Hausdorff operators associated with the Sturm–Liouville operator

In the present paper, we introduce the Hausdorff operators associated with the Sturm–Liouville operator \(\Delta :=\frac{\text{ d}^2}{\text{ d }x^2}+\frac{A'(x)}{A(x)}\frac{\text{ d }}{\text{ d }x}\) $\mathrm{\Delta }:=\frac{{\text{d}}^2}{\text{d}{x}^2}+\frac{A^{\prime }(x)}{A(x)}\frac{\text{d}}{\text{d}x}$ , where A is a nonnegative function satisfying certain conditions; and we prove the boundedness of the Sturm–Liouville Hausdorff operators in space \(L^2(\mathbb {R}_+,A(x)\text{ d }x)\) $L^2({\mathbb{R}}_{+},A(x)\text{d}x)$ . 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.