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).
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