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

Aparajita Dasgupta, Prerna Gulia, Sanjoy Pusti, Sundaram Thangavelu

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 Calderón'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.

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

Abstract

In this article, we establish an analogue of Pitt's inequality for the Strichartz Fourier transform on the Heisenberg group . By exploiting the scalar-valued formulation of the transform and the framework of decreasing rearrangements, we derive weighted - estimates of Pitt type. In particular, we obtain sufficient conditions for the validity of such inequalities via weighted Hardy inequalities and Calderón'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 . These results extend the classical Euclidean theory of Pitt's inequality to the non-commutative, nilpotent setting of for the sub-Laplacian and conformal Laplacian. Here we highlight the role of Laguerre functions in harmonic analysis on the Heisenberg group.
Paper Structure (14 sections, 27 theorems, 239 equations)

This paper contains 14 sections, 27 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decreasing rearrangement and Lorentz spaces
  4. Calderón's operator and Weighted Hardy's inequalities
  5. Pitt's Inequality for Strichartz Fourier Transform
  6. An uncertainty principle
  7. Necessary Condition for Pitt's inequality
  8. Pitt's inequality for polynomial type weights
  9. Weight on the function side
  10. Weight on the Fourier transform side
  11. Sufficient conditions on weights
  12. Necessary conditions on weights
  13. Paley's Inequality
  14. Hardy's inequality and Pitt's inequality

Key Result

Theorem 1.1

Let $1 \leq p \leq q \leq \infty$. If $p < \infty$ and $(p,q) \neq (2,2)$, and if then inequality pitts inequality Rn holds. Moreover, in the special case $p = q = 2$, if condition Rn pitts cond1 is satisfied together with then pitts inequality Rn remains valid.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.1.1
  • Theorem 2.2.1
  • Theorem 2.2.2
  • Theorem 3.1
  • proof
  • ...and 31 more