The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-5, author = {Lasha Ephremidze}, title = {On the generalization of the Stein-Weiss theorem for the ergodic Hilbert transform}, journal = {Studia Mathematica}, volume = {157}, year = {2003}, pages = {67-75}, zbl = {1016.28012}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-5} }
Lasha Ephremidze. On the generalization of the Stein-Weiss theorem for the ergodic Hilbert transform. Studia Mathematica, Tome 157 (2003) pp. 67-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-5/