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

Content Provider	Springer Nature Link
Author	Okikiolu, Kate
Copyright Year	2007
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 Δ g . 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 Δ g −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 Δ g −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.
Starting Page	1655
Ending Page	1684
Page Count	30
File Format	PDF
ISSN	1016443X
Journal	Geometric and Functional Analysis
Volume Number	17
Issue Number	5
e-ISSN	14208970
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2007-11-29
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Extremal metrics sharp logarithmic Hardy-Littlewood-Sobolev inequality conformal geometry spectral invariant trace of the Green function Analysis
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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