Embedding Theorems for M\"untz spaces

TL;DR

This paper investigates the conditions under which the M"untz space embeddings into L1 spaces are bounded or compact, focusing on quasilacunary sequences and introducing a class of measures to characterize these properties.

Contribution

It provides a comprehensive analysis of the embedding properties of M"untz spaces, especially for quasilacunary sequences, and introduces a new class of measures to characterize boundedness and compactness.

Findings

Characterization of boundedness and compactness of embeddings for quasilacunary sequences

Introduction of 'sublinear' measures for embedding analysis

Recovers results on weighted composition operators from M"untz spaces

Abstract

We discuss boundedness and compactness properties of the embedding $M_\Lambda^1\subset L^1(\mu)$, where $M_\Lambda^1$ is the closure of the monomials $x^{\lambda_n}$ in $L1([0,1])$ and $\mu$ 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 $\Lambda$. Finally, we show how one can recapture some of Al Alam's results on boundedness and essential norm of weighted composition operators from $M_\Lambda^1$ to $L1([0,1])$.