Chapter IX. $L^{p}$ Spaces

Abstract

This chapter extends the theory of the spaces $L^1$, $L^2$, and $L^{\infty}$ to include a whole family of spaces $L^p$, $1\leq p\leq\infty$, 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 $1\leq p \lt \infty$ when the underlying measure is $\sigma$-finite. For one thing this identification makes Alaoglu's Theorem in Chapter V concrete enough so as to be quite useful.

Section 6 establishes the Riesz–Thorin Convexity Theorem, which asserts that linear operators that are bounded between two pairs of $L^p$ spaces are bounded between suitable intermediate pairs of $L^p$ spaces as well. Immediate corollaries include the Hausdorff–Young Theorem concerning the Euclidean Fourier transform and Young's inequality concerning convolution of functions in two $L^p$ spaces.

Section 7 discusses the Marcinkiewicz Interpolation Theorem, which allows one to reinterpret bounded sublinear operators between two pairs of $L^p$ spaces as bounded between suitable 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 7 goes on to use each of these corollaries to derive interesting consequences.