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2020 Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces
Paata Ivanisvili, Connor Mooney
Anal. PDE 13(5): 1591-1603 (2020). DOI: 10.2140/apde.2020.13.1591

Abstract

Motivated by the inequality f+g22f22+2fg1+g22, Carbery (2009) raised the question of what is the “right” analogue of this estimate in Lp for p2. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an Lp version of this inequality by providing upper bounds for f+gpp in terms of the quantities fpp, gpp and fgp2p2 when p(0,1][2,), and lower bounds when p(,0)(1,2), thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for f+gpp also when p(,0)(1,2) and lower bounds when p(0,1][2,). For p[1,2] we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of f+gpp for p, p0, are the best possible in terms of the quantities fpp, gpp and fgp2p2, and we characterize the equality cases.

Citation

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Paata Ivanisvili. Connor Mooney. "Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces." Anal. PDE 13 (5) 1591 - 1603, 2020. https://doi.org/10.2140/apde.2020.13.1591

Information

Received: 11 February 2019; Revised: 2 May 2019; Accepted: 11 June 2019; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07271840
MathSciNet: MR4149072
Digital Object Identifier: 10.2140/apde.2020.13.1591

Subjects:
Primary: 42B20 , 42B35 , 47A30

Keywords: $L^p$ spaces , Bellman function , concave envelopes , triangle inequality

Rights: Copyright © 2020 Mathematical Sciences Publishers

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