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Embeddings of Müntz Spaces in $L^{\infty }(\unicode[STIX]{x1D707})$

  • Ihab Al Alam (a1) and Pascal Lefèvre (a2)
Abstract

In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$ , where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$ , where $\unicode[STIX]{x1D707}_{1}$ , $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$ .

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This work is part of the project CEDRE ESFO. The authors would like to thank the program PHC CEDRE and the Lebanese University for their support.

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References
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[1] Al Alam, I., Essential norm of weighted composition operator on Müntz spaces . J. Math. Anal. Appl. 358(2009), 273280. https://doi.org/10.1016/j.jmaa.2009.04.042.
[2] Al Alam, I., Gaillard, L., Habib, G., Lefèvre, P., and Maalouf, F., Essential norms of Cesàro operators on L p and Cesàro spaces . J. Math. Anal. Appl. 467(2018), no. 2, 10381065. https://doi.org/10.1016/j.jmaa.2018.07.038.
[3] Borwein, P. and Erdélyi, T., Polynomials and polynomial inequalities . Graduate Texts in Mathematics, 161. Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-0793-1.
[4] Chalendar, I., Fricain, E., and Timotin, D., Embeddings theorems for Müntz spaces . Ann. Inst. Fourier 61(2011), 22912311 https://doi.org/10.5802/aif.2674.
[5] Gaillard, L. and Lefèvre, P., Lacunary Müntz spaces: isomorphisms and Carleson embeddings. arxiv:1701.05807.
[6] Schwartz, L., Etude des sommes d’exponentielles. Hermann, Paris, 1959.
[7] Waleed Noor, S., Embeddings of Müntz spaces: Composition operators . Integral Equations Operator Theory 73(2012), 589602. https://doi.org/10.1007/s00020-012-1965-9.
[8] Waleed Noor, S. and Timotin, D., Embeddings of Müntz spaces: The Hilbertian case . Proc. Amer. Math. Soc. 141(2013), 20092023. https://doi.org/10.1090/S0002-9939-2012-11681-8.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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