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Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality

Frank, Rupert L. and Lieb, Elliott H. (2010) Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality. Calculus of Variations and Partial Differential Equations, 39 (1-2). pp. 85-99. ISSN 0944-2669. http://resolver.caltech.edu/CaltechAUTHORS:20170510-143424777

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Abstract

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00526-009-0302-xDOIArticle
http://link.springer.com/article/10.1007%2Fs00526-009-0302-xPublisherArticle
https://arxiv.org/abs/0904.4275arXivDiscussion Paper
Additional Information:© 2009 The Authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Received: 28 April 2009; Accepted: 25 November 2009; First Online: 23 December 2009. We are grateful to E. Carlen for pointing out that the conformal invariance of the HLS functional and the conventional reflection positivity through planes imply the inversion positivity through spheres. This allows us to circumvent our original, direct but complicated proof, which uses properties of Gegenbauer polynomials. Support through DFG grant FR 2664/1-1 (R.F.) and U.S. NSF grant PHY 0652854 (R.F. and E.L.) is gratefully acknowledged.
Funders:
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)FR 2664/1-1
NSFPHY-0652854
Classification Code:Mathematics Subject Classification (2000): 39B62; Secondary 26A33; 26D10; 46E35
Record Number:CaltechAUTHORS:20170510-143424777
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170510-143424777
Official Citation:Frank, R.L. & Lieb, E.H. Calc. Var. (2010) 39: 85. doi:10.1007/s00526-009-0302-x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77354
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 May 2017 22:24
Last Modified:12 May 2017 22:24

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