Weighted Norm Inequalities for the Strichartz Fourier transform on the Heisenberg Group

TL;DR

This paper extends Pitt's inequality to the Heisenberg group for the Strichartz Fourier transform, deriving weighted inequalities, uncertainty principles, and Paley inequalities, thus generalizing classical Euclidean harmonic analysis to a non-commutative setting.

Contribution

It introduces weighted Pitt's inequalities for the Strichartz Fourier transform on the Heisenberg group, including necessary and sufficient conditions, and applies these to uncertainty principles and Paley inequalities.

Findings

Established weighted $L^p$-$L^q$ estimates for the transform.

Derived necessary conditions using Laguerre functions and Bessel zeros.

Extended Euclidean Pitt's inequality to the Heisenberg group setting.

Abstract

In this article, we establish an analogue of Pitt's inequality for the Strichartz Fourier transform on the Heisenberg group $\mathbb{H}^n$. By exploiting the scalar-valued formulation of the transform and the framework of decreasing rearrangements, we derive weighted $L^p$-$L^q$ estimates of Pitt type. In particular, we obtain sufficient conditions for the validity of such inequalities via weighted Hardy inequalities and Calder\'{o}n's interpolation method, and we also prove necessary conditions in the case of radial weights, using structural properties of Laguerre functions and zeros of Bessel function. As an application, we deduce an uncertainty principle of Heisenberg-Pauli-Weyl type in this setting and establish a Paley inequality for the Strichartz Fourier transform. We also derive Pitt's inequality using Hardy's inequality for the case $p=q=2$. These results extend the classical Euclidean theory of Pitt's inequality to the non-commutative, nilpotent setting of $\mathbb{H}^n$ for the sub-Laplacian and conformal Laplacian. Here we highlight the role of Laguerre functions in harmonic analysis on the Heisenberg group.