L^p Spaces

Abstract

This chapter extends the theory of the spaces L¹, L², and L^∞ to include a whole family of spaces L^p, 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 L^p. 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 L^p 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 L^p 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 L^p spaces as bounded between intermediate pairs of L^p 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.