An Lp bound for the Riesz and Bessel potentials of orthonormal functions

Elliott H. Lieb

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Abstract

Let ψ1, ...,ψN be orthonormal functions in Rd and let ui = (-Δ)- 1 2ψi, or ui = (-Δ + 1)- 1 2ψi, and let p(x) = ∑|ui(x)|2. Lp bounds are proved for p, an example being ∥p∥p ≤ AdN 1 p for d ≥ 3, with p = d(d - 2)-1. The unusual feature of these bounds is that the orthogonality of the ψi, yields a factor N 1 p instead of N, as would be the case without orthogonality. These bounds prove some conjectures of Battle and Federbush (a Phase Cell Cluster Expansion for Euclidean Field Theories, I, 1982, preprint) and of Conlon (Comm. Math. Phys., in press).

Original languageEnglish (US)
Pages (from-to)159-165
Number of pages7
JournalJournal of Functional Analysis
Volume51
Issue number2
DOIs
StatePublished - Apr 1983

All Science Journal Classification (ASJC) codes

  • Analysis

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