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Elsevier, Journal of Functional Analysis, 8(259), p. 2045-2072

DOI: 10.1016/j.jfa.2010.06.005

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A logarithmic Hardy inequality

Journal article published in 2009 by Manuel del Pino, Jean Dolbeault ORCID, Stathis Filippas, Achilles Tertikas
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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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.