Abstract
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.
| Original language | English |
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| Pages (from-to) | 2045-2072 |
| Number of pages | 28 |
| Journal | Journal of Functional Analysis |
| Volume | 259 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - 15 Oct 2010 |
Funding
J.D. has been supported by the ECOS contract no. C05E09 and the ANR grants IFO, EVOL and CBDif, and thanks the Department of Mathematics of the University of Crete and the Depar-tamento de Ingeniería Matemática of the University of Chile for their warm hospitality. M.d.P. has also been supported by grants Fondecyt 1070389, Anillo ACT125 CAPDE and Fondo Basal CMM. This work is also part of the MathAmSud NAPDE project (M.d.P. & J.D.). Plots have been done with Mathematica™. The authors thank a referee for his careful reading and wise suggestions.
Keywords
- Caffarelli-Kohn-Nirenberg inequalities
- Emden-Fowler transformation
- Hardy inequality
- Hardy-Sobolev inequalities
- Interpolation
- Logarithmic Sobolev inequality
- Radial symmetry
- Scale invariance
- Sobolev inequality
- Symmetry breaking
ASJC Scopus subject areas
- Analysis