Linear canonical Sturm—Liouville Hausdorff and convolution type operators

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

Key words and phrases. Canonical Sturm—Liouville transform, canonical Sturm—Liouville Hausdorff operators, canonical Sturm—Liouville convolution operators.