Abstract
This chapter extends the theory of the spaces L1, L2, and L∞ to include a whole family of spaces Lp, 1 <- p <- ∞, in order to be able to capture finer quantitative facts about the size of measurable functions and the effect of linear operators on such functions.
Sections 1–2 give the basics about Lp. For general measure spaces these consist of Hölder’s inequality, Minkowski’s inequality, a completeness theorem, and related results. For Euclidean space they include also facts about convolution.
Sections 3–4 develop some tools that at first may seem quite unrelated to Lp spaces but play a significant role in Section 5. These are the Radon-Nikodym Theorem and two decomposition theorems for additive set functions. The Radon-Nikodym Theorem gives a sufficient condition for writing a measure as a function times another measure.
Section 5 identifies the space of continuous linear functionals on Lp for 1 <- p < ∞ when the underlying measure is σ-finite. For one thing this identification makes Alaoglu’s Theorem in Chapter V concrete enough so as to be quite useful.
Section 6 discusses the Marcinkiewicz Interpolation Theorem, which allows one to reinterpret suitably bounded operators between two pairs of Lp spaces as bounded between intermediate pairs of Lp spaces as well. The theorem has immediate corollaries for the Hardy-Littlewood maximal function and an approximation to the Hilbert transform, and Section 6 goes on to use each of these corollaries to derive interesting consequences.
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© 2005 Anthony W. Knapp
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(2005). Lp Spaces. In: Basic Real Analysis. Cornerstones. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4441-5_9
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DOI: https://doi.org/10.1007/0-8176-4441-5_9
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3250-2
Online ISBN: 978-0-8176-4441-3
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