Weighted inequalities and uncertainty principles for the $(\mathbf{k},\mathbf{a})$-generalized Fourier transform

Abstract

We obtain several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for the $(k,a)$-generalized Fourier transform recently investigated at length by Ben Saï d, Kobayashi, and Ø rsted. We also obtain a number of weighted inequalities - in particular Pitt's inequality - that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg-Pauli-Weyl principle for $L^p$-functions, local Cowling-Price-type inequalities, Donoho-Stark-type inequalities and qualitative extensions. We finally use the Hausdorff-Young inequality as a means to obtain entropic uncertainty inequalities.

Updated references, minor corrections, adjustments to range of parameters in Pitt's inequality. Final version, accepted for publication in International J. Math

References in corpus (3)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• Generalized uncertainty inequalities
• Hardy--Littlewood inequalities for the Heckman--Opdam transform