Hausdorff operators associated with the Sturm-Liouville operator

In the present paper, we introduce the Hausdorff operators associated with the Sturm-Liouville operator Delta:=d2dx2+A '(x)A(x)ddx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta :=\frac{\text{ d}<^>2}{\text{ d }x<^>2}+\frac{A'(x)}{A(x)}\frac{\text{ d }}{\text{ d }x}$$\end{document}, where A is a nonnegative function satisfying certain conditions; and we prove the boundedness of the Sturm-Liouville Hausdorff operators in space L2(R+,A(x)dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(\mathbb {R}_+,A(x)\text{ d }x)$$\end{document}. We investigate continuous Sturm-Liouville wavelet transform, and obtain some useful results. The relation between Sturm-Liouville wavelet transform and Sturm-Liouville Hausdorff operator is also established. The properties of the adjoint Sturm-Liouville Hausdorff operator are discussed.