Abstract
The necessary and sufficient conditions for k-uniform convexity in Orlicz-Lorentz function spaces equipped with the Orlicz norm and generated by N-functions as well as any non-increasing weight sequences are given. Moreover, Some tools useful in the proofs of the main results are also provided. Besides, in the proof process, we use the reachability of the infimum in Orlicz norm formula to obtain a function working like supporting functional, which makes us complete the proof even though the characterization of dual space is still unclear.
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References
J. Cerdá, H. Hudzik, A. Kamińska, M. Mastłyo, Geometric property of symmetric spaces with application to Orlicz-Lorentz spaces, Positivity, 2(4)130(1998) 311-337.
H. Hudzik, A. Kamińska, M. Mastyło, On the dual of Orlicz-Lorentz space, Proc. Amer. Math. Soc. 130(6)(2002)1645-1654.
W. Gong, D. Zhang, Monotonicity in Orlicz-lorentz sequence spaces equipped with the orlicz norm, Acta Mathematica Scientia, 36(6)(2016)1577-1589.
B. Chen, W. Gong, Uniform normal structure and uniform non-squareness of Orlicz-Lorentz function spaces endowed with the Orlicz norm, Annals of Functional Analysis 12(2)(2021).
A. Kamińska, Uniform convexity of generalized Lorentz spaces, Archiv der Mathematik, 56(2)(1991)181-188.
J. Wang, Y. Chen, Rotundity and uniform rotundity of Orlicz-Lorentz spaces equipped with the Orlicz norm, Mathematische Nachrichten 38(1)(2012)131-896.
P. Lin, K-uniform rotundity of Lorentz-Orlicz spaces, Journal of Mathematical Analysis and Applications, 204(1)(1996)29-45.
J. Raymond, Kadec-Klee property and fixed points, Journal of Functional Analysis, 266(8)(2014)5429-5438.
Y. Cui, L. Zhao, Kadec-Klee property in Musielak-Orlicz function spaces euipped with the Orlicz norm, Aequationes Mathematicae, 96(1)(2021)167-184.
C. Bennett, R. Sharpley, Interpolation of Operators, Vol. 129, 1953.
S. Chen, Geometry of Orlicz space, Dissertation Mathematicae 356.
D. Ferreyra, F. Levis, M. Gareis, Extended best polynomial approximation operator in Orlicz-Lorentz spaces, Mathematische Nachrichtenl, 2021.
A. Kamińska, K. Lésnik, Y. Raynaud, Dual spaces to Orlicz-Lorentz spaces, 160 Studia Mathematica, 222(3)(2014)229-261.
A. Kaminska, Y. Raymaud, Isomorphic copies in the lattice e and its symmetrization \(e^{*}\) with applications to Orlicz-Lorentz spaces, Journal of Functional Analysis 267(1)(2009)271-331.
F. Levis, Weak inequalities for maximal functions in Orlicz-Lorentz spaces and applications, Journal of Approximation Theory, 162(2)(2010)239-251.
C. Wu, L. Ren, Strict convexity of Orlicz-Lorentz space with Orlicz norm, Journal of Mathematics, 19(2)(1999)235-240.
F. Smithies, M. Krasnoselśkii, Y. Rutickii, Convex functions and Orlicz spaces, Vol. 47, Mathematical Association, 1963.
Acknowledgements
Yunan Cui gratefully acknowledges the support of NSF of CHINA (11871181).
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Communicated by Pradipta Bandyopadhyay.
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Wang, D., Cui, Y. K-uniform convexity in Orlicz-Lorentz function space equipped with the Orlicz norm. Indian J Pure Appl Math 54, 1025–1032 (2023). https://doi.org/10.1007/s13226-022-00319-5
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DOI: https://doi.org/10.1007/s13226-022-00319-5