Non-linear ground state representations and sharp Hardy inequalities |
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| Authors: | Rupert L Frank Robert Seiringer |
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| Institution: | a Department of Mathematics, Princeton University, Washington Road, Princeton, NJ 08544, USA
b Department of Physics, Princeton University, P.O. Box 708, Princeton, NJ 08544, USA |
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| Abstract: | We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces. |
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| Keywords: | Hardy inequality Sobolev embedding Ground state substitution Fractional Sobolev spaces |
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