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Inequalities for Lᵖ-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality

Carlen, Eric A. and Frank, Rupert L. and Ivanisvili, Paata and Lieb, Elliott H. (2021) Inequalities for Lᵖ-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality. Journal of Geometric Analysis, 31 (4). pp. 4051-4073. ISSN 1050-6926. doi:10.1007/s12220-020-00425-y.

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In 2006 Carbery raised a question about an improvement on the naïve norm inequality ∥f+g∥^p_p ≤ 2^(p−1)(∥f∥^p_p+∥g∥^p_p) for two functions f and g in Lᵖ of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p−1) is not needed. Carbery’s question concerns a proposed interpolation between the two situations for p > 2 with the interpolation parameter measuring the overlap being ∥fg∥_(p/2). Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all real p ≠ 0.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Carlen, Eric A.0000-0003-2613-187X
Frank, Rupert L.0000-0001-7973-4688
Lieb, Elliott H.0000-0001-5843-3587
Additional Information:© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 06 November 2019; Published 25 May 2020. This paper may be reproduced, in its entirety, for non-commercial purposes. Work partially supported by NSF grants DMS-1501007 and DMS-1764254 (E.A.C.), DMS-1363432 (R.L.F.), DMS-1856486 (P.I.), PHY-1265118 (E.H.L.). Open Access funding provided by Projekt DEAL. We thank Anthony Carbery for useful correspondence.
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Subject Keywords:Lᵖ space; Minkowski’s inequality; Convexity
Issue or Number:4
Record Number:CaltechAUTHORS:20200526-144324896
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Official Citation:Carlen, E.A., Frank, R.L., Ivanisvili, P. et al. Inequalities for Lᵖ-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality. J Geom Anal 31, 4051–4073 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103470
Deposited By: Tony Diaz
Deposited On:26 May 2020 22:22
Last Modified:30 Apr 2021 17:38

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