On Lipschitz-Lorentz spaces and their Zygmund classes

Abstract

Let G be a metrizable locally compact abelian group. We prove that ($L _1$(G), lip($alpha,pq$), $widetilde {lip(alpha,pq)}$, ($L _1$(G), Lip ( $alpha$, pq)) and Lip ($alpha$ , pq) are isometrically isomorphic, where Lip ( $alpha$$, pq) and lip ($alpha$ , pq) denote the Lipschitz-Lorentz spaces defined on G, ($L _1$(G),A) is the space of multipliers from $L _1$(G) to A and $widetilde {lip(alpha,pq)}$ denotes the relative completion of lip ( $alpha$, pq). Also, we characterize the space of multipliers from Lorentz spaces to the Lipschitz-Lorentz-Zygmund classes L$Lambda_*$($alpha$ , pq;G) and L$lambda_*$( $alpha$, pq;G).