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

Abstract

We obtain several versions of the Hausdorff–Young and Hardy–Littlewood inequalities for the [Formula: see text]-generalized Fourier transform recently investigated at length by Ben Saïd, Kobayashi, and Ørsted. 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 [Formula: see text]-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.