A logarithmic Hardy inequality
Del Pino, Manuel; Dolbeault, Jean; Filippas, Stathis; Tertikas, Achilles (2010), A logarithmic Hardy inequality, Journal of Functional Analysis, 259, 8, p. 2045-2072. http://dx.doi.org/10.1016/j.jfa.2010.06.005
Type
Article accepté pour publication ou publiéExternal document link
http://arxiv.org/abs/0912.0590v1Date
2010Journal name
Journal of Functional AnalysisVolume
259Number
8Publisher
Elsevier
Pages
2045-2072
Publication identifier
Metadata
Show full item recordAbstract (EN)
We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden–Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved by radial functions, in some range of the parameters.Subjects / Keywords
Hardy inequality; Sobolev inequality; Interpolation; Logarithmic Sobolev inequality; Hardy–Sobolev inequalities; Caffarelli–Kohn–Nirenberg inequalities; Scale invariance; Emden–Fowler transformation; Radial symmetry; Symmetry breakingRelated items
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