Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 16 (2021), 149 -- 192

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THE HILBERT TRANSFORM

Edisson Arley Arcos and René Erlin Castillo

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 [Math Mech. 8, 1959] and we use it combined with the Cavalieri principle to obtain an exact formula for the Lp-norm of H(χE).

2020 Mathematics Subject Classification: 44A15; 42A85; 42A75
Keywords: Hilbert transform; Fourier transform; Cauchy principal value.

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Edisson Arley Arcos
Departamento de Matemáticas, Universidad Nacional de Colombia
e-mail: earcosb@unal.edu.co

René Erlin Castillo
Departamento de Matemáticas, Universidad Nacional de Colombia
e-mail: recastillo@unal.edu.co


http://www.utgjiu.ro/math/sma