Nonembedding of Lorentz sequence spaces into Lorentz function spaces

Abstract

Let $0<p,q<\infty$ with $p\ne q$, and suppose that $p<2$ or $q\le 1$. We prove that the Lorentz sequence space ${\ell }_{p,q}$ does not isomorphically embed into $L_{p,q}(0,1)$. Consequently, $L_{p,q}(0,\infty )$ does not isomorphically embed into $L_{p,q}(0,1)$.