The Hilbert transform

Abstract

The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly make it one of the most important linear operator in harmonic analysis. This is an expository paper about the Hilbert transform aimed to anyone that has even scratched the surface of the theory of integration, and functional analysis as well as a basic rudiments of Fourier transform. We provide a systematic (Although by no means complete) account of the basic results on the Hilbert transform. We want to point out that we present a friendly proof of the remarkable result due to Stein and Weiss (1959) (see [24]) and we use it combined with the Cavalieri principle to obtain an exact formula for the Lp-norm of H(χE).