L^p boundedness of the Hilbert transform

The Hilbert transform is essentially the 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 L(R), and strongly bounded on L(R) for 1 < p < ∞, and provide a selfcontained derivation of the same using real-variable techniques. This note is partly based on the expositions on the Hilbert transform in [1, 3].