Embeddings between sequence variable Lebesgue spaces, strict and finitely strict singularity

Abstract

In this paper, we provide necessary and sufficient conditions under which two sequence variable Lebesgue spaces ${\ell }_{p_n}$ and ${\ell }_{q_n}$ are equivalent and also describe conditions under which the natural embeddings $id:{\ell }_{p_n}\to {\ell }_{q_n}$ are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding $id$ and show how they depend on the exponents $p_n$ and $q_n$.

18 pages