On Titchmarsh’s Theorem for the Canonical Bessel Transform: Weighted Sobolev Spaces versus Classical Lipschitz Classes

Abstract

Abstract This paper establishes Titchmarsh-type theorems for the Canonical Bessel Transform (CBT) in weighted Sobolev spaces associated with Lipschitz and Dini–Lipschitz type conditions. We characterize the decay and integrability properties of the CBT in the spaces $$\mathcal {L}_{p,\nu }$$ . Our approach is based on the classical Titchmarsh formulation involving second-order difference operators, which is consistent with the standard framework of Lipschitz and Sobolev spaces. Consequently, we improve several previously known results obtained via simplified difference operators and extend classical one-dimensional Titchmarsh-type theorems to the setting of Canonical Bessel analysis.