Sharp constants related to the triangle inequality in Lorentz spaces

Abstract

We study the Lorentz spaces $L^{p,s}(R,\mu )$ in the range $1<p<s\le \infty$, for which the standard functional $$||f|{|}_{p,s}=({\int }_0^{\infty }(t^{1/p}{f}^{\ast }(t){)}^s\frac{dt}{t}{)}^{1/s}$$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$||f|{|}_{(p,s)}=\inf \{\sum _k||f_k|{|}_{p,s}\},$$ where the infimum is taken over all finite representations $f={\sum }_k{f}_k.$ We also prove that the decomposition norm and the dual norm $$||f|{|}_{p,s}^{\prime }=\sup \left\{{\int }_{R}fgd\mu :||g|{|}_{p^{\prime },s^{\prime }}=1\right\}$$ agree for all values $p,s>1$.

24 pages