On exceptional sets of the Hilbert transform (Q1750324)

The author continues his study of exceptional sets for classical operators. In his previous works the results were obtained for operators satisfying the localization property. In the study of the Hilbert transform in the paper under review, the earlier elaborated approach does not work, since the Hilbert transform does not possess the localization property. It is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also contains a real variable proof of the main lemma in the Kahane-Katznelson's work on divergence of Fourier series as a consequence of the author's general lemma.