247A, Notes 1: Rearrangement-invariant spaces

Disclaimer: due to current events, I have not been able to devote as much time to lecture notes preparation as I would have liked, so I apologize in advance for the unpolished nature of the text below, which has been largely recycled from previous lecture notes I have written. This is the first set of lecture notes for my graduate course 247A, “Fourier analysis”. The course name is rather general, but I will focus the course not on the Fourier transform per se, but on the closely related topic of real variable harmonic analysis, with a particular emphasis on Calderón–Zygmund theory, which underlies basic tools in PDE such as the theory of Sobolev spaces. To avoid confusion at the outset, let us make the distinction between real-variable harmonic analysis and abstract harmonic analysis, which are only distantly related to each other despite the similar names. Abstract harmonic analysis, roughly speaking, is the extension of the classical theory of the Fourier transform to other…