Vol. 13, No. 5, 2020

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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

Motivated by the inequality f + g22 f22 + 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.

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triangle inequality, $L^p$ spaces, concave envelopes, Bellman function
Mathematical Subject Classification 2010
Primary: 42B20, 42B35, 47A30
Received: 11 February 2019
Revised: 2 May 2019
Accepted: 11 June 2019
Published: 27 July 2020
Paata Ivanisvili
Department of Mathematics
University of California
Irvine, CA
United States
Connor Mooney
Department of Mathematics
University of California
Irvine, CA
United States