Lacunary Müntz spaces: isomorphisms and Carleson embeddings  [ Espace lacunaire de Müntz : isomorphismes et plongements de Carleson ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, p. 2215-2251
Dans cet article, nous montrons que M Λ p est presque isométrique à p , et ce de façon naturelle, lorsque Λ est lacunaire avec une raison grande. Par ailleurs, notre approche permet aussi d’étudier les mesures de Carleson pour les espaces Müntz M Λ p lorsque Λ 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 Λ se comporte comme une suite géométrique.
In this paper we prove that M Λ p is almost isometric to p in the canonical way when Λ 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 Λ p when Λ 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 Λ behaves like a geometric sequence the results are sharp, and we get some characterizations.
Reçu le : 2017-01-17
Révisé le : 2017-07-22
Accepté le : 2017-11-07
Publié le : 2018-11-23
DOI : https://doi.org/10.5802/aif.3207
Classification:  30B10,  47B10,  47B38
Mots clés: Espaces de Müntz, plongements de Carleson, suites lacunaires, classes de Schatten
@article{AIF_2018__68_5_2215_0,
     author = {Gaillard, Lo\"\i c and Lef\`evre, Pascal},
     title = {Lacunary M\"untz spaces: isomorphisms and Carleson embeddings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     pages = {2215-2251},
     doi = {10.5802/aif.3207},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_5_2215_0}
}
Gaillard, Loïc; Lefèvre, Pascal. Lacunary Müntz spaces: isomorphisms and Carleson embeddings. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2215-2251. doi : 10.5802/aif.3207. https://aif.centre-mersenne.org/item/AIF_2018__68_5_2215_0/

[1] Al Alam, Ihab; Habib, Georges; Lefèvre, Pascal; Maalouf, Fares Essential norms of Volterra and Cesàro operators on Müntz Spaces, Colloq. Math., Tome 151 (2018) no. 2, pp. 157-169 | Zbl 06857948

[2] Al Alam, Ihab; Lefèvre, Pascal Essential norms of weighted composition operators on L 1 Müntz Spaces, Serdica Math. J., Tome 40 (2014) no. 3, pp. 241-260 | MR MR3288126

[3] Borwein, Peter; Erdélyi, Tamàs Polynomials and polynomial inequalities, Springer (1995) | MR MR1367960 | Zbl 0840.26002

[4] Chalendar, Isabelle; Fricain, Emmanuel; Timotin, Dan Embeddings theorems for Müntz Spaces, Ann. Inst. Fourier, Tome 61 (2011) no. 6, pp. 2291-2311 | MR MR2976312 | Zbl 1255.46013

[5] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 43 (1995), xv+474 pages | Zbl 1139.47021

[6] Godefroy, Gilles Unconditionality in spaces of smooth functions, Arch. Math., Tome 92 (2009) no. 6, pp. 476-484 | MR MR2506948 | Zbl 1189.46008

[7] Gurariy, Vladimir I.; Lusky, Wolfgang Geometry of Müntz spaces and related questions, Springer, Lecture Notes in Math., Tome 1870 (2005), xiv+172 pages | MR MR2190706 | Zbl 1094.46003

[8] Gurariy, Vladimir I.; Matsaev, Vladimir I Lacunary power sequences in the spaces C and L p , Am. Math. Soc., Transl., Tome 72 (1966), pp. 9-21 | MR MR0190703 | Zbl 0187.37505

[9] Ludkovsky, Sergey V.; Lusky, Wolfgang On the geometry of Müntz spaces, J. Funct. Spaces (2015), 787291, 7 pages (Art. ID 787291, 7 p.) | MR MR3361114 | Zbl 1332.46023

[10] Noor, S. Waleed; Timotin, Dan Embeddings of Müntz spaces: the Hilbertian Case, Proc. Am. Math. Soc., Tome 141 (2013) no. 6, pp. 2009-2023 | MR MR3034427 | Zbl 1282.46026

[11] Werner, Douglas A remark about Müntz spaces (http://page.mi.fu-berlin.de/werner/preprints/muentz.pdf )

[12] Wojtaszczyk, Przemyslaw Banach spaces for analysts, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 25 (1991), xiii+382 pages | Zbl 0724.46012