L^p boundedness of the Hilbert transform

TL;DR

This paper reviews the boundedness properties of the Hilbert transform on L^p spaces, emphasizing its importance in harmonic analysis and related fields, and provides a self-contained proof of these classical results.

Contribution

It offers a self-contained derivation of the L^p boundedness of the Hilbert transform using real-variable techniques, revisiting established results with clarity.

Findings

Hilbert transform is weakly bounded on L^1(R)

Hilbert transform is strongly bounded on L^p(R) for 1<p<∞

Provides a self-contained proof using real-variable methods

Abstract

The Hilbert transform is essentially the \textit{only} singular operator in one dimension. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulation theory, wavelet theory, aerofoil design, dispersion relations and high-energy physics, to name a few. In this monograph, we revisit some of the established results concerning the global behavior of the Hilbert transform, namely that it is is weakly bounded on $\eL^1(\R)$, and strongly bounded on $\eL^p(\R)$ for $1 < p <\infty$, and provide a self-contained derivation of the same using real-variable techniques.