On exceptional sets of Hilbert transform

TL;DR

This paper investigates the properties of exceptional sets related to the Hilbert transform on the real line, demonstrating that null sets can be exceptional and offering a real variable perspective on Fourier series divergence.

Contribution

It establishes that any null set can be an exceptional set for the Hilbert transform of an indicator function and introduces a real variable approach to Kahane-Katsnelson theorem.

Findings

Null sets are shown to be exceptional sets for the Hilbert transform.

Provides a real variable approach to Fourier series divergence.

Expands understanding of exceptional sets in harmonic analysis.

Abstract

We prove several theorems concerning the exceptional sets of Hilbert transform on the real line. In particular, it is proved that any null set is exceptional set for the Hibert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.