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$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

Pritam Ganguly, Jayanta Sarkar

TL;DR

This work establishes an $L^p$ Heisenberg–Pauli–Weyl uncertainty inequality for the group Fourier transform on broad classes of two-step MW groups, covering the range $1\le p\le 2$ with a focus on operator-valued representations. The authors deploy the dilation structure of nonisotropic two-step nilpotent groups and spectral properties of the sub-Laplacian, realised as a scaled Hermite operator $H(\eta(\lambda))$ in the Fourier side, to bound the Schatten-$p'$ norms of $\mathcal{F}(f)(\lambda)H(\eta(\lambda))^{\beta/2}$. The main result sharpens all previous $L^p$ HPW inequalities in this setting and is novel for $p=1$ on non-stratified two-step MW groups, while extending beyond stratified cases to encompass a wider class of groups. The paper provides a comprehensive treatment across the three regimes $p=1$, $1<p<2$, and $p=2$, combining representation theory, dilation analysis, and Schatten-class estimates to yield a quantitative uncertainty principle in a noncommutative, non-Euclidean context with potential implications for harmonic analysis on nilpotent groups and related PDEs.

Abstract

This article presents the $L^p$-Heisenberg-Pauli-Weyl uncertainty inequality for the group Fourier transform on a broad class of two-step nilpotent Lie groups, specifically the two-step MW groups. This inequality quantitatively demonstrates that on two-step MW groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known $L^p$-Heisenberg-Pauli-Weyl uncertainty inequalities for $1 \leq p < 2$ within the realm of two-step nilpotent Lie groups.

$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

TL;DR

This work establishes an Heisenberg–Pauli–Weyl uncertainty inequality for the group Fourier transform on broad classes of two-step MW groups, covering the range with a focus on operator-valued representations. The authors deploy the dilation structure of nonisotropic two-step nilpotent groups and spectral properties of the sub-Laplacian, realised as a scaled Hermite operator in the Fourier side, to bound the Schatten- norms of . The main result sharpens all previous HPW inequalities in this setting and is novel for on non-stratified two-step MW groups, while extending beyond stratified cases to encompass a wider class of groups. The paper provides a comprehensive treatment across the three regimes , , and , combining representation theory, dilation analysis, and Schatten-class estimates to yield a quantitative uncertainty principle in a noncommutative, non-Euclidean context with potential implications for harmonic analysis on nilpotent groups and related PDEs.

Abstract

This article presents the -Heisenberg-Pauli-Weyl uncertainty inequality for the group Fourier transform on a broad class of two-step nilpotent Lie groups, specifically the two-step MW groups. This inequality quantitatively demonstrates that on two-step MW groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known -Heisenberg-Pauli-Weyl uncertainty inequalities for within the realm of two-step nilpotent Lie groups.

Paper Structure

This paper contains 11 sections, 9 theorems, 152 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Basic Notations
  4. Two-step nilpotent Lie groups
  5. Harmonic analysis on two-step MW groups
  6. The sub-Laplacian
  7. Proof of main results
  8. Proof for $p=1$ case
  9. Proof for $1<p<2$ case
  10. Proof for $p=2$ case.
  11. Schatten class norms

Key Result

Theorem 1.1

Let $G$ be a stratified Lie group equipped with a sub-Laplacian $L$, and let $|\cdot|$ denote a homogeneous norm. Assume that $\beta,\:\delta\in(0,\infty)$, $p,\:r\in(1,\infty)$, $s \geq 1$, and that Then, there exists a constant $C>0$ such that for any $f \in C_c^\infty(G)$, the following holds:

Theorems & Definitions (14)

  • Theorem 1.1: CCR
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 4 more