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

Abstract

In 2006 Carbery raised a question about an improvement on the naïve norm inequality $\Vert f+g{\Vert }_p^p\le 2^{p-1}(\Vert f{\Vert }_p^p+\Vert g{\Vert }_p^p)$ 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 $\Vert fg{\Vert }_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.

18 pages. This version has added material on cases of equality