Lipschitz and Dini-Lipschitz functions for the modified Whittaker transform

Abstract

In this work, we introduce Lipschitz and Dini-Lipschitz classes of functions associated with the Whittaker
operator $L:=-\frac{1}{4}\left[x^2\frac{\mbox{d}^2}{\mbox{d}x^2}+(x^{-1}+(3-4\alpha)x)\frac{\mbox{d}}{\mbox{d}x}\right]$,
and we prove two versions of Titchmarsh's theorems for the modified Whittaker transform $F_W$. An application to the Whittaker multipliers is given. The Whittaker translation associated with the operator $L$ plays an important role in establishing the results of this paper.