An L p Bound for the Riesz and Bessel Potentials of Orthonormal Functions

Let $${\Psi _1},...,{\Psi _N}$$ be orthonormal functions in Rd and let $${u_1} = {( - \Delta )^{ - 1/2}}{\Psi _i}$$ or $${u_1} = {( - \Delta + 1)^{ - 1/2}}{\Psi _i}$$ and let $$p(x) = {\sum {\left| {{u_i}(x)} \right|} ^2}$$ . Lp bounds are proved for p, an example being for $${\left\| P \right\|_P} \le {A_d}{N^{1/p}}for{\rm{ d}} \ge {\rm{3, with p = d(d - 2}}{{\rm{)}}^{ - 1}}$$ . 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).