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

摘要

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 Said, Kobayashi, and Orsted. 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.