On boundedness of the Hilbert transform on Marcinkiewicz spaces

Abstract

We study boundedness properties of the classical (singular) Hilbert transform (Hf)(t) = p:v: 1

Z R f(s) t 􀀀 s ds acting on Marcinkiewicz spaces. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Questions involving the H arise therefore from the utilization of complex methods in Fourier analysis, for example. In particular, the H plays the crucial role in questions of norm-convergence of Fourier series and Fourier integrals. We consider the problem of what is the least rearrangement-invariant Banach function space F(R) such that H : M (R) ! F(R) is bounded for a fixed Marcinkiewicz space M (R): We also show the existence of optimal rearrangement-invariant Banach function range on Marcinkiewicz spaces. We shall be referring to the space F(R) as the optimal range space for the operator H restricted to the domain M (R) '0 (R): Similar constructions have been studied by J.Soria and P.Tradacete for the Hardy and Hardy type operators [1]. We use their ideas to obtain analogues of their some results for the H on Marcinkiewicz spaces.