Abstract
In this chapter we study the so-called weak Lebesgue spaces which are one of the first generalizations of the Lebesgue spaces and a prototype of the so-called Lorentz spaces which will be studied in a subsequent chapter. In the framework of weak Lebesgue spaces we will study, among other topics, embedding results, convergence in measure, interpolation results, and the question of normability of the space. We also show a Fatou type lemma for weak Lebesgue spaces as well as the completeness of the quasi-norm. The Lyapunov inequality and the Hölder inequality are shown to hold.
Mathematics is the most beautiful and most powerful creation of the human spirit. Stefan Banach
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References
R. E. Castillo, F. Vallejo Narvaez, and J.C. Ramos Fernández. Multiplication and composition operators on weak l p spaces. Bull. Malays. Math. Sci. Soc., 38(3):927–973, 2015.
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© 2016 Springer International Publishing Switzerland
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Castillo, R., Rafeiro, H. (2016). Weak Lebesgue Spaces. In: An Introductory Course in Lebesgue Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-30034-4_5
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DOI: https://doi.org/10.1007/978-3-319-30034-4_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30032-0
Online ISBN: 978-3-319-30034-4
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