Lacunary Müntz spaces: isomorphisms and Carleson embeddings

Abstract

In this paper we prove that $M_{\Lambda }^p$ is almost isometric to ${\ell }^p$ in the canonical way when $\Lambda$ is lacunary with a large ratio. On the other hand, our approach can be used to study also the Carleson measures for Müntz spaces $M_{\Lambda }^p$ when $\Lambda$ is lacunary. We give some necessary and some sufficient conditions to ensure that a Carleson embedding is bounded or compact. In the hilbertian case, the membership to Schatten classes is also studied. When $\Lambda$ behaves like a geometric sequence the results are sharp, and we get some characterizations.