Embedding Theorems for Müntz spaces

Abstract

We discuss boundedness and compactness properties of the embedding $M_Λ^1\subset L^1(μ)$, where $M_Λ^1$ is the closure of the monomials $x^{λ_n}$ in $L1([0,1])$ and $μ$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of "sublinear" measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $Λ$. Finally, we show how one can recapture some of Al Alam's results on boundedness and essential norm of weighted composition operators from $M_Λ^1$ to $L1([0,1])$.

21 pages