On Local Properties of Singular Integrals

Let γ be a regular curve. We study the local properties of singular integrals in the class of functions Hαα+βt0,γ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\alpha }^{\alpha +\beta }\left({t}_{0},\upgamma \right).$$\end{document} We obtain a strengthening of the Plemelj–Privalov theorem for functions from the class Hαα+βt0,γ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\alpha }^{\alpha +\beta }\left({t}_{0},\upgamma \right).$$\end{document} It is proved that, at the point t0 of increased smoothness for α+β < 1, there is only a logarithmic loss.