Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

@article{Carlen2018InequalitiesF,
title={Inequalities for \$L^p\$-norms that sharpen the triangle inequality and complement Hanner's Inequality},
author={E. Carlen and R. Frank and P. Ivanisvili and E. Lieb},
journal={arXiv: Functional Analysis},
year={2018}
}

In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ 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 $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type… CONTINUE READING