Hausdorff operators associated with the linear canonical Sturm-Liouville transform

Fethi Soltani; Maher Aloui

doi:10.56754/0719-0646.2801.179

DOI:

https://doi.org/10.56754/0719-0646.2801.179

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.

Keywords

Canonical Sturm-Liouville transform , canonical Sturm-Liouville convolution , canonical Sturm-Liouville Hausdorff operators , canonical Sturm-Liouville wavelet transform

Mathematics Subject Classification:

44A05 , 44A20 , 47G10

Fethi Soltani Faculté des Sciences de Tunis, Laboratoire d’Analyse Mathématique et Applications LR11ES11, Université de Tunis El Manar, Tunis 2092, Tunisia. https://orcid.org/0000-0001-7519-0994
Maher Aloui Ecole Nationale d’Ingénieurs de Carthage, Université de Carthage, Tunis 2035, Tunisia. https://orcid.org/0009-0007-6380-0979

Pages: 179–204
Date Published: 2026-01-27
Vol. 28 No. 1 (2026)

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