Berezin Transform, Mellin Transform and Toeplitz Operators

We consider functions of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_1\bar g_1+h}$$\end{document} in the range of the Berezin transform B, where f1 and g1 are holomorphic on the unit disk \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb D}$$\end{document}, and h is either harmonic or of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_2\bar g_2}$$\end{document} for some holomorphic functions f2 and g2 on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb D}$$\end{document}. First, by using the Mellin transform, we complement Ahern’s Theorem (Ahern in J Funct Anal 215:206–216, 2004) by proving that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u\in L^1}$$\end{document} and B(u) is harmonic, then u is harmonic. Secondly, we extend Ahern’s Theorem when h is harmonic, and give very precise relations between f1 and f2, g1 and g2 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h=f_2\bar g_2}$$\end{document} and g2(z) = zn with n ≥ 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.