One result on boundedness of the Hilbert transform in Marcinkiewics spaces

Abstract

In mathematics and in signal theory, the Hilbert transform is an important linear operator that takes a real-valued function and produces another real-valued function. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Also, it is a widely used tool in signal processing. The Cauchy integral is a figurative way to motivate the Hilbert transform. The complex view helps us to relate the Hilbert transform to something more concrete and understandable. Moreover, the Hilbert transform is closely connected with many operators in harmonic analysis such as Laplace and Fourier transforms which have numerous application in partial and ordinary differential equations. In this paper, we study boundedness properties of the classical (singular) Hilbert transform acting on Marcinkiewicz spaces. More precisely, we obtain if and only if condition for boundedness of the Hilbert transform in Marcinkiewicz function spaces.