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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 publié
External document link
http://arxiv.org/abs/0912.0590v1
Date
2010
Journal name
Journal of Functional Analysis
Volume
259
Number
8
Publisher
Elsevier
Pages
2045-2072
Publication identifier
http://dx.doi.org/10.1016/j.jfa.2010.06.005
Metadata
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Author(s)
Del Pino, Manuel
Dolbeault, Jean cc
Filippas, Stathis
Tertikas, Achilles
Abstract (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 breaking

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