthonormal Sobolev estimates with fractal measures

Neal Bez·Keith M. Rogers·Shunya Toyoshima
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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 $-\Delta-\mu$, 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.

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BibTeXRIS

Neal Bez, Keith M. Rogers, Shunya Toyoshima. 2026-07-17. Orthonormal Sobolev estimates with fractal measures. https://arxiv.org/abs/2607.15826

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