# Lacunary M\"untz spaces: isomorphisms and Carleson embeddings

## TL;DR

This paper investigates the structure of lacunary M"untz spaces, showing they are nearly isometric to ll^p, and explores Carleson embeddings, providing conditions for boundedness, compactness, and Schatten class membership.

## Contribution

It establishes isomorphisms between lacunary Mfntz spaces and ll^p, and analyzes Carleson measures and embeddings with sharp characterizations for geometric sequences.

## Key findings

- Lacunary Mfntz spaces are almost isometric to ll^p for large lacunary ratios.
- Provides necessary and sufficient conditions for Carleson embedding boundedness and compactness.
- Characterizes Schatten class membership in the Hilbertian case for geometric sequences.

## Abstract

In this paper we prove that $M^p_\Lambda$ 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 M\"untz spaces $M^p_\Lambda$ 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.