Atomic Characterization of L_1 And The Lorentz-Bochner Space L^X(p,1) for With Some Applications

Abstract

In the 1950s, G. G Lorentz introduced the spaces $\Lambda(\alpha)$ and $M(\alpha)$, for $0<\alpha<1$ and showed that the dual of $\Lambda(\alpha)$ is equivalent to $M(\alpha)$ in his paper titled 'Some New Functional Spaces'. Indeed, Lorentz mentioned that for the excluded value $\alpha=1$, the space $\Lambda(1)$ is $L_1$ and $M(1)$ is $L_\infty$. In 2010, De Souza motivated by a theorem by Guido Weiss and Elias Stein on operators acting on $\Lambda(\alpha)$, showed that there is a simple characterization for the space $\Lambda(\alpha)$ for $0<\alpha<1$. The theorem by Stein and Weiss is an immediate consequence of the new characterization by De Souza. In this work, we seek to investigate the decomposition of $L_1$ which is the case $\alpha=1$, and also extend the result to the well-known Lorentz-Bochner space $L^X(p,1)$ for $p\geq1$, and $X$ is a Banach space, that is, the Lorentz space of vector-valued functions. As a by product, we will use these new characterizations to study some operators defined on these spaces into some well-known Banach spaces.