Lacunary Müntz spaces: isomorphisms and Carleson embeddings

Gaillard, Loïc; Lefèvre, Pascal

Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2215-2251.

Dans cet article, nous montrons que $M_{\Lambda }^p$ est presque isométrique à ${\ell }^p$, et ce de façon naturelle, lorsque $\Lambda$ est lacunaire avec une raison grande. Par ailleurs, notre approche permet aussi d’étudier les mesures de Carleson pour les espaces Müntz $M_{\Lambda }^p$ lorsque $\Lambda$ est lacunaire. Nous donnons des conditions nécessaires et des conditions suffisantes qui permettent d’assurer qu’un plongement de Carleson est borné ou compact. Dans le cadre hilbertien, nous étudions aussi l’appartenance de ce plongement aux classes de Schatten. Nous obtenons des caractérisations complètes lorsque $\Lambda$ se comporte comme une suite géométrique.

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 ensuring 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.

Classification : 30B10, 47B10, 47B38
Keywords: Müntz spaces, Carleson embeddings, lacunary sequences, Schatten classes
Mot clés : Espaces de Müntz, plongements de Carleson, suites lacunaires, classes de Schatten

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