Weighted inequalities and uncertainty principles for the \((k,a)\)-generalized Fourier transform (Q2797903)

Let \(\mathcal{R}\) be a root system of \({\mathbb{R}}^N\) and \(k : \mathcal{R}\to\mathbb{C}\) a multiplicity function. Let \(\Delta_k\) be the Dunkl-Laplacian and \(\Delta_{k,a}=\|x\|^{2-a}\Delta_k-\|x\|^a\) \((a>0)\) the \(a\)-deformed Dunkl-harmonic oscillator introduced by Ben Saïd, Kobayashi and Orsted in [\textit{S. Ben Saïd} et al., C. R., Math., Acad. Sci. Paris 347, No. 19--20, 1119--1124 (2009; Zbl 1176.43003)] and [\textit{S. Ben Saïd} et al., Compos. Math. 148, No. 4, 1265--1336 (2012; Zbl 1255.43004)]. By using this operator, the \((k,a)\)-generalized Fourier transform \({\mathcal{F}}_{k,a}\) is defined as NEWLINE\[NEWLINE {\mathcal{F}}_{k,a}=\exp\Big[\frac{i\pi}{2}\Big(\frac{1}{a}(2\langle k \rangle+N+a-2\Big)\Big] \exp\Big[\frac{i\pi}{2a}(\|x\|^{2-a}\Delta_k-\|x\|^a)\Big]. NEWLINE\]NEWLINE This operator is a unitary on \(L^2({\mathbb{R}}^N, \|a\|^{a-2}\vartheta_kdx)\). In this paper the author obtains the radial projection of \({\mathcal{F}}_{k,a}\) (Hankel transform), several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for \({\mathcal{F}}_{k,a}\), and furthermore, several versions of the Heisenberg-Pauli-Weyl uncertainty principle, Pitt's inequality, etc. Although they are established in a general scheme, the proof of the Heisenberg-Pauli-Weyl inequality for \({\mathcal{F}}_{k,a}\) based on Hirschman's entropic inequality is new.