Sharpening the triangle inequality : envelopes between L2 and Lp spaces

Ivanisvili, Paata; Mooney, Connor

doi:10.2140/apde.2020.13.1591

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Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

Paata Ivanisvili and Connor Mooney

Vol. 13 (2020), No. 5, 1591–1603

DOI: 10.2140/apde.2020.13.1591

Abstract

Motivated by the inequality $∥ f + g ∥_{2}^{2} \leq ∥ f ∥_{2}^{2} + 2 ∥ f g ∥_{1} + ∥ g ∥_{2}^{2}$ , Carbery (2009) raised the question of what is the “right” analogue of this estimate in $L^{p}$ for $p \neq 2$ . Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version of this inequality by providing upper bounds for $∥ f + g ∥_{p}^{p}$ in terms of the quantities $∥ f ∥_{p}^{p}$ , $∥ g ∥_{p}^{p}$ and $∥ f g ∥_{p ∕ 2}^{p ∕ 2}$ when $p \in (0, 1] \cup [2, \infty)$ , and lower bounds when $p \in (- \infty, 0) \cup (1, 2)$ , 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 $∥ f + g ∥_{p}^{p}$ also when $p \in (- \infty, 0) \cup (1, 2)$ and lower bounds when $p \in (0, 1] \cup [2, \infty)$ . For $p \in [1, 2]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $∥ f + g ∥_{p}^{p}$ for $p \in ℝ$ , $p \neq 0$ , are the best possible in terms of the quantities $∥ f ∥_{p}^{p}$ , $∥ g ∥_{p}^{p}$ and $∥ f g ∥_{p ∕ 2}^{p ∕ 2}$ , and we characterize the equality cases.

Keywords

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

Mathematical Subject Classification 2010

Primary: 42B20, 42B35, 47A30

Milestones

Received: 11 February 2019

Revised: 2 May 2019

Accepted: 11 June 2019

Published: 27 July 2020

Authors

Paata Ivanisvili
Department of Mathematics University of California Irvine, CA United States

Connor Mooney
Department of Mathematics University of California Irvine, CA United States

Vol. 13, No. 5, 2020