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

Abstract

Motivated by the inequality $\parallel f+g{\parallel }_2^2\le \parallel f{\parallel }_2^2+2\parallel fg{\parallel }_1+\parallel g{\parallel }_2^2$, Carbery (2009) raised the question of what is the “right” analogue of this estimate in $L^p$ for $p\ne 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^p$ version of this inequality by providing upper bounds for $\parallel f+g{\parallel }_p^p$ in terms of the quantities $\parallel f{\parallel }_p^p$, $\parallel g{\parallel }_p^p$ and $\parallel fg{\parallel }_{p∕2}^{p∕2}$ when $p\in \left(0,1\right]\cup \left[2,\infty \right)$, and lower bounds when $p\in \left(-\infty ,0\right)\cup \left(1,2\right)$, 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 $\parallel f+g{\parallel }_p^p$ also when $p\in \left(-\infty ,0\right)\cup \left(1,2\right)$ and lower bounds when $p\in \left(0,1\right]\cup \left[2,\infty \right)$. For $p\in \left[1,2\right]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $\parallel f+g{\parallel }_p^p$ for $p\in ℝ$, $p\ne 0$, are the best possible in terms of the quantities $\parallel f{\parallel }_p^p$, $\parallel g{\parallel }_p^p$ and $\parallel fg{\parallel }_{p∕2}^{p∕2}$, and we characterize the equality cases.