Sharp constants related to the triangle inequality in Lorentz spaces

TL;DR

This paper determines the optimal constants in the triangle inequality for Lorentz spaces' quasi-norms and explores the relationship between the decomposition norm and the dual norm, providing new insights into their equivalence.

Contribution

It finds the best constant for the triangle inequality in Lorentz spaces and establishes the equivalence of the decomposition and dual norms.

Findings

Optimal triangle inequality constant derived for Lorentz space quasi-norms.

Decomposition norm and dual norm are shown to be equivalent for all p,s > 1.

Provides a new understanding of the structure of 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 $$ ||f||_{p,s}=(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s} $$ 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: $$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\}, $$ 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 $$ ||f||_{p,s}'= \sup\left\{\int_R fg d\mu: ||g||_{p',s'}=1\right\} $$ agree for all values $p,s>1$.