Generalized Carleson Embeddings of M{ü}ntz Spaces

This paper establishes Carleson embeddings of M{ü}ntz spaces $M^q_Λ$ into weighted Lebesgue spaces $L^p(\mathrm{d}μ)$, where $μ$ is a Borel regular measure on $[0,1]$ satisfying $μ([1-\varepsilon])\lesssim \varepsilon^β$. In the case $β\geqslant 1$ we show that such measures are exactly the ones for which Carleson embeddings $L^{\frac{p}β} \hookrightarrow L^p(\mathrm{d}μ)$ hold. The case $β\in (0,1)$ is more intricate but we characterize such measures $μ$ 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.