Hausdorff operators associated with the linear canonical Sturm-Liouville transform

Abstract

In the present paper, we introduce the canonical Sturm-Liouville operator \(L^M:=\frac{{\rm d}^2}{{\rm d}x^2}+\left(\frac{A'(x)}{A(x)}-2i\frac{a}{b}x\right)\frac{{\rm d}}{{\rm d}x} -\left(\frac{a^2}{b^2}x^2+i\frac{a}{b}x\frac{A'(x)}{A(x)}+i\frac{a}{b}\right)\), where \(A\) is a nonnegative function satisfying certain conditions. We prove the boundedness of the canonical Sturm-Liouville Hausdorff operators on the space \(L^p(\mathbb{R}_+,A(x)\,{\rm d}x)\), \(p\in [1,\infty)\). We investigate canonical Sturm-Liouville wavelet transform, and obtain some useful results. The relation between the canonical Sturm-Liouville wavelet transform and canonical Sturm-Liouville Hausdorff operator is also established. The properties of the adjoint canonical Sturm-Liouville Hausdorff operators are further discussed. The harmonic analysis associated with the operator \(L^M\) plays an important role in establishing the results of this paper.