Lipschitz and Dini–Lipschitz functions for the Sturm–Liouville transform

Abstract

In this work, we introduce Lipschitz and Dini–Lipschitz classes of functions associated with the Sturm–Liouville operator L := d 2 d x 2 + A ′ ( x ) A ( x ) d d x , where A is a nonnegative function satisfying certain conditions; and we prove two versions of Titchmarsh's theorems for the Sturm–Liouville transform F . The harmonic analysis associated with the operator L plays an important role in establishing the results of this paper.