Some qualitative uncertainty principles for the Fractional Dunkl Transform

Abstract

The fractional Dunkl transform (FrDT) is a natural extension of the classical Dunkl transform \(\mathcal {D}_\mu \). In this paper, we establish two qualitative uncertainty principles associated with the FrDT. The first result is a Cowling–Price-type theorem, in which we study the decay properties of two fractional Dunkl transformations for two different angles \(\alpha \) and \(\gamma \), assuming the angular difference satisfies \(\gamma - \alpha \ne n\pi \) for all \(n \in {\mathbb {Z}}\). The second result is an \(L^p\)–\(L^q\) version of Morgan’s theorem in the context of the FrDT. These results generalize classical uncertainty principles by imposing joint constraints on the decay behavior of a function and its fractional Dunkl transform.