An analog of Titchmarsh's theorem and Dini Lipschitz theorem for the Mehler-Fock-Clifford transform

Using a generalized dual translation operator, we obtain an analog of Titchmarsh's theorem and the analogu of Dini Lipschitz theorem for the Mehler-Fock-Clifford transform for functions in f is an element of L1(I;x-12dx)boolean AND L2(I;x-12dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in L<^>1(I;x<^>{-\frac{1}{2}}dx) \cap L<^>2(I;x<^>{-\frac{1}{2}}dx)$$\end{document} where I=]14,+infinity[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=]\frac{1}{4},+\infty [$$\end{document}.