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

TL;DR

This paper develops weighted inequalities and uncertainty principles for the $(k,a)$-generalized Fourier transform, extending classical results and deriving new inequalities with applications to quantum mechanics and signal analysis.

Contribution

It introduces new weighted inequalities and uncertainty principles for the $(k,a)$-generalized Fourier transform, expanding the theoretical framework of harmonic analysis.

Findings

Derived Hausdorff-Young inequalities for the $(k,a)$-transform

Established weighted inequalities including Pitt's inequality

Extended uncertainty principles such as Heisenberg and entropic inequalities

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 Sa\"i d, Kobayashi, and {\O} 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.