Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1199-1209, 2021

An example of a reflexive Lorentz Gamma space with trivial Boyd and Zippin indices

Alexei Karlovich, Eugene Shargorodsky

Received August 14, 2020. Published online June 25, 2021.

Abstract: 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 $\Gamma_{p,w}'$.

Keywords: Lorentz Gamma space; reflexivity; Boyd indices; Zippin indices

Classification MSC: 46E30, 42B25

DOI: 10.21136/CMJ.2021.0355-20

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] C. Bennett, R. Sharpley: Interpolation of Operators. Pure and Applied Mathematics 129. Academic Press, Boston (1988). DOI 10.1016/s0079-8169(08)x6053-2 | MR 0928802 | Zbl 0647.46057
[2] D. W. Boyd: The Hilbert transform on rearrangement-invariant spaces. Can. J. Math. 19 (1967), 599-616. DOI 10.4153/CJM-1967-053-7 | MR 0212512 | Zbl 0147.11302
[3] M. Ciesielski: Relationships between $K$-monotonicity and rotundity properties with application. J. Math. Anal. Appl. 465 (2018), 235-258. DOI 10.1016/j.jmaa.2018.05.008 | MR 3806700 | Zbl 1402.46010
[4] A. Gogatishvili, R. Kerman: The rearrangement-invariant space $\Gamma_{p,\phi}$. Positivity 18 (2014), 319-345. DOI 10.1007/s11117-013-0246-4 | MR 3215181 | Zbl 1311.46025
[5] A. Gogatishvili, L. Pick: Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat., Barc. 47 (2003), 311-358. DOI 10.5565/PUBLMAT_47203_02 | MR 2006487 | Zbl 1066.46023
[6] A. Kamińska, L. Maligranda: On Lorentz spaces $\Gamma_{p,w}$. Isr. J. Math. 140 (2004), 285-318. DOI 10.1007/BF02786637 | MR 2054849 | Zbl 1068.46019
[7] S. G. Krejn, Yu. I. Petunin, E. M. Semenov: Interpolation of Linear Operators. Translations of Mathematical Monographs 54. American Mathematical Society, Providence (1982). DOI 10.1090/mmono/054 | MR 0649411 | Zbl 0493.46058
[8] L. Maligranda: Indices and interpolation. Diss. Math. 234 (1985), 1-49. MR 0820076 | Zbl 0566.46038
[9] L. Pick, A. Kufner, O. John, S. Fučík: Function Spaces. Volume 1. De Gruyter Series in Nonlinear Analysis and Applications 14. Walter de Gruyter, Berlin (2013). DOI 10.1515/9783110250428 | MR 3024912 | Zbl 1275.46002
[10] E. Sawyer: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96 (1990), 145-158. DOI 10.4064/sm-96-2-145-158 | MR 1052631 | Zbl 0705.42014
[11] M. Zippin: Interpolation of operators of weak type between rearrangement invariant function spaces. J. Funct. Anal. 7 (1971), 267-284. DOI 10.1016/0022-1236(71)90035-8 | MR 0412793 | Zbl 0224.46038

Affiliations: Alexei Karlovich (corresponding author), Centro de Matemática e Aplicações, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal, e-mail: oyk@fct.unl.pt; Eugene Shargorodsky, Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom; Technische Universität Dresden, Fakultät Mathematik, 01062 Dresden, Germany, e-mail: eugene.shargorodsky@kcl.ac.uk

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
free access

···

Digital Mathematics Library
free access