Boundedness of the Hilbert Transform on Rearrangement Invariant Spaces

Abstract

In this self-contained review, the aspects about the applications of decreasing rearrangement techniques to the analysis of pointwise estimates for the Hilbert transform are analyzed. We make a consistent revision of these techniques in the proof of the $L_{p}$-boundedness of the Hilbert transform. This is a celebrated theorem due to M. Riesz.

Keywords

• Hilbert transform
• distribution function
• decreasing rearregement
• Hardy's inequality