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Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds
| Content Provider | Semantic Scholar |
|---|---|
| Author | Okikiolu, Kate |
| Copyright Year | 2006 |
| Abstract | 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. |
| Starting Page | 1655 |
| Ending Page | 1684 |
| Page Count | 30 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00039-007-0636-5 |
| Volume Number | 17 |
| Alternate Webpage(s) | https://arxiv.org/pdf/math/0603717v2.pdf |
| Alternate Webpage(s) | http://xxx.lanl.gov/PS_cache/math/pdf/0603/0603717v2.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0603717v2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00039-007-0636-5 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
