Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds

Abstract

Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g \(\in\) Γ, denote the area element by dV and the Laplace–Beltrami operator by Δg. We define the Robin mass m(x) at the point x \(\in\) M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δg−1 is then defined by trace Δ−1 = ∫M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let \(\Delta_{S^{2},V}\) be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g \(\in\) Γ with trace Δg−1 < trace\(\Delta_{S^{2},V}^{-1}\) then the minimum of trace Δ−1 over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ−1 over Γ is equal to trace \(\Delta_{S^{2},V}^{-1}\). In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have \(A_{F^{2/n}g} = F^{-1}A_{g}\). In this case the role of \(\Delta_{S^{2},V}\) is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem.