A variety of uncertainty principles for the Hankel-Stockwell transform

In this work, we establish \(L^p\) local uncertainty principle for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. Next, By combining these principles and the techniques of Donoho-Stark we present uncertainty principles of concentration type in the \(L^p\) theory, when \(1< p\leqslant2\). Finally, Pitt's inequality and Beckner's uncertainty principle are proved for this transform.

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

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.

The class Bp for weighted generalized Fourier transform inequalities

In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.