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An example of a reflexive Lorentz Gamma space with trivial Boyd and Zippin indices
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 4, pp. 1199-1209
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We show that for every $p\in (1,\infty )$ there exists a weight $w$ such that the Lorentz Gamma space $\Gamma _{p,w}$ is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space $\Gamma _{p,w}$ and on its associate space $\Gamma _{p,w}'$.
We show that for every $p\in (1,\infty )$ there exists a weight $w$ such that the Lorentz Gamma space $\Gamma _{p,w}$ is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space $\Gamma _{p,w}$ and on its associate space $\Gamma _{p,w}'$.
DOI : 10.21136/CMJ.2021.0355-20
Classification : 42B25, 46E30
Keywords: Lorentz Gamma space; reflexivity; Boyd indices; Zippin indices 
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Karlovich, Alexei; Shargorodsky, Eugene. An example of a reflexive Lorentz Gamma space with trivial Boyd and Zippin indices. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 4, pp. 1199-1209. doi: 10.21136/CMJ.2021.0355-20

[1] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics 129. Academic Press, Boston (1988). | DOI | MR | JFM

[2] Boyd, D. W.: The Hilbert transform on rearrangement-invariant spaces. Can. J. Math. 19 (1967), 599-616. | DOI | MR | JFM

[3] Ciesielski, M.: Relationships between $K$-monotonicity and rotundity properties with application. J. Math. Anal. Appl. 465 (2018), 235-258. | DOI | MR | JFM

[4] Gogatishvili, A., Kerman, R.: The rearrangement-invariant space $\Gamma_{p,\phi}$. Positivity 18 (2014), 319-345. | DOI | MR | JFM

[5] Gogatishvili, A., Pick, L.: Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat., Barc. 47 (2003), 311-358. | DOI | MR | JFM

[6] Kamińska, A., Maligranda, L.: On Lorentz spaces $\Gamma_{p,w}$. Isr. J. Math. 140 (2004), 285-318. | DOI | MR | JFM

[7] Krejn, S. G., Petunin, Yu. I., Semenov, E. M.: Interpolation of Linear Operators. Translations of Mathematical Monographs 54. American Mathematical Society, Providence (1982). | DOI | MR | JFM

[8] Maligranda, L.: Indices and interpolation. Diss. Math. 234 (1985), 1-49. | MR | JFM

[9] Pick, L., Kufner, A., John, O., Fučík, S.: Function Spaces. Volume 1. De Gruyter Series in Nonlinear Analysis and Applications 14. Walter de Gruyter, Berlin (2013). | DOI | MR | JFM

[10] Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96 (1990), 145-158. | DOI | MR | JFM

[11] Zippin, M.: Interpolation of operators of weak type between rearrangement invariant function spaces. J. Funct. Anal. 7 (1971), 267-284. | DOI | MR | JFM

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