Generalized Carleson Embeddings of M{ü}ntz Spaces

Abstract

This paper establishes Carleson embeddings of M{ü}ntz spaces $M_{\Lambda }^q$ into weighted Lebesgue spaces $L^p(\mathrm{d}\mu )$, where $\mu$ is a Borel regular measure on $[0,1]$ satisfying $\mu ([1-\epsilon ])\lesssim {\epsilon }^{\beta }$. In the case $\beta ⩾1$ we show that such measures are exactly the ones for which Carleson embeddings $L^{\frac{p}{\beta }}\hookrightarrow L^p(\mathrm{d}\mu )$ hold. The case $\beta \in (0,1)$ is more intricate but we characterize such measures $\mu$ in terms of a summability condition on their moments. Our proof relies on a generalization of $L^p$ estimates {à} la Gurariy-Macaev in the weighted $L^p$ spaces setting, which we think can be of interest in other contexts.