Generalized Carleson Embeddings of Müntz Spaces

This paper establishes Carleson embeddings of M{\"u}ntz spaces MqΛ into weighted Lebesgue spaces Lᵖ(dμ), where μ is a Borel regular measure on $[0,1]$ satisfying μ([1-ε]) εβ. In the case β 1 we show that such measures are exactly the ones for which Carleson embeddings Lp/β Lᵖ(dμ) hold. The case β ∈ (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 Lᵖ estimates {\`a} la Gurariy-Macaev in the weighted Lᵖ spaces setting, which we think can be of interest in other contexts.